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
Models of Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.5.3 Construction of Multifactor Regression
Models of Inflation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
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
4.1.3 Analysis of the Dependence of the Optimal
Parametric Control Law on Values of
Uncontrolled Parameters Based on the CGE Model
with the Knowledge Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
4.2 National Economic Evolution Control Based
on a Computable Model of General Equilibrium
of Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.2.1 Model Description, Parametric Identification,
and Retrospective Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.2.2 Finding Optimal Parametric Control Laws
on the Basis of the CGE Model
of the Economic Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
xii Contents
4.2.3 Analysis of the Dependence of the Optimal
Parametric Control Law on Values
of Uncontrolled Parameters on the Basis
of the CGE Model of Economic Sectors . . . . . . . . . . . . . . . . . . 214
4.3 National Economic Evolution Control Based
on a Computable Model of General Equilibrium
with the Shady Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.3.1 Model Description, Parametric Identification,
and Retrospective Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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.
4 1 Elements of Parametric Control Theory of Market Economic Development
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
6 1 Elements of Parametric Control Theory of Market Economic Development
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.
8 1 Elements of Parametric Control Theory of Market Economic Development
(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.
1.3 Approach to Choosing (Synthesis) Optimal Laws of Parametric Control. . . 9
(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.
10 1 Elements of Parametric Control Theory of Market Economic Development
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).
1.3 Approach to Choosing (Synthesis) Optimal Laws of Parametric Control. . . 11
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.
12 1 Elements of Parametric Control Theory of Market Economic Development
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
1.3 Approach to Choosing (Synthesis) Optimal Laws of Parametric Control. . . 13
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)
under constraints (1.11).
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.
14 1 Elements of Parametric Control Theory of Market Economic Development
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.
16 1 Elements of Parametric Control Theory of Market Economic Development
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.
18 1 Elements of Parametric Control Theory of Market Economic Development
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).
20 1 Elements of Parametric Control Theory of Market Economic Development
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)
1.6 Examples of the Application of Parametric Control Theory 21
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.
22 1 Elements of Parametric Control Theory of Market Economic Development
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 Examples of the Application of Parametric Control Theory 23
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
24 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6.2.1 Model Description
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
þ Xð24ÞXð24ÞXð24Þ
2
þ Xð36ÞXð36ÞXð36Þ
2
þ Xð48ÞXð48ÞXð48Þ
2
þ Kð12ÞKð12ÞKð12Þ
2
þ Kð24ÞKð24ÞKð24Þ
2
þ Kð36ÞKð36ÞKð36Þ
2
þ Kð48ÞKð48ÞKð48Þ
2!!min
(1.27)
1.6 Examples of the Application of Parametric Control Theory 25
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
26 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6 Examples of the Application of Parametric Control Theory 27
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
1.6.3.1 Model Description
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
28 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6 Examples of the Application of Parametric Control Theory 29
The parametric identification criterion is as follows:
K ¼ 1
8
xð1Þ xð1Þxð1Þ
2
þ xð2Þ xð2Þxð2Þ
2
þ xð3Þ xð3Þxð3Þ
2
þ xð4Þ xð4Þxð4Þ
2
þ yð1Þ yð1Þyð1Þ
2
þ yð2Þ yð2Þyð2Þ
2
þ yð3Þ yð3Þyð3Þ
2
þ yð4Þ yð4Þyð4Þ
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
30 1 Elements of Parametric Control Theory of Market Economic Development
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.
The problem of choosing the optimal parametric control law at the level of one
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)
under the constraints
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)
1.6 Examples of the Application of Parametric Control Theory 31
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
32 1 Elements of Parametric Control Theory of Market Economic Development
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 Examples of the Application of Parametric Control Theory 33
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
1.6.4.1 Model Description
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)
34 1 Elements of Parametric Control Theory of Market Economic Development
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;
1.6 Examples of the Application of Parametric Control Theory 35
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
36 1 Elements of Parametric Control Theory of Market Economic Development
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.
1.6 Examples of the Application of Parametric Control Theory 37
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.
The problem of choosing the optimal parametric control law at the level of one
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)
under the constraints
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).
38 1 Elements of Parametric Control Theory of Market Economic Development
– 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
1.6 Examples of the Application of Parametric Control Theory 39
under constraints (1.55).
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
40 1 Elements of Parametric Control Theory of Market Economic Development
under constraints (1.55).
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
1.6 Examples of the Application of Parametric Control Theory 41
#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
42 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6 Examples of the Application of Parametric Control Theory 43
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)
44 1 Elements of Parametric Control Theory of Market Economic Development
under the constraints
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
1.6 Examples of the Application of Parametric Control Theory 45
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
46 1 Elements of Parametric Control Theory of Market Economic Development
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)
1.6 Examples of the Application of Parametric Control Theory 47
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)
48 1 Elements of Parametric Control Theory of Market Economic Development
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:
1.6 Examples of the Application of Parametric Control Theory 49
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
50 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6 Examples of the Application of Parametric Control Theory 51
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
52 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6.6.1 Model Description
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
1.6 Examples of the Application of Parametric Control Theory 53
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)
54 1 Elements of Parametric Control Theory of Market Economic Development
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;
1.6 Examples of the Application of Parametric Control Theory 55
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
56 1 Elements of Parametric Control Theory of Market Economic Development
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.
Proof Let us first prove that the semitrajectory of the flux f starting from any point
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
DMiðtÞMiðt0Þ þ constib;
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
1.6 Examples of the Application of Parametric Control Theory 57
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)
under the constraints
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)
58 1 Elements of Parametric Control Theory of Market Economic Development
under constraints (1.106).
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
under constraints (1.106).
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.
1.6 Examples of the Application of Parametric Control Theory 59
– On the basis of the mathematical model of the interaction between nonidentical
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.
– 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 (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.
– On the basis of the mathematical model of the interaction between nonidentical
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
60 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6 Examples of the Application of Parametric Control Theory 61
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
62 1 Elements of Parametric Control Theory of Market Economic Development
1Þ Ui1;b ¼ ki1;b
DMiðtÞMiðt0Þ þ constib;
2Þ Ui2;b ¼ ki2;b
DMiðtÞMiðt0Þ þ constib;
3Þ Ui3;b ¼ ki3;b
DpiðtÞpiðt0Þ þ constib;
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)
1.6 Examples of the Application of Parametric Control Theory 63
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:
64 1 Elements of Parametric Control Theory of Market Economic Development
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
DMiðtÞMiðt0Þ þ constib;
2Þ Ui2;b ¼ ki2;b
DMiðtÞMiðt0Þ þ constib;
3Þ Ui3;b ¼ ki3;b
DpiðtÞpiðt0Þ þ constib;
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
1.6 Examples of the Application of Parametric Control Theory 65
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
66 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6.7.1 Model Description
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)
1.6 Examples of the Application of Parametric Control Theory 67
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);
68 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6 Examples of the Application of Parametric Control Theory 69
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.
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 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
70 1 Elements of Parametric Control Theory of Market Economic Development
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.
1.6 Examples of the Application of Parametric Control Theory 71
The problem is solved in two stages:
– 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).
72 1 Elements of Parametric Control Theory of Market Economic Development
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
1.6 Examples of the Application of Parametric Control Theory 73
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
74 1 Elements of Parametric Control Theory of Market Economic Development
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
A.A. Ashimov et al., Macroeconomic Analysis and Economic PolicyBased on Parametric Control, DOI 10.1007/978-1-4614-1153-6_2,# Springer Science+Business Media, LLC 2012
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
inmillionsoftenge(inpricesoftheyear2000)
Iisthevolumeofinvestm
entto
thecapital
assetin
millionsoftenge(inpricesoftheyear2000)
Gisthelevel
ofpublicexpensesin
millionsoftenge(inpricesoftheyear2000)
NXisthenet
exportin
millionsoftenge(inpricesoftheyear2000)
Pisthelevel
ofprices(calculatedforthebaseoftheyear2000)
Tisthetaxationlevel
inmillionsoftenge(inpricesoftheyear2000)
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
icsoftheRepublicofKazakhstan
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
78 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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:
Dependent variable: C_
Method: least squares
Date: 06/22/09 Time: 03:21
Sample: 2000 2008
Included observations: 9
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
R-squared 0.993252 Mean dependent var 2,408,695.
