Macroeconomic Analysis and Parametric Control of a Regional Economic Union 3030322041, 9783030322045

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Abdykappar A. Ashimov  Yuriy V. Borovskiy · Dmitry A. Novikov  Bakyt T. Sultanov · Mukhit A. Onalbekov

Macroeconomic Analysis and Parametric Control of a Regional Economic Union

Macroeconomic Analysis and Parametric Control of a Regional Economic Union

Abdykappar A. Ashimov • Yuriy V. Borovskiy Dmitry A. Novikov • Bakyt T. Sultanov Mukhit A. Onalbekov

Macroeconomic Analysis and Parametric Control of a Regional Economic Union

Abdykappar A. Ashimov Kazakh National Research Technical University after K.I. Satpayev Almaty City, Kazakhstan

Yuriy V. Borovskiy Kazakh National Research Technical University after K.I. Satpayev Almaty City, Kazakhstan

Dmitry A. Novikov V.A.Trapeznikov Institute of Control Sciences (Russian Academy of Sciences) Moscow, Russia

Bakyt T. Sultanov Kazakh National Research Technical University after K.I. Satpayev Almaty City, Kazakhstan

Mukhit A. Onalbekov Kazakh National Research Technical University after K.I. Satpayev Almaty City, Kazakhstan

ISBN 978-3-030-32204-5 ISBN 978-3-030-32205-2 https://doi.org/10.1007/978-3-030-32205-2

(eBook)

© Springer Nature Switzerland AG 2020 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. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

As indicated by recurrent economic crises, there is a vital need for macroeconomic analysis and coordinated optimal economic policy at different levels, from separate countries to the global economy. In fact, macroeconomic mathematical models are an important tool for the macroeconomic analysis and assessment of an efficient policy to control the evolution of national economies. To this effect, it is crucial (a) to establish conditions under which computer simulations with mathematical models are applicable to real macroeconomic systems and (b) to study the properties of dynamic optimization models based on corresponding mathematical models of real macroeconomic systems. This book is dedicated to solving the listed problems within the theory of parametric control, further developing the original approach pioneered in the monographs [3, 24]. It presents a refined framework of parametric control that includes new numerical estimation methods for smooth mappings as well as new sufficient conditions for a continuous dependence of the optimality criteria values on uncontrolled functions in optimal parametric control design and choice problems. In addition, the book contains new applications of parametric control based on three global multiple-country dynamic macroeconomic models of the following classes: the computable general equilibrium (CGE) model; the dynamic stochastic general equilibrium (DSGE) model; and finally, the hybrid econometric model. The material of this book can be used for assessing model-based recommendations on different types of optimal macroeconomic policy at the level of separate countries, economic unions, and the global economy. Chapter 1 considers the refined framework of parametric control, in particular, • The structure and basic components of this theory; • A parameter identification algorithm for large-scale macroeconomic models; • Numerical estimation methods for the weak structural stability of dynamic models; • Numerical estimation methods for the stability of differentiable mappings and their seeds; v

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Introduction

• The statements of dynamic optimization problems on the design and choice of optimal parametric control laws from a given finite collection of algorithms for continuous- or discrete-time dynamic systems; • Sufficient conditions (theorems) for the existence of solutions of the above dynamic optimization problems; • Sufficient conditions (theorems) for a continuous dependence of optimality criteria values on uncontrolled functions or parameters in the above dynamic optimization problems; • Bifurcation points for extremals of the dynamic optimization problem on the choice of optimal parametric control laws from a given finite collection of algorithms; • Sufficient conditions (theorem) for the existence of all bifurcation points for the extremals of the above dynamic optimization problem; • An example of parametric control of cyclic dynamics based on Kondratiev’s cycle model, including an estimation of all bifurcation points in this parametric control problem; • An example of parametric control for attractor suppression processes based on the Lorenz model. The results presented below differ from those of the well-known books on variational calculus problems: book [9], in which parametric disturbances were adopted for obtaining sufficient optimality conditions with construction of S-functions and elimination of associated constraints; book [17], in which the stability of solutions of variational calculus problems (Ulam’s problem) was studied by obtaining regularity conditions under which the objective functional of the disturbed problem was the minimum point close to that of the objective functional of the original problem; and book [4], in which the existence conditions of a bifurcation point were established in a variational calculus problem with an objective functional in the Sobolev space that depended on a scalar parameter. Next, Chapter 2 builds the global multiple-country dynamic computable general equilibrium model based on [43] as well as performs macroeconomic analysis and solves a series of parametric control problems within this model. The new model relies on a conceptual description of the global economy. The sets of social accounting matrices (SAMs) for 9 global regions on the horizon 2004–2022 are adopted for the data binding of this model. The SAM sets were obtained using the developed algorithms that involved the GTAP Data Base, the national input-output tables, as well as the retrospective [16] and forecasting [80] data on basic macroeconomic indexes and international trade. The applicability conditions of computer simulations with the mathematical model to the real macroeconomic systems are tested with the methods described in Chapter 1. Chapter 2 also gives some examples of comparative and scenario macroeconomic analysis on retrospective and forecasting periods within this model. In particular, the following scenarios are studied: increase of effective rates for some taxes and duties; tightening of international economic sanctions; establishment of a new monetary union; collapse of the Eurasian Economic Union (EAEU) or the Commonwealth of Independent States (CIS); and some

