Macroeconomic Analysis and Parametric Control of a National Economy

294 Pages·2013·3.3 MB·English

by  Abdykappar A. Ashimov, Bahyt T. Sultanov, Zheksenbek M. Adilov, Yuriy V. Borovskiy, Dmitriy A. Novikov, Rakhman A. Alshanov, Askar A. Ashimov (auth.)

Macroeconomic Analysis and Parametric Control
of a National Economy
Abdykappar A. Ashimov l Bahyt T. Sultanov
Zheksenbek M. Adilov l Yuriy V. Borovskiy
Dmitriy A. Novikov l Rakhman A. Alshanov
Askar A. Ashimov
Macroeconomic Analysis
and Parametric Control
of a National Economy
Abdykappar A. Ashimov Bahyt T. Sultanov
Kazakh National Technical University Kazakh National Technical University
National Academy of Sciences of the State Scientific and Technical Program
Republic of Kazakhstan Almaty City, Kazakhstan
Almaty City, Kazakhstan
Yuriy V. Borovskiy
Zheksenbek M. Adilov Kazakh National Technical University
Kazakh National Technical University State Scientific and Technical Program
Almaty City, Kazakhstan Almaty City, Kazakhstan
Dmitriy A. Novikov Rakhman A. Alshanov
Institute of Control Sciences RAS Kazakh National Technical University
Moscow, Russia Almaty City, Kazakhstan
Askar A. Ashimov
Kazakh National Technical University
State Scientific and Technical Program
Almaty City, Kazakhstan
ISBN 978-1-4614-4459-6 ISBN 978-1-4614-4460-2 (eBook)
DOI 10.1007/978-1-4614-4460-2
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2012948194
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Contents
1 Elements of Parametric Control Theory of Market
Economic Development. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Components of Parametric Control Theory
of Market Economic Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Methods of Analysis of the Stability and Structural
Stability of Mathematical Models of National
Economic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Development of the Methods for Evaluating
Stability Indicators of Mathematical Models . . . . . . . . . . . . . . . . . . 3
1.2.2 Development of Methods for Evaluating Weak
Structural Stability of a Discrete-Time Dynamical
System (Semi-cascade) Based on the
Robinson Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Approach to Synthesis and Choice (in the Environment
of a Given Finite Set of Algorithms) of Optimal Laws
of Parametric Control of a National Economic System’s
Development. Existence Conditions for a Solution
to Respective Variational Calculus Problems. Conditions
of Influence of Uncontrolled Parameters to These Problems. . . . . . . . . . 9
1.3.1 Analysis of the Existence Conditions for a Solution
of the Variational Calculus Problem of Synthesis
and Choice (in the Environment of a Given Finite
Set of Algorithms) of Optimal Laws of Parametric
Control of a Continuous-Time Deterministic
Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 Analysis of the Existence Conditions for a Solution
of the Variational Calculus Problem of Synthesis
and Choice (in the Environment of a Given Finite
Set of Algorithms) of Optimal Laws of Parametric
Control of a Discrete-Time Deterministic
Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
v
vi Contents
1.3.3 Analysis of the Existence Conditions for a Solution
of the Variational Calculus Problem of Synthesis
and Choice (in the Environment of a Given Finite
Set of Algorithms) of Optimal Laws of Parametric
Control of a Discrete-Time Stochastic
Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.4 Analysis of the Influence of Uncontrolled
Parameter Variations (Parametric Disturbances)
on the Solution of the Variational Calculus Problem
of Synthesis and Choice of Optimal Parametric
Control Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Algorithm of Application of Parametric Control Theory
and Rules of Interaction Between Persons Making
Decisions on Elaboration and Realization of the Effective
State Economic Policy on the Basis of an Information
System for Decision-Making Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Algorithm of the Application of Parametric
Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 Rules of Interaction for Decision Makers on the
Formulation and Implementation of an Effective
Public Economic Policy Based on the Information
Decision Support System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5 Examples for Application of Parametric Control Theory . . . . . . . . . . . . 36
1.5.1 Mathematical Model of the Neoclassical Theory
of Optimal Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.5.2 One-Sector Solow Model of Economic Growth. . . . . . . . . . . . . . 42
1.5.3 Richardson Model for the Estimation of Defense Costs . . . . . 46
1.5.4 Mathematical Model of a National Economic
System Subject to the Influence of the Share
of Public Expense and the Interest Rate
of Government Loans on Economic Growth . . . . . . . . . . . . . . . . . 50
1.5.5 Mathematical Model of the National Economic
System Subject to the Influence of International
Trade and Currency Exchange on Economic Growth. . . . . . . . 70
1.5.6 Forrester’s Mathematical Model of Global Economy. . . . . . . . 83
1.5.7 Turnovsky’s Monetary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
1.5.8 Endogenous Jones’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Contents vii
2 Methods of Macroeconomic Analysis and Parametric Control
of Equilibrium States in a National Economy . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.1 Macroeconomic Analysis of a National Economic State
Based on IS, LM, and IS-LM Models, Keynesian
All-Economy Equilibrium. Analysis of the Influence
of Instruments on Equilibrium Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.1.1 Construction of the IS Model and Analysis
of the Influence of Economic Instruments. . . . . . . . . . . . . . . . . . 118
2.1.2 Macroeconomics of Equilibrium Conditions
in the Money Market. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.1.3 Macroestimation of the Mutual Equilibrium State