Adjusted R-squared 0.991003 S.D. dependent var 676,711.2
S.E. of regression 64,187.03 Akaike info criterion 25.23819
Sum squared resid 2.47E + 10 Schwarz criterion 25.30393
Log likelihood �110.5719 F-statistic 441.6029
Durbin–Watson stat 1.160812 Prob(F-statistic) 0.000000
2.1 Factor Modeling of the Aggregate Demand in a National Economy. . . 79
Estimation of the third model:
Dependent variable: C_
Method: least squares
Date: 06/22/09 Time: 03:22
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.994548 Mean dependent var 2,408,695.
Adjusted R-squared 0.991278 S.D. dependent var 676,711.2
S.E. of regression 63,200.72 Akaike info criterion 25.24712
Sum squared resid 2.00E + 10 Schwarz criterion 25.33478
Log likelihood �109.6120 F-statistic 304.0587
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
80 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
Dependent variable: I
Method: least squares
Date: 06/22/09 Time: 03:26
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.912614 Mean dependent var 1,353,980.
Adjusted R-squared 0.860182 S.D. dependent var 539,367.1
S.E. of regression 201,681.3 Akaike info criterion 27.56787
Sum squared resid 2.03E + 11 Schwarz criterion 27.65552
Log likelihood �120.0554 F-statistic 17.40578
Durbin–Watson stat 0.909211 Prob(F-statistic) 0.004453
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
Dependent variable: I
Method: least squares
Date: 06/22/09 Time: 02:19
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
E �15,783.44 2,159.018 �7.310470 0.0002
C 3,053,359. 241,485.3 12.64408 0.0000
R-squared 0.884188 Mean dependent var 1,353,980.
Adjusted R-squared 0.867644 S.D. dependent var 539,367.1
S.E. of regression 196,226.0 Akaike info criterion 27.40505
Sum squared resid 2.70E + 11 Schwarz criterion 27.44888
Log likelihood �121.3227 F-statistic 53.44297
Durbin–Watson stat 1.256380 Prob(F-statistic) 0.000161
2.1 Factor Modeling of the Aggregate Demand in a National Economy. . . 81
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:
Dependent variable: I
Method: least squares
Date: 10/25/09 Time: 23:14
Sample (adjusted): 2001 2008
Included observations: 8 after adjustments
Variable Coefficient Std. error t-Statistic Prob.
I(�1) 1.105468 0.047283 23.37958 0.0000
R-squared 0.847928 Mean dependent var 1,458,352.
Adjusted R-squared 0.847928 S.D. dependent var 469,497.4
S.E. of regression 183,087.1 Akaike info criterion 27.18978
Sum squared resid 2.35E + 11 Schwarz criterion 27.19971
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
82 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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:
Autocorrelation Partial correlation AC PAC Q-Stat Prob
. |***** | . |***** | 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
Method: least squares
Date: 06/22/09 Time: 02:32
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
G-1 1.000002 1.85E-07 5,398,845. 0.0000
R-squared 1.000000 Mean dependent var 557,249.8
Adjusted R-squared 1.000000 S.D. dependent var 190,471.4
S.E. of regression 0.325331 Akaike info criterion 0.696492
Sum squared resid 0.846722 Schwarz criterion 0.718406
Log likelihood �2.134213 Durbin–Watson stat 0.151288
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
2.1 Factor Modeling of the Aggregate Demand in a National Economy. . . 83
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
Dependent variable: G
Method: least squares
Date: 06/22/09 Time: 02:45
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
E �5,838.985 385.8181 �15.13404 0.0000
C 1,185,925. 43,153.60 27.48148 0.0000
R-squared 0.970344 Mean dependent var 557,249.8
Adjusted R-squared 0.966107 S.D. dependent var 190,471.4
S.E. of regression 35,065.72 Akaike info criterion 23.96097
Sum squared resid 8.61E + 09 Schwarz criterion 24.00479
Log likelihood �105.8243 F-statistic 229.0391
Durbin–Watson stat 2.616388 Prob(F-statistic) 0.000001
Gt ¼ �5838:985� Eþ 1185925
Dependent variable: G
Method: least squares
Date: 06/22/09 Time: 03:30
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
Y 0.094588 0.004507 20.98709 0.0000
C 85,435.24 24,030.68 3.555257 0.0093
R-squared 0.984356 Mean dependent var 557,249.8
Adjusted R-squared 0.982121 S.D. dependent var 190,471.4
S.E. of regression 25,468.22 Akaike info criterion 23.32138
Sum squared resid 4.54E + 09 Schwarz criterion 23.36521
Log likelihood �102.9462 F-statistic 440.4579
Durbin–Watson stat 3.063158 Prob(F-statistic) 0.000000
Let us give preference to the second model:
G ¼ 0:0945881296� Y þ 85435:23751:
84 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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:
Autocorrelation Partial correlation AC PAC Q-Stat Prob
.****| . | .****| . | 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
Method: least squares
Date: 07/10/09 Time: 10:27
Sample (adjusted): 2002 2008
Included observations: 7 after adjustments
Variable Coefficient Std. error t-Statistic Prob.
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
Adjusted R-squared 0.738526 S.D. dependent var 22,132.43
S.E. of regression 11,317.32 Akaike info criterion 21.74101
Sum squared resid 6.40E + 08 Schwarz criterion 21.72556
Log likelihood �74.09354 Durbin–Watson stat 3.148337
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
2.1 Factor Modeling of the Aggregate Demand in a National Economy. . . 85
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:
Dependent variable: G
Method: least squares
Date: 10/25/09 Time: 23:31
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
T 1.303832 0.091964 14.17769 0.0000
C 113,994.6 33,652.48 3.387406 0.0116
R-squared 0.966347 Mean dependent var 557,249.8
Adjusted R-squared 0.961540 S.D. dependent var 190,471.4
S.E. of regression 37,353.89 Akaike info criterion 24.08739
Sum squared resid 9.77E + 09 Schwarz criterion 24.13122
Log likelihood �106.3933 F-statistic 201.0069
Durbin–Watson stat 1.662048 Prob(F-statistic) 0.000002
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:
Dependent variable: I
Method: least squares
Date: 10/26/09 Time: 00:00
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
T 3.445681 0.564900 6.099630 0.0005
C 182,574.0 206,715.3 0.883214 0.4064
R-squared 0.841649 Mean dependent var 1,353,980.
Adjusted R-squared 0.819027 S.D. dependent var 539,367.1
S.E. of regression 229,451.7 Akaike info criterion 27.71790
Sum squared resid 3.69E + 11 Schwarz criterion 27.76173
Log likelihood �122.7306 F-statistic 37.20548
Durbin–Watson stat 0.696745 Prob(F-statistic) 0.000491
86 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
Method: least squares
Date: 06/22/09 Time: 02:51
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
E �12,972.00 3,471.498 �3.736716 0.0073
C 1,851,510. 388,285.7 4.768423 0.0020
R-squared 0.666079 Mean dependent var 454,833.9
Adjusted R-squared 0.618376 S.D. dependent var 510,739.4
S.E. of regression 315,512.9 Akaike info criterion 28.35492
Sum squared resid 6.97E + 11 Schwarz criterion 28.39874
Log likelihood �125.5971 F-statistic 13.96305
Durbin–Watson stat 2.270347 Prob(F-statistic) 0.007294
2.1 Factor Modeling of the Aggregate Demand in a National Economy. . . 87
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
Included observations: 9
Autocorrelation Partial correlation AC PAC Q-Stat Prob
. |** . | . |** . | 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
Dependent variable: NX
Method: least squares
Date: 10/26/09 Time: 00:16
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
T 3.001217 0.721258 4.161085 0.0042
C �565,470.2 263,931.9 �2.142485 0.0694
R-squared 0.712107 Mean dependent var 454,833.9
Adjusted R-squared 0.670980 S.D. dependent var 510,739.4
S.E. of regression 292,961.5 Akaike info criterion 28.20660
Sum squared resid 6.01E + 11 Schwarz criterion 28.25043
Log likelihood �124.9297 F-statistic 17.31463
Durbin–Watson stat 2.269880 Prob(F-statistic) 0.004236
88 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
2. NX ¼ �15295� Eþ 2231761
Dependent variable: NX
Method: least squares
Date: 10/26/09 Time: 00:22
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
E �15,294.99 3,917.023 �3.904749 0.0059
C 2,231,761. 466,379.4 4.785292 0.0020
R-squared 0.685352 Mean dependent var 454,833.9
Adjusted R-squared 0.640402 S.D. dependent var 510,739.4
S.E. of regression 306,272.5 Akaike info criterion 28.29547
Sum squared resid 6.57E + 11 Schwarz criterion 28.33930
Log likelihood �125.3296 F-statistic 15.24706
Durbin–Watson stat 2.395862 Prob(F-statistic) 0.005863
Note that this regression is somewhat better in the main indexes in comparison
to the previous one.