Introduction

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others. Finally, Chapter 2 considers a series of parametric control problems focusing on economic growth, food security, reduction of trade gap and regional development disproportions as well as on the economic structural adjustment of the Republic of Kazakhstan. Chapter 3 introduces the developed global multiple-country dynamic stochastic general equilibrium (DSGE) model as well as computer simulations performed with it. A conceptual description of the global economy within this model is given, which includes two types of households, several types of firms (producing sectors), secondlevel banks and state consisting of central bank and government. The existence of solutions of the agents’ optimization problems is proved using the finite and infinitedimensional Kuhn–Tucker theorems [74]. The model is adapted to the goals of research—the observed values of its endogenous parameters. More specifically, some parameters are calibrated while the other estimated using a two-stage Bayesian algorithm. In addition, the applicability conditions of computer simulations with the model to real macroeconomic systems are tested using six methods, including parametric control methods. The tested model is employed for mid-term forecasting and macroeconomic analysis. Chapter 3 also presents problem statements and solutions for three parametric control problems focusing on economic growth and volatility suppression for different macroeconomic indexes at the levels of the Republic of Kazakhstan, the EAEU and the global economy, respectively. Finally, the applicability of these solutions is tested. Chapter 4 presents the following results: the global multi-country hybrid econometric model built on the basis of the suggested algorithm; testing of the applicability conditions of computer simulations to real macroeconomic systems with econometric modeling and parametric control methods; problem statements and solutions for mid-term forecasting and macroeconomic analysis; statements and solutions for a series of parametric control problems; testing of the implementability of optimal solutions (parametric control laws) as well as a study of relationships between optimal criteria values and uncontrolled factors in parametric control problems. The authors are grateful to D.T. Aidarkhanov for his help with computer simulations.

Contents

1

Parametric Control of Macroeconomic Systems: Basic Components of Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Structure of Theory of Parametric Control . . . . . . . . . . . . . . . . . 1.2 Parameter Identification Algorithm for Large-Scale Macroeconomic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Numerical Estimation Methods for some Types of Stability of Mathematical Macroeconomic Models . . . . . . . . . 1.3.1 Numerical Estimation Methods for Weak Structural Stability of Dynamic Models . . . . . . . . . . . . . . 1.3.2 Numerical Estimation Methods for Stability Indexes of Model Mappings . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Numerical Estimation Methods for Stability of Differentiable Model Mappings . . . . . . . . . . . . . . . . . 1.3.4 Numerical Estimation Methods for Stability of Germs of Differential Model Mappings . . . . . . . . . . . . 1.4 Some Properties of Solutions of Parametric Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Sufficient Conditions for the Existence of Solutions of Optimal Parametric Control Design and Choice Problems . . . . . . . . . . . . . . . . . . . . . 1.4.2 Sufficient Conditions for a Continuous Dependence of Optimal Criteria Values on Uncontrolled Functions in Optimal Parametric Control Design and Choice Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Sufficient Conditions for the Existence of Bifurcation Points for Extremals of Optimal Parametric Control Choice Problems . . . . . . . . . . . . . . . . 1.5 Example: Parametric Control of Cyclic Dynamics Based on Kondratiev’s Cycle Model . . . . . . . . . . . . . . . . . . . . .

