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 # Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) 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