3. NX ¼ 3455� PP$� 479230
Dependent variable: NX
Method: least squares
Date: 10/26/09 Time: 00:24
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
PP$ 3,454.733 485.5019 7.115796 0.0002
C �479,229.6 145,787.3 �3.287183 0.0134
R-squared 0.878545 Mean dependent var 454,833.9
Adjusted R-squared 0.861194 S.D. dependent var 510,739.4
S.E. of regression 190,284.3 Akaike info criterion 27.34356
Sum squared resid 2.53E + 11 Schwarz criterion 27.38738
Log likelihood �121.0460 F-statistic 50.63456
Durbin–Watson stat 2.353455 Prob(F-statistic) 0.000191
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:
2.1 Factor Modeling of the Aggregate Demand in a National Economy. . . 89
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
90 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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)
The statistical characteristics of model (2.3) are as follows: the determination
coefficient R2 ¼ 0.994; the standard error Se ¼ 69.2; the approximation coefficient
A ¼ 11.47%; the Fisher statistics F ¼ 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)
92 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
The statistical characteristics of model (2.4) are as follows: the determination
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
2.2 Macroeconomic Analysis of the National Economic State Based. . . 93
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
94 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.2 Macroeconomic Analysis of the National Economic State Based. . . 95
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)
The statistical characteristics of model (2.9) are as follows: the determination
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
96 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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)
Y (Gross National Income)
actual point 2007: GNI = 11371.07; i=13.6
actual point 2007: GNI = 11374.29; i=15.3
LM 2007
LM 2008
Fig. 2.4 Plots of models LM2007 and LM2008
2.2 Macroeconomic Analysis of the National Economic State Based. . . 97
� 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)
Y (Gross National Income)
actual point 2007: GNI = 11371.07; i=13.6
IS 2007
LM 2008
actual point 2007: GNI = 11374.29; i=15.3
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
98 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.2 Macroeconomic Analysis of the National Economic State Based. . . 99
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%
100 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.2 Macroeconomic Analysis of the National Economic State Based. . . 101
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
102 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
icsoftheRepublicofKazakhstan
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
millionsoftenge(inpricesoftheyear2000)
Iistheinvestm
entvolumein
milliontenge(inpricesoftheyear2000)
Gisthelevel
ofpublicexpensesin
millionsoftenge(inpricesoftheyear2000)
NXisthenet
exportin
millionsoftenge(inpricesoftheyear2000)
Risthereal
interestratecalculatedbytheFisher
equationwithuse
ofthecurrentinflationlevel(theconsumer
price
index
isusedas
theinflationindex;the
meancreditinterestrate
isusedas
thenominal
interestrate)
Eisthereal
interestratecalculatedbytheFisher
equationwithuse
ofthecurrentinflationlevel(theconsumer
price
index
isusedas
theinflationindex;the
meancreditinterestrate
isusedas
thenominal
interestrate)
M2,
M3aretherespectivemoney
aggregates
inmillionsoftenge(inpricesoftheyear2000)PPistheprice
ofonetonofUralsoilin
tenge(inpricesofthe
year2000)
104 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
Tab
le2.11
Correlationmatrixofindexes
ofthemacroeconomic
dynam
icsoftheRepublicofKazakhstan
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
2.3 Long-Term IS–LM Model and Mundell–Flemming Model 105
Dependent variable: Y
Method: least squares
Date: 11/27/09 Time: 21:35
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.985236 Mean dependent var 4,988,095.
Adjusted R-squared 0.980315 S.D. dependent var 1,997,879.
S.E. of regression 280,309.4 Akaike info criterion 28.18638
Sum squared resid 4.71E + 11 Schwarz criterion 28.25212
Log likelihood �123.8387 F-statistic 200.1997
Durbin–Watson stat 2.869894 Prob(F-statistic) 0.000003
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
Dependent variable: Y
Method: least squares
Date: 11/27/09 Time: 21:40
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.915521 Mean dependent var 4,988,095.
Adjusted R-squared 0.887361 S.D. dependent var 1,997,879.
S.E. of regression 670,522.1 Akaike info criterion 29.93070
Sum squared resid 2.70E + 12 Schwarz criterion 29.99644
Log likelihood �131.6882 F-statistic 32.51176
Durbin–Watson stat 1.164057 Prob(F-statistic) 0.000603
106 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
Dependent variable: Y
Method: least squares
Date: 11/27/09 Time: 21:44
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
R �335,340.1 125,105.8 �2.680452 0.0315
C 7,074,628. 925,242.8 7.646239 0.0001
R-squared 0.506515 Mean dependent var 4,988,095.
Adjusted R-squared 0.436017 S.D. dependent var 1,997,879.
S.E. of regression 1,500,383. Akaike info criterion 31.47347
Sum squared resid 1.58E + 13 Schwarz criterion 31.51730
Log likelihood �139.6306 F-statistic 7.184823
Durbin–Watson stat 1.672475 Prob(F-statistic) 0.031519
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
2.3 Long-Term IS–LM Model and Mundell–Flemming Model 107
Dependent variable: Y
Method: least squares
Date: 11/27/09 Time: 22:40
Sample (adjusted): 2001 2008
Included observations: 8 after adjustments
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.978132 Mean dependent var 5,286,619.
Adjusted R-squared 0.969384 S.D. dependent var 1,909,217.
S.E. of regression 334,061.3 Akaike info criterion 28.55603
Sum squared resid 5.58E + 11 Schwarz criterion 28.58582
Log likelihood �111.2241 F-statistic 111.8211
Durbin–Watson stat 2.273508 Prob(F-statistic) 0.000071
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
Method: least squares
Date: 11/28/09 Time: 01:38
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
Y 0.249065 0.010246 24.30966 0.0000
R �48,136.29 7,386.492 �6.516799 0.0003
(continued)
108 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
R-squared 0.971418 Mean dependent var 930,412.0
Adjusted R-squared 0.967335 S.D. dependent var 688,173.9
S.E. of regression 124,377.4 Akaike info criterion 26.49316
Sum squared resid 1.08E + 11 Schwarz criterion 26.53699
Log likelihood �117.2192 Durbin–Watson stat 1.247213
M3 ¼ 0:3236175168� Y � 54199:73886� R
Dependent variable: M_3
Method: least squares
Date: 11/28/09 Time: 01:40
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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.
Adjusted R-squared 0.954984 S.D. dependent var 880,468.3
S.E. of regression 186,808.9 Akaike info criterion 27.30669
Sum squared resid 2.44E + 11 Schwarz criterion 27.35052
Log likelihood �120.8801 Durbin–Watson stat 1.412212
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
2.3 Long-Term IS–LM Model and Mundell–Flemming Model 109
Dependent variable: Y
Method: least squares
Date: 11/28/09 Time: 01:54
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.937605 Mean dependent var 4,988,095.
Adjusted R-squared 0.916806 S.D. dependent var 1,997,879.
S.E. of regression 576,254.6 Akaike info criterion 29.62769
Sum squared resid 1.99E + 12 Schwarz criterion 29.69343
Log likelihood �130.3246 F-statistic 45.08056
Durbin–Watson stat 2.277324 Prob(F-statistic) 0.000243
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
Dependent variable: Y
Method: least squares
Date: 11/28/09 Time: 01:56
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.934467 Mean dependent var 4,988,095.