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Example: Parametric Control of Nonlinear Dynamic System Based on Lorenz Model and Estimation of Bifurcation Points for Extremals . . . . . . . . . . . . . . . . . . . . . . .

Macroeconomic Analysis and Parametric Control Based on Global Multi-country Dynamic Computable General Equilibrium Model (Model 1) . . . . . . . . . . . . . . . . . . . . . . 2.1 Modeling Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Conceptual Description of the Global Economy . . . . . . . . 2.1.2 Building of Model 1 and Solution Algorithm . . . . . . . . . . 2.2 Adaptation of Model 1 to the Goals of Research . . . . . . . . . . . . . 2.2.1 Choice of Economic Regions and Sectors, Retrospective and Forecasting Periods for Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Initial Database and Calibration of Model 1 . . . . . . . . . . . 2.3 Applicability Testing of Model 1 . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Macroeconomic Analysis Based on Model 1 . . . . . . . . . . . . . . . 2.4.1 Comparative Snapshot Analysis Based on Model 1 . . . . . 2.4.2 Scenario Analysis Based on Model 1 . . . . . . . . . . . . . . . 2.5 Parametric Control Based on Model 1: A Series of Problem Statements and Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Problems of Economic Growth, Food Security, Reduction of Trade Gap, and Regional Development Disproportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Structural Adjustment Problem at a Sectoral Level . . . . . . 2.5.3 Testing of Solutions Applicability . . . . . . . . . . . . . . . . . . 2.5.4 Study of Solutions Dependence on Uncontrolled Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Macroeconomic Analysis and Parametric Control Based on Global Dynamic Stochastic General Equilibrium Model (Model 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conceptual Description of the Global Economy Within Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Prerequisites for Global Economy Description . . . . . . . . . 3.1.2 Conceptual Description of Household’s Behavior . . . . . . 3.1.3 Conceptual Description of Firm’s Behavior . . . . . . . . . . . 3.1.4 Conceptual Description of Second-Level Bank’s Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Conceptual Description of State’s Behavior . . . . . . . . . . . 3.1.6 Balance and Auxiliary Equations . . . . . . . . . . . . . . . . . . 3.2 Nonlinear Modeling and Parameter Estimation . . . . . . . . . . . . . . 3.2.1 Nonlinear Model 2 and Estimation of Its Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Building of Linear Model 2 . . . . . . . . . . . . . . . . . . . . . .

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3.2.3

3.3

3.4

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Adaptation to the Goals of Research, Parameter Identification and Calibration of Linear Model 2 . . . . . . . 3.2.4 Adjustment of Parameter Values for Nonlinear Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applicability Testing of Model 2 . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Testing with Stability Indexes . . . . . . . . . . . . . . . . . . . . . 3.3.2 Testing with Retrospective Forecasting . . . . . . . . . . . . . . 3.3.3 Testing with Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Testing with Impulse Responses . . . . . . . . . . . . . . . . . . . 3.3.5 Testing with Local Sensitivity Analysis . . . . . . . . . . . . . . 3.3.6 Testing with Marginal Likelihood Comparison . . . . . . . . Mid-term Forecasting and Macroeconomic Analysis Based on Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Knoware and Software for Forecasting and Macroeconomic Analysis . . . . . . . . . . . . . . . . . . . . . 3.4.2 Solution of Basic Mid-term Forecasting Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Solution of Snapshot Macroeconomic Analysis Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Solution of Scenario Macroeconomic Analysis Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parametric Control Based on Model 2: A Series of Problem Statements and Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Problem Statements for Parametric Control . . . . . . . . . . . 3.5.2 Solutions of Parametric Control Problems . . . . . . . . . . . . 3.5.3 Applicability Testing of Solutions . . . . . . . . . . . . . . . . . . 3.5.4 Study of Solutions Dependence on Uncontrolled Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Macroeconomic Analysis and Parametric Control Based on Global Multi-country Hybrid Econometric Model (Model 3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Building of Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Applicability Testing of Model 3 . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Knowhow and Software for Macroeconomic Analysis and Mid-term Forecasting . . . . . . . . . . . . . . . . . 4.3.2 Examples of Mid-term Forecasts Based on Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Examples of Comparative Macroeconomic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Examples of Scenario Macroeconomic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Parametric Control Based on Model 3: A Series of Problem Statements and Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Problem Statements for Parametric Control . . . . . . . . . . . 4.4.2 Solutions of Parametric Control Problems . . . . . . . . . . . . 4.4.3 Applicability Testing of Solutions . . . . . . . . . . . . . . . . . . 4.4.4 Study of Solutions Dependence on Uncontrolled Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