in Wealth and Money Markets. Analysis of the
Influence of Economic Instruments. . . . . . . . . . . . . . . . . . . . . . . . . 124
2.1.4 Macroestimation of the Equilibrium State
on the Basis of the Keynesian Model
of Common Economic Equilibrium. Analysis
of the Influence of Economic Instruments. . . . . . . . . . . . . . . . . . 125
2.1.5 Parametric Control of an Open Economic
State Based on the Keynesian Model. . . . . . . . . . . . . . . . . . . . . . . 127
2.2 Macroeconomic Analysis and Parametric Control
of the National Economic State Based on the Model
of a Small Open Country. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.2.1 Construction of the Model of an Open Economy
of a Small Country and Estimation of Its Equilibrium
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.2.2 Influence of Economic Instruments on Equilibrium
Solutions and Payment Balance States . . . . . . . . . . . . . . . . . . . . . 136
2.2.3 Parametric Control of an Open Economy State
Based on a Small Country Model . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3 Parametric Control of Cyclic Dynamics of Economic Systems. . . . . . . 141
3.1 Mathematical Model of the Kondratiev Cycle. . . . . . . . . . . . . . . . . . . . . . 141
3.1.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.1.2 Estimating the Robustness of the Kondratiev
Cycle Model Without Parametric Control . . . . . . . . . . . . . . . . . . 143
3.1.3 Parametric Control of the Evolution of Economic
Systems Based on the Kondratiev Cycle Model . . . . . . . . . . . 144
3.1.4 Estimating the Structural Stability of the Kondratiev
Cycle Mathematical Model with Parametric Control . . . . . . 147
3.1.5 Analysis of the Dependence of the Optimal Value
of Criterion K on the Parameter for the Variational
Calculus Problem Based on the Kondratiev
Cycle Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
viii Contents
3.2 Goodwin Mathematical Model of Market Fluctuations
of a Growing Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.2.2 Analysis of the Structural Stability of the Goodwin
Mathematical Model Without Parametric Control . . . . . . . . . 149
3.2.3 Problem of Choosing Optimal Parametric Control
Laws on the Basis of the Goodwin
Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.2.4 Analysis of the Structural Stability of the Goodwin
Mathematical Model with Parametric Control . . . . . . . . . . . . . 153
3.2.5 Analysis of the Dependence of the Optimal
Parametric Control Law on Values of the
Uncontrolled Parameter of the Goodwin
Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
4 Macroeconomic Analysis and Parametric Control
of Economic Growth of a National Economy Based
on Computable Models of General Equilibrium. . . . . . . . . . . . . . . . . . . . . . . 157
4.1 National Economic Evolution Control Based on a
Computable Model of General Equilibrium of
Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.1.1 Model Description, Parametric Identification,
and Retrospective Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.1.2 Macroeconomic Analysis on the Basis of the
Computable Model of General Equilibrium
of Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.1.3 Finding Optimal Parametric Control Laws
on the Basis of the CGE Model
of Economic Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4.2 National Economic Evolution Control Based on the
Computable Model of General Equilibrium with the
Knowledge Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.2.1 Model Description, Parametric Identification,
and Retrospective Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.2.2 Estimation of the Macroeconomic Theory Provisions
on the Basis of the Computable Model of
General Equilibrium with the Knowledge Sector . . . . . . . . . . 236
4.2.3 Finding Optimal Parametric Control Laws Based
on the CGE Model with the Knowledge Sector . . . . . . . . . . . . 238
Contents ix
4.3 National Economic Evolution Control Based
on the Computable Model of General Equilibrium
with the Shady Sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
4.3.1 Model Description, Parametric Identification,
and Retrospective Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
4.3.2 Finding the Optimal Values of the Adjusted
Parameters on the Basis of the CGE Model
in the Shady Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
5 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
About The Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Chapter 1
Elements of Parametric Control Theory
of Market Economic Development
1.1 Components of Parametric Control Theory
of Market Economic Development
The application of mathematical models of a national economy is an important
subject area for the analysis of an effective public policy in the area of the economic
growth [73].
Many dynamical systems, including the national economic system [33, 30], after
some transformations, can be described by the following systems of nonlinear
ordinary differential equations:

x ðtÞ ¼ f ðxðtÞ; uðtÞ; aÞ; (1.1)
with the initial condition
xðt0Þ ¼ x0: (1.2)
Here t is the time, t 2 ½t0; t0 þ TŠ; T>0, is a fixed number;
m
x ¼ xðtÞ 2 R is the state of system (1.1), (1.2);
m
x0 2 R is the initial state of the system (deterministic vector);
q
u ¼ uðtÞ 2 R is the vector of controlled (regulated) parameters; the functions uðtÞ
and their derivatives are to be uniformly bounded;
s
a 2 A R is the vector of uncontrolled parameters; and A is an open connected set.
For a solution to system (1.1), (1.2) to exist, let’s assume that the vector function
f satisfies the Lipschitz condition and the following linear constraints on its growth
rate:
jf ðx; u; aÞj cð1 þ jxjÞ;
A.A. Ashimov et al., Macroeconomic Analysis and Parametric Control 1
of a National Economy, DOI 10.1007/978-1-4614-4460-2_1,
# Springer Science+Business Media New York 2013