Adjusted R-squared 0.912623 S.D. dependent var 1,997,879.
S.E. of regression 590,565.3 Akaike info criterion 29.67675
Sum squared resid 2.09E + 12 Schwarz criterion 29.74249
Log likelihood �130.5454 F-statistic 42.77860
Durbin–Watson stat 1.517209 Prob(F-statistic) 0.000281
YBP ¼ cþ d1 � Rþ d2 � PPþ e;
Y ¼ �134497:9641� Rþ 224:5877659 � PP� 505075:5724
110 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
Dependent variable: Y
Method: least squares
Date: 11/28/09 Time: 01:57
Sample: 2000 2008
Included observations: 9
Variable Coefficient Std. error t-Statistic Prob.
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
R-squared 0.919141 Mean dependent var 4,988,095.
Adjusted R-squared 0.892188 S.D. dependent var 1,997,879.
S.E. of regression 655,997.4 Akaike info criterion 29.88690
Sum squared resid 2.58E + 12 Schwarz criterion 29.95264
Log likelihood �131.4911 F-statistic 34.10173
Durbin–Watson stat 1.704565 Prob(F-statistic) 0.000529
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.
2.3 Long-Term IS–LM Model and Mundell–Flemming Model 111
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”
112 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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]:
114 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.4 Macroeconomic Analysis and Parametric Control. . . 115
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)
with the determination coefficient R2 ¼ 0.62 and the approximation coefficient
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)
with the determination coefficient R2 ¼ 0.88 and the approximation coefficient
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.
116 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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;
2.4 Macroeconomic Analysis and Parametric Control. . . 117
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)
118 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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%).
2.4 Macroeconomic Analysis and Parametric Control. . . 119
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
120 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.4 Macroeconomic Analysis and Parametric Control. . . 121
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
122 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.4 Macroeconomic Analysis and Parametric Control. . . 123
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).
124 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
126 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 127
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
128 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 129
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
130 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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)
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 131
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
132 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
Regression statistics
Multiple R 0.090896264
Squared R 0.008262131
Normalized squared R �0.157027514
Standard error 0.039556997
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 1 7.82155E-05 7.82155E-05 0.04998577 0.830505924
Residual 6 0.009388536 0.001564756
Total 7 0.009466752
Coefficients Standard error t-statistics P-Level
Y-intersection 0.102447184 0.023546608 4.350825545 0.00481825
Monetary aggregate 0.009707317 0.043418618 0.223574986 0.83050592
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 133
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
Regression statistics
Multiple R 0.393505471
Squared R 0.154846556
Normalized squared R 0.013987648
Standard error 0.036516795
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 1 0.001465894 0.001465894 1.09930255 0.334805228
Residual 6 0.008000858 0.001333476
Total 7 0.009466752
Coefficients Standard error t-statistics P-Level
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
Regression statistics
Multiple R 0.363609741
Squared R 0.132212044
Normalized squared R �0.012419282
Standard error 0.037002552
Observations 8
(continued)
134 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
Variance analysis
df SS MS F Significance of F
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
Regression statistics
Multiple R 0.448455855
Squared R 0.201112654
Normalized squared R 0.067964763
Standard error 0.035503211
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 1 0.001903884 0.001903884 1.51044566 0.265081063
Residual 6 0.007562868 0.001260478
Total 7 0.009466752
Coefficients Standard
error
t-statistics P-Level
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
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 135
Regression statistics
Multiple R 0.68692623
Squared R 0.471867645
Normalized squared R 0.383845586
Standard error 0.028866641
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 1 0.004467054 0.004467054 5.36078854 0.059829763
Residual 6 0.004999698 0.000833283
Total 7 0.009466752
Coefficients Standard error t-statistics P-Level
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
Regression statistics
Multiple R 0.325197844
Squared R 0.105753638
Normalized squared R �0.043287422
Standard error 0.037562411
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 1 0.001001143 0.001001143 0.70956042 0.431878752
Residual 6 0.008465608 0.001410935
Total 7 0.009466752
Coefficients Standard
error
t-statistics P-Level
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
136 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
Regression statistics
Multiple R 0.082469621
Squared R 0.006801238
Normalized squared R �0.158731889
Standard error 0.039586121
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 1 6.43856E-05 6.43856E-05 0.04108687 0.846069149
Residual 6 0.009402366 0.001567061
Total 7 0.009466752
Coefficients Standard error t-statistics P-Level
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
Regression statistics
Multiple R 0.544831904
Squared R 0.296841803
Normalized squared R 0.17964877
Standard error 0.033308217
Observations 8
(continued)
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 137
Variance analysis
df SS MS F Significance of F
Regression 1 0.002810128 0.002810128 2.53293047 0.162598324
Residual 6 0.006656624 0.001109437
Total 7 0.009466752
Coefficients Standard error t-statistics P-Level
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
Regression statistics
Multiple R 0.200867874
Squared R 0.040347903
Normalized squared R �0.119594113
Standard error 0.038911841
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 1 0.000381964 0.000381964 0.25226581 0.633380857
Residual 6 0.009084788 0.001514131
Total 7 0.009466752
Coefficients Standard error t-statistics P-Level
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.
138 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
Regression statistics
Multiple R 0.983824162
Squared R 0.967909982
Normalized squared R 0.755073974
Standard error 0.025399041
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)
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 139
Variance analysis
df SS MS F Significance of F
Regression 3 0.097290321 0.032430107 50.2705633 0.001240905
Residual 5 0.003225556 0.000645111
Total 8 0.100515878
Coefficients Standard error t-statistics P-Level
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
Regression statistics
Multiple R 0.983349812
Squared R 0.966976853
Normalized squared R 0.794806328
Standard error 0.023520738
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 2 0.097196527 0.048598263 87.8453696 0.000127375
Residual 6 0.003319351 0.000553225
Total 8 0.100515878
Coefficients Standard error t-statistics P-Level
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
140 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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
Regression statistics
Multiple R 0.959674566
Squared R 0.920975272
Normalized squared R 0.689365381
Standard error 0.039857847
Observations 8
Variance analysis
df SS MS F Significance of F
Regression 3 0.092572638 0.030857546 19.4237782 0.007568807
Residual 5 0.00794324 0.001588648
Total 8 0.100515878
Coefficients Standard
error
t-statistics P-Level
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 Modeling of Inflationary Processes by Means of Regression Analysis. . . 141
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
Regression statistics
Multiple R 0.810422603
Squared R 0.656784795
Normalized squared R 0.427974658
Standard error 0.030184345
Observations 6
Variance analysis
df SS MS F Significance of F
Regression 2 0.005230478 0.002615239 2.870435745 0.201071158
Residual 3 0.002733284 0.000911095
Total 5 0.007963762
Coefficients Standard error t-statistics P-Level
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
Regression statistics
Multiple R 0.812898732
Squared R 0.660804348
Normalized squared R �0.356782608
Standard error 0.030588451
Observations 5
(continued)
142 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
Variance analysis
df SS MS F Significance of F
Regression 3 0.001822794 0.000607598 0.649383666 0.697175376
Residual 1 0.000935653 0.000935653
Total 4 0.002758447
Coefficients Standard error t-statistics P-Level
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
Regression statistics
Multiple R 0.810354519
Squared R 0.656674446
Normalized squared R 0.313348891
Standard error 0.021760577
Observations 5
Variance analysis
df SS MS F Significance of F
Regression 2 0.001811402 0.000905701 1.912687353 0.343325554
Residual 2 0.000947045 0.000473523
Total 4 0.002758447
Coefficients Standard error t-statistics P-Level
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
2.5 Modeling of Inflationary Processes by Means of Regression Analysis. . . 143
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.
144 2 Macroeconomic Analysis and Parametric Control of Equilibrium. . .