About the Authors

Abdykappar A. Ashimov is one of the leading scientists in the Republic of Kazakhstan. Over the years, he held positions of the Professor and Department Head, and later President of the Kazakh National Polytechnic Institute, and Principal Investigator of numerous research projects. He should be credited for providing guidance to the entire generation of Kazakh engineers and scientists in the areas of automatic control and cybernetics. As the Chairman of the State Attestation Commission, he established key regulations in the areas of science and education in the Republic of Kazakhstan. Currently, he is the Director of the Institute of Information and Computational Technologies and a full member of the National Academy of Sciences of the Republic of Kazakhstan. His current research includes the theory of parametric control, theory of automatic control of systems with variable configuration, statistical theory of automatic systems with dynamic pulse frequency modulation, and theory of database engineering. He is an Author of approximately 500 publications, monographs, and patents. His contributions are recognized by numerous awards and medals of the Republic of Kazakhstan. Yuriy V. Borovskiy is Associate Professor at the Kazakh National Technical University and the Chief Scientist and Principal Investigator of various projects funded by the Ministry of Education and Science of the Republic of Kazakhstan. Earlier, he was employed as an Associate Professor at the Kazakh-British Technical University. He is an Author of 220 publications, including monographs and textbooks, a member of the International Informatization Academy (Kazakhstan), and a recipient of several Best Conference Paper Awards. Dmitry A. Novikov is the Director of the Institute of Control Sciences and a Professor and Department Head at the Institute of Physics and Technology, Moscow, Russia. He is a Principal Investigator of a number of research projects in the area of organizational control. He has authored 28 books and over 200 journal and conference papers. He is an Organizer/Cochairman/Committee Cochairman of an ongoing Control Conference in Russian Federation. xiii

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About the Authors

Bakyt T. Sultanov is employed at the Kazakh National Technical University in Almaty, Kazakhstan, and serves as a consultant on a number of state scientifictechnical projects. In addition, he is the Mayor of the City of Nur-Sultan, Kazakhstan. His earlier positions included Minister of Finance and Deputy Prime Minister of the Republic of Kazakhstan. He is an Author of 5 books, 21 book chapters, and 40 conference papers. Mukhit A. Onalbekov is a Senior Researcher at the Kazakh National Technical University. He has experience in software development, stock trading, and asset management. He has published 6 journal and 12 conference papers.

Chapter 1

Parametric Control of Macroeconomic Systems: Basic Components of Theory

As is well-known [1, 6, 12], the applicability of computer simulations with mathematical models to real systems is connected with the sensitivity of simulation results based on such a model to its small changes (in some sense) or to small changes of its exogenous variables. For example, a small change of an autonomous dynamic model can be represented as a small variation of a corresponding vector field (continuoustime systems) or of an invertible mapping (discrete-time systems); a small change of an arbitrary mathematical model can be represented as a small variation of the mappings defined by it. The low sensitivity of an autonomous dynamic model to its small changes can be characterized using the concept of a (weak) structural stability of such a model. The low sensitivity of all maps defined by a mathematical model to its small changes can be characterized by the concept of stability of such mappings. The fundamental theoretical results in this field—sufficient conditions for the structural (and weak structural) stability of autonomous dynamic systems as well as sufficient conditions for the stability of smooth mappings—were established in [6, 71]. However, until now these results have not been applied in mathematical modeling due to the absence of appropriate numerical estimation algorithms. For estimating the optimal values of macroeconomic tools with a dynamic mathematical model, first it is necessary to verify the existence of a solution of a corresponding dynamic optimization problem and also a continuous dependence of optimal criterion values on uncontrolled exogenous variables of the model. Chapter 1 describes the following original results of the authors: • Numerical parameter identification methods for large-scale macroeconomic models. • Numerical estimation methods for (a) the weak structural stability of dynamic mathematical models, (b) the stability indexes of model mappings, and (c) the weak structural stability of model mappings and their germs. • Theorems (sufficient conditions) for (a) the existence of solutions of the optimal parametric control design and choice problems, (b) a continuous dependence of © Springer Nature Switzerland AG 2020 A. A. Ashimov et al., Macroeconomic Analysis and Parametric Control of a Regional Economic Union, https://doi.org/10.1007/978-3-030-32205-2_1

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

optimal criterion values on uncontrolled functions in the optimal parametric control design and choice problems, and (c) the existence of all bifurcation points for the extremals of the optimal parametric control choice problems. • Examples illustrating the efficiency of parametric control for Kondratiev’s cycle model and the Lorenz model.

1.1

Structure of Theory of Parametric Control

As a matter of fact, • The solution of a continuous- or discrete-time dynamic system, which may contain the vectors of controlled parameters (economic policy tools u) and uncontrolled parameters (a), depends on the initial conditions and parameters (coefficients) of this system. • The solution of a static system depends on the parameters (coefficients) of this system. • Analysis results obtained for an autonomous dynamic system are applicable to a modeled object only if the system is structurally stable (or robust) [1]. • Analysis results obtained for a (static or dynamic) model are applicable to a modeled object only if the model mappings are stable [6]. • It is necessary to satisfy constraints on the stability indexes of mappings defined by a macroeconomic model (dynamic or static system) under small disturbances of its input parameters or initial statistical data used for parameter identification [12]. In view of all these circumstances, the authors suggested the following basic components of the theory of parametric control [3]: 1. The methods for forming the set (library) of macroeconomic mathematical models. These models are oriented toward the description of various (specific) socioeconomic situations. 2. The methods for estimating the robustness (structural stability) of dynamic mathematical models (including their stability indexes) and also of the mappings defined by the mathematical models of a national economic system from the library (without parametric control). If the estimation procedure determines that a model under study is structurally instable or the model mappings are instable or the stability indexes of these mappings exceed given thresholds, then the model is moved to the reserved list of the library. 3. The methods for choosing and designing parametric control laws for a macroeconomic system based on its dynamic mathematical models. The methods for formulating and solving parametric control problems in form of corresponding mathematical programming problems based on static mathematical models of a macroeconomic system.

1.2 Parameter Identification Algorithm for Large-Scale Macroeconomic Models

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4. The methods for estimating the robustness (structural stability) of dynamic mathematical models, including stability indexes. The methods for estimating the structural stability of model mappings of a macroeconomic system (with parametric control). 5. The approach to adjusting the parametric control constraints of a macroeconomic system if the estimation procedure determines that a model under study is structurally instable or the model mappings are instable or the stability indexes of these mappings exceed given thresholds. 6. The methods for studying the influence of uncontrolled parameters and functions (uncontrolled factors) on the solutions of the variational calculus problems on the design and choice of optimal parametric control laws from a given finite collection of algorithms. The methods for studying all bifurcation points for the extremals of the variational calculus problems on the choice of optimal parametric control laws. The methods for studying the influence of changes of uncontrolled factors on the solutions of the mathematical programming problems based on static mathematical models. 7. The approach to recommending economic policy rules with parametric control laws of a macroeconomic system based on the dependence of optimal criterion values on uncontrolled factors in corresponding parametric control problems.

1.2

Parameter Identification Algorithm for Large-Scale Macroeconomic Models

While developing the first component of the theory of parametric control, the authors suggested the following parameter identification algorithm for large-scale macroeconomic models. For a discrete dynamic macroeconomic model, the parameter identification problem is to estimate its unknown parameters (the unknown values of its exogenous functions and also the unknown initial values of its dynamic equations) by minimizing a certain objective function. The latter characterizes the deviations of the model’s output variables from their observed counterparts (i.e., available statistical data for a past period t ¼ t1, t1 + 1, . . ., t2). Actually, this problem is reduced to minimization of a function of several variables (parameters) in some closed domain D of the Euclidean space subject to constraints imposed on the values of endogenous model variables (constraints E) and also on the desired values of the parameters (constraints F). If this domain has a high dimension N, standard optimization methods may be inefficient due to the presence of several local minima of the objective function. The algorithm below takes into account all these parameter identification features of macroeconomic models, thereby avoiding the abovementioned problem of “local minimums.” The constraints E were determined by the economic sense of endogenous model N  Y    variables (e.g., nonnegativity). The domain D ¼ ai , bi , where ai , bi denoted i¼1