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:
A.A. Ashimov et al., Macroeconomic Analysis and Economic PolicyBased on Parametric Control, DOI 10.1007/978-1-4614-1153-6_3,# Springer Science+Business Media, LLC 2012
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
148 3 Parametric Control of Cyclic Dynamics of Economic Systems
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
Fig. 3.4 Capital productivity ratio without parametric control and with use of law 3, optimal in the
sense of criterion 2
3.1 Mathematical Model of the Kondratiev Cycle 149
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.
Fig. 3.5 Capital productivity ratio without parametric control and with use of law 4, optimal in the
sense of criterion 3
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
Fig. 3.6 Capital productivity ratio without parametric control and with use of law 4, optimal in the
sense of criterion 4
150 3 Parametric Control of Cyclic Dynamics of Economic Systems
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
sense of criterion 4
3.1 Mathematical Model of the Kondratiev Cycle 151
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
152 3 Parametric Control of Cyclic Dynamics of Economic Systems
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:
154 3 Parametric Control of Cyclic Dynamics of Economic Systems
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)
under the constraints
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).
The problem is solved in two stages:
– In the first stage, the optimal values of the coefficients ci for each law Ui are
determined by enumerating their values in the respective intervals (quantized
with a small step size) minimizing K under constraints (3.9).
– In the second stage, the optimal law regulating parameter k is chosen based on
the 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.
3.2 Goodwin Mathematical Model of Market Fluctuations of Growing Economies 155
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
156 3 Parametric Control of Cyclic Dynamics of Economic Systems
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
3.2 Goodwin Mathematical Model of Market Fluctuations of Growing Economies 157
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.
158 3 Parametric Control of Cyclic Dynamics of Economic Systems
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):
3.2 Goodwin Mathematical Model of Market Fluctuations of Growing Economies 159
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
160 3 Parametric Control of Cyclic Dynamics of Economic Systems
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.
3.2 Goodwin Mathematical Model of Market Fluctuations of Growing Economies 161
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,
A.A. Ashimov et al., Macroeconomic Analysis and Economic PolicyBased on Parametric Control, DOI 10.1007/978-1-4614-1153-6_4,# Springer Science+Business Media, LLC 2012
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;
166 4 Parametric Control of Economic Growth of a National Economy. . .
– 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_2l The share of the budget for purchasing the labor force at the price P__1l.
CO_p1_2z The share of the budget for purchasing knowledge-provision services at the price
P__1z.
CO_p1_2r The share of the budget for purchasing educational services at the price P__1r.
CO_p1_2n The share of the budget for purchasing innovation products at the price P__1n.
CO_p1_2i The share of the budget for purchasing investment products at the price P__1i.
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.
CA_r_2 The dimension coefficient of the production function.
(continued)
4.1 National Economic Evolution Control Based on a Computable. . . 167
CA_k_2 The coefficient of capital assets of the production function.
CA_l_2 The coefficient of labor of the production function.
Calpha__2 The coefficient of the costs of knowledge-provision services of the production
function.
Cbeta__2 The coefficient of the costs of educational services of the production function.
Cgamma__2 The coefficient of the costs of innovation products of the production function.
CA_0_2 The rate of depreciation for capital assets.
CR__2 The share of retired capital assets.
4.1.1.5 Sector 3
CO_p1_3l The share of the budget for purchasing the labor force at the price P__1l.
CO_p1_3z The share of the budget for purchasing knowledge-provision services at the price
P__1z.
CO_p1_3r The share of the budget for purchasing educational services at the price P__1r.
CO_p1_3n The share of the budget for purchasing innovation products at the price P__1n.
CO_p1_3i The share of the budget for purchasing investment products at the price P__1i.
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.
CA_r_3 The dimension coefficient of the production function.
CA_k_3 The coefficient of capital assets of the production function.
CA_l_3 The coefficient of labor of the production function.
Calpha__3 The coefficient of the costs of knowledge-provision services of the production
function.
Cbeta__3 The coefficient of the costs of educational services of the production function.
Cgamma__3 The coefficient of the costs of innovation products of the production function.
CA_0_3 The rate of depreciation for capital assets.
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_p1_4r The share of the budget for purchasing educational services at the price P__1r.
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.
168 4 Parametric Control of Economic Growth of a National Economy. . .
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 National Economic Evolution Control Based on a Computable. . . 169
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.
Ceta__2 The iteration constant applied in the case of the exogenous price.
Ceta__3 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_1l The demand of the labor force at the price P__1l.
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)
170 4 Parametric Control of Economic Growth of a National Economy. . .
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_p3_2l The share of the budget for purchasing the labor force at the price P__3l.
VO_t_2 The share of the budget for paying taxes to the consolidated budget.
VO_f_2 The share of the budget for paying taxes to the off-budget funds.
VO_s_2 The share of the retained budget.
VY__2 The value added produced by the sector.
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_p3_2l The demand of the labor force at the price P__3l.
VD_p1_2l The demand of the labor force at the price P__1l.
VD_p1_2z The demand of knowledge-provision services at the price P__1z.
VD_p1_2r The demand of educational services at the price P__1r.
VD_p1_2n The demand of innovation products at the price P__1n.
VD_p1_2i The demand of investment products at the price P__1i.
VY_p_2 The gain in current prices.
VB__2 The budget of the sector.
VB_b_2 The balance of banking accounts.
VK__2 The capital assets of the sector.
4.1.1.14 Sector 3
VO_p3_3l The share of the budget for purchasing the labor force at the price P__3l.
VO_t_3 The share of the budget for paying taxes to the consolidated budget.
VO_f_3 The share of the budget for paying taxes to the off-budget funds.
VO_s_3 The share of the retained budget.
VY__3 The value added produced by the sector.
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.
VD_p3_3l The demand of the labor force at the price P__3l.
VD_p1_3l The demand of the labor force at the price P__1l.
VD_p1_3z The demand of knowledge-provision services at the price P__1z.
VD_p1_3r The demand of educational services at the price P__1r.
(continued)
4.1 National Economic Evolution Control Based on a Computable. . . 171
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__3 The budget of the sector.
VB_b_3 The balance of bank accounts.
VK__3 The capital assets of the sector.
4.1.1.15 Sector 4
VO_tax_4 The share of the budget for discharging the income tax.
VO_s_4 The share of the retained budget.
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).
VW_3_2 The wages of the employees of Sector 2 (the state-owned enterprises).
VW_1_2 The wages of the employees of Sector 2 (the privately owned enterprises).
VW_3_3 The wages of the employees of Sector 3 (the state-owned enterprises).
VW_1_3 The wages of the employees of Sector 3 (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.
VD_p1_5z The demand of knowledge-provision services at the price P__1z.
VD_p1_5r The demand of educational services at the price P__1r.
VD_p1_5n The demand of innovation products at the price P__1n.
VG_s_1 The expenditures of the consolidated budget aimed at backing Sector 1.
VG_s_2 The expenditures of the consolidated budget aimed at backing Sector 2.
VG_s_3 The expenditures of the consolidated budget aimed at backing Sector 3.
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.
172 4 Parametric Control of Economic Growth of a National Economy. . .
4.1.1.17 General Part of the Model
VP__1l The price of the labor force.
VP__1c The price of final products for the households.
VP__1g The price of final products for the economic agent 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 National Economic Evolution Control Based on a Computable. . . 173
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)
174 4 Parametric Control of Economic Growth of a National Economy. . .
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)
4.1 National Economic Evolution Control Based on a Computable. . . 175
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)
176 4 Parametric Control of Economic Growth of a National Economy. . .
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)
4.1 National Economic Evolution Control Based on a Computable. . . 177
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].
178 4 Parametric Control of Economic Growth of a National Economy. . .
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:
VO�p3�2l½t;Qþ 1� ¼ VO�p3�2l½t;Q� � Ceta��2þ VO�p3�2l½t;Q�� VI��l½t;Q� � ð1� Ceta��2Þ: (4.49)
4.1 National Economic Evolution Control Based on a Computable. . . 179
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.
The production function equation is given by
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:
The demand of the labor force at governmental prices:
VD�p3�2l½t� ¼ VO�p3�2l½t� � VB��2½t�ð Þ=CP��3l½t�: (4.51)
The demand of the labor force at market prices:
VD�p1�2l½t� ¼ CO�p1�2l½t� � VB��2½t�ð Þ=VP��1l½t�: (4.52)
The demand of knowledge-provision services:
VD�p1�2z½t� ¼ CO�p1�2z½t� � VB��2½t�ð Þ=VP��1z½t�: (4.53)
The demand of educational services:
VD�p1�2r½t� ¼ CO�p1�2r½t� � VB��2½t�ð Þ=VP��1r½t�: (4.54)
The demand of innovation products:
VD�p1�2n½t� ¼ CO�p1�2n½t� � VB��2½t�ð Þ=VP��1n½t� (4.55)
The demand of investment products:
VD�p1�2i½t� ¼ CO�p1�2i½t� � VB��2½t�ð Þ=VP��1i½t�: (4.56)
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)
180 4 Parametric Control of Economic Growth of a National Economy. . .
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)
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_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)
The capital assets are determined by
VK��2½tþ 1� ¼ 1� CR��2½t�ð Þ � VK��2½t� þ VD�p1�2i½t�: (4.62)
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 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].
4.1 National Economic Evolution Control Based on a Computable. . . 181
The share of budget for discharging the single social tax to off-budget funds
is described as
VO�f�2½t� ¼ VW�3�2½t� þ VW�1�2½t�ð Þ � CT�esn½t�ð Þ=VB��2½t�: (4.64)
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:
VO�p3�3l½t;Qþ 1� ¼ VO�p3�3l½t;Q� � Ceta��3þ VO�p3�3l½t;Q�� VI��l½t;Q� � ð1� Ceta��3Þ: (4.66)
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.
The production function equation is given by
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)
The following formulas determine the demand of the production factors in the
other branches of the economy:
The demand of the labor force at governmental prices:
VD�p3�3l½t� ¼ VO�p3�3l½t� � VB��3½t�ð Þ=CP��3l½t�: (4.68)
The demand of the labor force at market prices:
182 4 Parametric Control of Economic Growth of a National Economy. . .
VD�p1�3l½t� ¼ CO�p1�3l½t� � VB��3½t�ð Þ=VP��1l½t�: (4.69)
The demand of knowledge-provision services:
VD�p1�3z½t� ¼ CO�p1�3z½t� � VB��3½t�ð Þ=VP��1z½t�: (4.70)
The demand of educational services:
VD�p1�3r½t� ¼ CO�p1�3r½t� � VB��3½t�ð Þ=VP��1r½t�: (4.71)
The demand of innovation products:
VD�p1�3n½t� ¼ CO�p1�3n½t� � VB��3½t�ð Þ=VP��1n½t�: (4.72)
The demand of investment products:
VD�p1�3i½t� ¼ CO�p1�3i½t� � VB��3½t�ð Þ=VP��1i½t�: (4.73)
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)
4.1 National Economic Evolution Control Based on a Computable. . . 183
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_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)
The capital assets are determined by
VK��3½tþ 1� ¼ 1� CR��3½t�ð Þ � VK��3½t� þ VD�p1�3i½t�: (4.81)
This formula calculates the volume of the 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 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
VO�f�3½t� ¼ VW�3�3½t� þ VW�1�3½t�ð Þ � CT�esn½t�ð Þ=VB��3½t�: (4.83)
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
184 4 Parametric Control of Economic Growth of a National Economy. . .
VD�p1�4c½t� ¼ CO�p1�4c½t� � VB��4½t�ð Þ=VP��1c½t�: (4.85)
The households’ demand for educational services:
VD�p1�4r½t� ¼ CO�p1�4r½t� � VB��4½t�ð Þ=VP��1r½t�: (4.86)
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)
4.1 National Economic Evolution Control Based on a Computable. . . 185
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 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)
186 4 Parametric Control of Economic Growth of a National Economy. . .
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:
VD�p1�5z½t� ¼ CO�p1�5z½t� � VB��5½t�ð Þ=VP��1z½t�: (4.102)
The educational service payment:
VD�p1�5r½t� ¼ CO�p1�5r½t� � VB��5½t�ð Þ=VP��1r½t�: (4.103)
The demand of the innovation products:
VD�p1�5n½t� ¼ CO�p1�5n½t� � VB��5½t�ð Þ=VP��1n½t�: (4.104)
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)
4.1 National Economic Evolution Control Based on a Computable. . . 187
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.;
188 4 Parametric Control of Economic Growth of a National Economy. . .
– 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
4.1 National Economic Evolution Control Based on a Computable. . . 189
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
190 4 Parametric Control of Economic Growth of a National Economy. . .
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)
The problem is solved in two stages:
– 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�:
4.1 National Economic Evolution Control Based on a Computable. . . 191
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
192 4 Parametric Control of Economic Growth of a National Economy. . .
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
4.1 National Economic Evolution Control Based on a Computable. . . 193
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)
194 4 Parametric Control of Economic Growth of a National Economy. . .
4.2 National Economic Evolution Control Basedon a Computable Model of General Equilibriumof Economic Branches
4.2.1 Model Description, Parametric Identification,and Retrospective Prediction
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.
Economic agent 26 is the outside world.
This model also includes 1,722 exogenous parameters and 1,104 endogenous
variables.
4.2.1.2 Exogenous Parameters of the Model
The exogenous parameters of the model include the following:
– The coefficients of the production functions of the sectors;
– 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 deposit interest rates;
– The various taxation rates;
– 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.
196 4 Parametric Control of Economic Growth of a National Economy. . .
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)
4.2 National Economic Evolution Control Based. . . 197
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.
4.2.1.6 Banking Sector
CP__bpercent The deposit interest rate for enterprises.
CP_h_bpercent The deposit interest rate for physical bodies.
4.2.1.7 General Part of the Model
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.
4.2.1.9 Endogenous Variables of the Model
The endogenous variables include the following:
– The budgets of the sectors and their various shares;
– The remainders of the agents’ budgets;
– The added produced values of the producing sectors;
– Demands and supplies of the various products and services;
– The gains of the sectors;
– 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 expenditures of the consolidated budget;
– 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;
198 4 Parametric Control of Economic Growth of a National Economy. . .
– 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 National Economic Evolution Control Based. . . 199
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.
4.2.1.13 Integral Indexes of the Model
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.
4.2.1.14 General Part of the Model
VP__il The price of the labor force in the ith branch.
VP__iz The price of the intermediate product produced by the ith branch, 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.
200 4 Parametric Control of Economic Growth of a National Economy. . .
4.2.1.16 Model Markets
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)
4.2 National Economic Evolution Control Based. . . 201
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.
202 4 Parametric Control of Economic Growth of a National Economy. . .
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:
4.2 National Economic Evolution Control Based. . . 203
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.
The production function equation is given by
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�p3�iz½t� ¼ ðCO�p3�iz� VB��i½t�Þ=VP��3z½t�; (4.141)
204 4 Parametric Control of Economic Growth of a National Economy. . .
VD�p4�iz½t� ¼ ðCO�p4�iz� VB��i½t�Þ=VP��4z½t�; (4.142)
VD�p5�iz½t� ¼ ðCO�p5�iz� VB��i½t�Þ=VP��5z½t�; (4.143)
VD�p6�iz½t� ¼ ðCO�p6�iz� VB��i½t�Þ=VP��6z½t�; (4.144)
VD�p7�iz½t� ¼ ðCO�p7�iz� VB��i½t�Þ=VP��7z½t�; (4.145)
VD�p8�iz½t� ¼ ðCO�p8�iz� VB��i½t�Þ=VP��8z½t�; (4.146)
VD�p9�iz½t� ¼ ðCO�p9�iz� VB��i½t�Þ=VP��9z½t�; (4.147)
VD�p10�iz½t� ¼ ðCO�p10�iz� VB��i½t�Þ=VP��10z½t�; (4.148)
VD�p11�iz½t� ¼ ðCO�p11�iz� VB��i½t�Þ=VP��11z½t�; (4.149)
VD�p12�iz½t� ¼ CO�p12�iz� VB��i½t�ð Þ=VP��12z½t�; (4.150)
VD�p13�iz½t� ¼ ðCO�p13�iz� VB��i½t�Þ=VP��13z½t�; (4.151)
VD�p14�iz½t� ¼ ðCO�p14�iz� VB��i½t�Þ=VP��14z½t�; (4.152)
VD�p15�iz½t� ¼ CO�p15�iz� VB��i½t�ð Þ=VP��15z½t�; (4.153)
VD�p16�iz½t� ¼ ðCO�p16�iz� VB��i½t�Þ=VP��16z½t�; (4.154)
VD�p17�iz½t� ¼ CO�p17�iz� VB��i½t�ð Þ=VP��17z½t�; (4.155)
VD�p18�iz½t� ¼ CO�p18�iz� VB��i½t�ð Þ=VP��18z½t�; (4.156)
VD�p19�iz½t� ¼ VO�p19�iz� VB��i½t�ð Þ=CP��19z½t�; (4.157)
VD�p20�iz½t� ¼ CO�p20�iz� VB��i½t�ð Þ=VP��20z½t�; (4.158)
VD�p21�iz½t� ¼ CO�p21�iz� VB��i½t�ð Þ=VP��21z½t�; (4.159)
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.
4.2 National Economic Evolution Control Based. . . 205
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.
206 4 Parametric Control of Economic Growth of a National Economy. . .
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);
2. The gain received in the current period;
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)
The capital assets are determined by
VK��i½tþ 1� ¼ 1� CR��i½t�ð Þ � VK��i½t� þ VD�p�in½t�: (4.171)
This formula calculates the volume of the capital assets taking into account their
retirement. An asset put into operation enters the formula with a plus sign.
The share of the budget of the producing agent for discharging the taxes to the
consolidated budget is given by
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 National Economic Evolution Control Based. . . 207
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)
The agent’s budget is formed from the following:
1. The funds in bank accounts (subject to interest on deposits);
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.
Let us proceed to the formulas determining the behavior of economic agent 24.
208 4 Parametric Control of Economic Growth of a National Economy. . .
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).
The dynamics of the banking account balance of the consolidated budget are
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 pension fund;
– The social insurance fund;
– The federal and territorial funds of obligatory medical insurance.
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)
The social transfers to the population:
VG�tr�24½t� ¼ CO�tr�24� VB��24½t�: (4.186)
4.2 National Economic Evolution Control Based. . . 209
The assets of the off-budget funds made available for the population:
VG�f�24½t� ¼ CO�f�24� VF��24½t�: (4.187)
This includes the assets of the pension fund and social insurance fund for paying
out the pensions and welfare payments.
4.2.1.20 Integral Indexes of the Model
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
210 4 Parametric Control of Economic Growth of a National Economy. . .
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.
4.2 National Economic Evolution Control Based. . . 211
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,
212 4 Parametric Control of Economic Growth of a National Economy. . .
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 National Economic Evolution Control Based. . . 213
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
214 4 Parametric Control of Economic Growth of a National Economy. . .
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 Model Description, Parametric Identification,and Retrospective Prediction
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.
216 4 Parametric Control of Economic Growth of a National Economy. . .
– 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.
Economic agent 7 is the outside world.
4.3.1.2 Exogenous Parameters of the Model
The exogenous parameters of the model include the following:
– The coefficients of the production functions of the sectors;
– 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 deposit interest rates;
– The issue of money;
– The various taxation rates;
– 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.
4.3 National Economic Evolution Control Based on a Computable Model. . . 217
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
economic agent 5 at the price P_2g.
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
products at the price P_2i.
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
economic agent 5 at the price P_2g.
E_2i_P2 The share of the market sector products for selling in the markets of investment
products at the price P_2i.
(continued)
218 4 Parametric Control of Economic Growth of a National Economy. . .
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 National Economic Evolution Control Based on a Computable Model. . . 219
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.
4.3.1.8 Banking Sector
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.
220 4 Parametric Control of Economic Growth of a National Economy. . .
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.
4.3.1.11 Endogenous Variables of the Model
The endogenous variables include the following:
– The budgets of the sectors and their various shares;
– 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 gains of the sectors;
– 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 expenditures of the consolidated budget;
– 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)
4.3 National Economic Evolution Control Based on a Computable Model. . . 221
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)
222 4 Parametric Control of Economic Growth of a National Economy. . .
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.
D_4c_P2 The households’ demand of final products at the price P_2c.
D_4c_P3 The households’ demand of final products at the price P_3c.
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 National Economic Evolution Control Based on a Computable Model. . . 223
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.
D_sl_P2 The total demand of the labor force at the price P_2l.
D_sl_P3 The total demand of the labor force at the price P_3l.
S_sl_P1 The total supply of the labor force at the price P_1l.
S_sl_P2 The total supply of the labor force at the price P_2l.
S_sl_P3 The total supply of the labor force at the price P_3l.
D_sc_P1 The total demand of final products at the price P_1c.
D_sc_P2 The total demand of final products at the price P_2c.
D_sc_P3 The total demand of final products at the price P_3c.
S_sc_P1 The total supply of final products at the price P_1c.
(continued)
224 4 Parametric Control of Economic Growth of a National Economy. . .
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.
4.3.1.18 Model Markets
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)
4.3 National Economic Evolution Control Based on a Computable Model. . . 225
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)
226 4 Parametric Control of Economic Growth of a National Economy. . .
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)
4.3 National Economic Evolution Control Based on a Computable Model. . . 227
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)
228 4 Parametric Control of Economic Growth of a National Economy. . .
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.
The production function equation is given by
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.
The demand of the labor force at governmental prices:
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)
The demand of investment products:
At governmental prices:
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)
4.3 National Economic Evolution Control Based on a Computable Model. . . 229
The following formulas determine the supply of products of the state sector.
The supply of final products for households:
At governmental prices:
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:
At governmental prices:
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:
At governmental prices:
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:
At governmental prices:
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)
The supply of exported products:
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)
230 4 Parametric Control of Economic Growth of a National Economy. . .
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)
The agent’s budget is formed from the following:
1. The funds in bank accounts (subject to interest on deposits);
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
consolidated budget is given by
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
funds is described as
O�1�f ½t� ¼ ðW�1½t� � T��esnÞ=B�1½t�: (4.256)
4.3 National Economic Evolution Control Based on a Computable Model. . . 231
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.
The production function equation is given by
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.
The following formulas determine the demand of the production factors in the
market sector.
The demand of the labor force at market prices:
D�2l�P2½t� ¼ ðO�2l�P2� B�2½t�Þ=P�2l½t�: (4.261)
The demand of capital products:
At governmental prices:
D�2k�P1½t� ¼ ðO�2k�P1½t� � B�2½t�Þ=P�1k; (4.262)
232 4 Parametric Control of Economic Growth of a National Economy. . .
At market prices:
D�2k�P2½t� ¼ ðO�2k�P2� B�2½t�Þ=P�2k½t�: (4.263)
The demand of investment products:
At governmental prices:
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)
The supply of exported products:
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)
4.3 National Economic Evolution Control Based on a Computable Model. . . 233
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)
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 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)
This formula calculates the volume of the capital assets taking into account their
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
consolidated budget is given by
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.
234 4 Parametric Control of Economic Growth of a National Economy. . .
The share of the budget for discharging the single social tax to the off-budget
funds is described as
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.
The production function equation is given by
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)
4.3 National Economic Evolution Control Based on a Computable Model. . . 235
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)
The agent’s budget is formed from the following:
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:
At governmental prices:
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:
236 4 Parametric Control of Economic Growth of a National Economy. . .
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)
The agent’s budget is formed from the following:
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;
4.3 National Economic Evolution Control Based on a Computable Model. . . 237
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.
Let us proceed to the formulas determining the behavior of economic agent 5.
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).
The dynamics of the banking account balance of the consolidated budget are
described by
B�5�b½tþ 1� ¼ ðO�5b�s½t� � B�5½t�Þ; (4.302)
238 4 Parametric Control of Economic Growth of a National Economy. . .
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 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
F�5�b½tþ 1� ¼ O�5f�s½t� � F�5½t�: (4.304)
The demand of the final products:
At governmental prices:
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)
The social transfers to the population:
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)
4.3 National Economic Evolution Control Based on a Computable Model. . . 239
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)
This includes the assets of the pension fund and social insurance fund for paying
out the pensions and welfare payments.
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.
4.3.1.26 Integral Indexes of the Model
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)
240 4 Parametric Control of Economic Growth of a National Economy. . .
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
4.3 National Economic Evolution Control Based on a Computable Model. . . 241
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;
242 4 Parametric Control of Economic Growth of a National Economy. . .
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 National Economic Evolution Control Based on a Computable Model. . . 243
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
244 4 Parametric Control of Economic Growth of a National Economy. . .
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 various taxation rates;
– 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
4.3 National Economic Evolution Control Based on a Computable Model. . . 245
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]
10 The share of the consolidated budget for purchasing
capital products
[0.129; 0.158]
11 The share of the consolidated budget for purchasing
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]
13 The share of products of the state sector for selling in
the markets of final products for economic agent
5 at exogenous prices
[0.039; 0.048]
14 The share of products of the state sector for selling in
the markets of final products for economic agent
5 at market prices
[0.039; 0.048]
15 The share of products of the state sector for selling in
the markets of investment products at exogenous
prices
[0.107; 0.131]
16 The share of products of the state sector for selling in
the markets of investment products at market
prices
[0.107; 0.131]
17 The share of products of the state sector for selling in
the markets of capital products at exogenous
prices
[0.200; 0.244]
18 The share of products of the state sector for selling in
the markets of capital products at market prices
[0.200; 0.244]
19 The share of products of the state sector for selling in
the markets of final products in foreign countries
[0.230; 0.281]
246 4 Parametric Control of Economic Growth of a National Economy. . .
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
4.3 National Economic Evolution Control Based on a Computable Model. . . 247
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
A.A. Ashimov et al., Macroeconomic Analysis and Economic PolicyBased on Parametric Control, DOI 10.1007/978-1-4614-1153-6_5,# Springer Science+Business Media, LLC 2012
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,
e-mail: [email protected].
Zheksenbek Makeevich Abdilov is a doctor of economic sciences, professor
at Kazakh National Technical University named after K.I. Satpaev, e-mail:
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,
e-mail: [email protected].
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
currency exchange rate, 87
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
aggregate consumer (households),
economic agent 4, 184–186
259
banking sector, 169
endogenous variables, 170–172
exogenous parameters, 166–169
general part, 169, 173
government, economic agent 5,
186–188
integral indexes, 173, 188–189
model agents, 164–166
model markets, 174–177
optimal parametric control laws, 189–194
other branches of the economy,
economic agent 3, 182–184
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
endogenous variables, 221–225
exogenous parameters, 217–219
finding, optimal values, 244–247
general part, 220
government, 220
integral indexes, 240–241
market sector, economic agent 2,
232–235
model constants, 221
model markets, 225–228
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
description, 145–146
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
aggregate consumer (households),
economic agent 23, 208
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
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
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
description, 152–153
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
regression equations, 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
regression equations, 111
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
aggregate consumer (households),
economic agent 4, 184–186
banking sector, 169
endogenous variables
Sector 1, 170–171
Sector 2, 171
Sector 3, 171–172
Sector 4, 172
Sector 5, 172
exogenous parameters
Sector 1, 167
Sector 2, 167–168
Sector 3, 168
Sector 4, 168
Sector 5, 169
general part, 169, 173
government, economic agent 5, 186–188
integral indexes, 173, 188–189
Index 261
Knowledge sector (cont.)model agents, 164–166
model markets, 174–177
optimal parametric control laws,
189–194
other branches of the economy, economic
agent 3, 182–184
science and education sector, economic
agent 1, 177–179
sector of innovation, economic agent 2,
179–182
technical parameters, 170
technical variable, 173
Kondratiev cycle, mathematical model
criterion K optimal value, variational
calculus problem, 151–152
description, 145–146
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
currency exchange rate, 116
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
statistical characteristics, 115
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
choosing optimal laws, 57–62
model description, 53–56
structural stability, 56–57, 62–65
neoclassic theory, optimal growth
bifurcation points, 24–25
choosing optimal laws, 21–22
model description, 19–20
structural stability, 20–21, 23
public expense and interest rate of
government loans
bifurcation points, 49–53
choosing optimal law, 37–43
model description, 34–36
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
statistical characteristics, 96
262 Index
values of multipliers, 95–96
velocity of money, 95
NNational economic evolution control based
CGE model
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
knowledge sector
aggregate consumer (households),
economic agent 4, 184–186
banking sector, 169
endogenous variables, 170–172
exogenous parameters, 166–169
general part, 169, 173
government, economic agent 5,
186–188
integral indexes, 173, 188–189
model agents, 164–166
model markets, 174–177
optimal parametric control laws,
189–194
other branches of the economy,
economic agent 3, 182–184
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, 219
aggregated consumer (households),
economic agent 4, 236–238
banking sector, economic agent 6, 240
economic agents, 215–217
endogenous variables, 221–225
exogenous parameters, 217–219
finding, optimal values, 244–247
general part, 220
government, 220
integral indexes, 240–241
market sector, economic agent 2,
232–235
model constants, 221
model markets, 225–228
outside world, economic agent 7, 240
parametric identification, 241–244
state, economic agent 5, 238–240
state sector, economic agent 1, 228–232
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
bifurcation points, 24–25
Index 263
Neoclassic theory (cont.)choosing optimal laws, 21–22
model description, 19–20
structural stability, 20–21, 23
OObjective functions, 12, 19
One-sector Solow model
choosing optimal laws, 26–27
dependence, 28
model description, 25
parameter estimation, 25–26
structural stability, 26, 27–28
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
public expense and interest rate of
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
choosing optimal laws, 70–73
model description, 66–70
structural stability, 70, 73–74
influence, uncontrolled parametric
disturbances, 14–16
neoclassical theory, optimal growth
bifurcation points, 24–25
choosing optimal laws, 21–22
model description, 19–20
structural stability analysis, 20–21, 23
one-sector Solow model
choosing optimal laws, 26–27
dependence, 28
model description, 25
parameter estimation, 25–26
structural stability, 26, 27–28
Richardson model, defense costs estimation
choosing optimal laws, 31–33
dependence, 33
model description, 28–29
parameters, 29–30
structural stability, 30, 33
variational calculus problem statement
continuous-time dynamical system, 8–9
continuous-time system, 6–8
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
continuous-time dynamical system, 8–9
continuous-time system, 6–8
components, 2–3
econometric methods, 1–2
influence, uncontrolled parametric
disturbances, 14–16
mathematical model, national economic
system (see Mathematical model,
national economic system)
neoclassic theory, optimal growth
bifurcation points, 24–25
choosing optimal laws, 21–22
model description, 19–20
structural stability, 20–21, 23
one-sector Solow model
choosing optimal laws, 26–27
dependence, 28
model description, 25
parameter estimation, 25–26
structural stability, 26, 27–28
Richardson model
choosing optimal laws, 31–33
dependence, 33
estimation, model parameters, 29–30
264 Index
model description, 28–29
structural stability, 30, 33
scenario approach, 2
Parametric identification, 189, 195, 210,
215, 241–244
RRichardson model, defense costs estimation
choosing optimal laws, 31–33
dependence, 33
model description, 28–29
parameters, 29–30
structural stability, 30, 33
SShady sector, CGE model
aggregated consumer (households),
economic agent 4, 219, 236–238
banking sector, economic agent 6, 240
economic agents, 215–217
endogenous variables, 221–225
exogenous parameters
Sector 1, 218
Sector 2, 218–219
Sector 3, 219
finding, optimal values, 244–247
general part, 220
government, 220
integral indexes, 240–241
market sector, economic agent 2, 232–235
model constants, 221
model markets, 225–228
outside world, economic agent 7, 240
parametric identification, 241–244
state, economic agent 5, 238–240
state sector, economic agent 1, 228–232
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
public expense and interest rate of
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
continuous-time dynamical system, 8–9
continuous-time system, 6–8
influence, uncontrolled parametric
disturbances, 14–16
Richardson mathematical model, 33
Solow mathematical model, 28
Index 265