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

the range of the parameter pi (i ¼ 1, . . ., N ), was considered as the domain associated with the constraints F for estimating the admissible values of the exogenous parameters. All parameters   with available observations were estimated within sufficiently small ranges ai , bi centered at their observed values (for a single observation) or within some ranges  covering these observed values (for multiple observations). Other ranges ai , bi for parameter estimation were selected using an indirect assessment of their admissible values based on other statistical data. The Nelder– Mead simplex direct search method [65] was employed in computer simulations to find the minimal values of a continuous multivariate function K: D ! R. Application of this algorithm for an initial point p1 2 D can be interpreted as a sequence {p1, p2, p3, . . .}, where K( pj+1)  K( pj), pj 2 D, j ¼ 1, 2,. . ., that converges to the local minimum p0 ¼ arg min K of the criterion K . In the description of the algorithm D

below, the point p0 is supposed to be found with sufficient accuracy. In view of a general assumption that the minimum points of two different functions did not coincide, two criteria were introduced for solving the parameter identification problem in the model under study: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  i 2 t2 X nA X u y ðt Þ  yi ðt Þ 1 K A ð pÞ ¼ t αi nα ðt 2  t 1 þ 1Þ t¼t i¼1 yi ðt Þ 1

and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  i 2 t2 X nB X u y ðt Þ  yi ðt Þ 1 t : β K B ð pÞ ¼ nβ ðt 2  t 1 þ 1Þ t¼t i¼1 i yi ðt Þ 1

Here the notations are the following: ft 1 , . . . , t 2 g as an identification interval; yi ðt Þ and yi ðt Þ as the calculated and observed values of the output model variables; K A ðpÞ as an auxiliary criterion; K B ðpÞ as a primary criterion; nB > nA , αi > 0, and βi > 0 as some weight coefficients evaluated PA P Bwhile solving the parameter identification problem; finally, ni¼1 αi ¼ nα and ni¼1 βi ¼ n β . The minimization problems of the criteria K A and K B will be called Problems A and B, respectively. The algorithm for solving the parameter identification problem includes several steps as follows. Step 1. For some vector of initial parameter values p1 2 D, solve Problems A and B in parallel to obtain the minimum points pA0 and pB0 of the criteria K A and K B , respectively. Step 2. If the inequality K B ðpB0 Þ < ε holds for a sufficiently small value ε, then the parameter identification problem of this model is successfully solved. Step 3. Otherwise, solve Problem A (using the point pB0 as the initial point p1 ) and Problem B (using the point pA0 as the initial point p1 ). Step 4. Get back to Step 2.

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

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Steps 1–3 with a sufficiently large number of repetitions (iterations) guarantee that the desired parameters will leave the neighborhoods of non-global minimum points of one criterion owing to the other criterion. Thus the parameter identification problem will be solved. While developing the second component of the theory of parametric control, the authors suggested the following estimation methods for the structural stability of a mathematical model and also for the stability of model mappings, including their stability indexes.

1.3 1.3.1

Numerical Estimation Methods for some Types of Stability of Mathematical Macroeconomic Models Numerical Estimation Methods for Weak Structural Stability of Dynamic Models

The methods for studying the robustness (structural stability) of mathematical models of national economic systems are based on: • Fundamental theoretical results on dynamic systems in the plane. • Methods for verifying the belonging of mathematical models to certain classes of structurally stable systems (Morse–Smale systems, Ω-robust systems, У-systems, systems with weak structural stability) At present, the theory of parametric control of market economic development disposes of several theorems on the structural stability of specific mathematical models (in particular, the neoclassical optimal growth model; the models of national economic systems considering the influence of the share of public expenses and of the interest rate of public loans on economic growth; the models of national economic systems considering the influence of international trade and exchange rates on economic growth) that have been formulated and proved using the abovementioned fundamental results. In addition to an analytical treatment of the structural stability of specific mathematical models (with or without parametric control), it is possible to develop computer simulation approaches to the structural stability of mathematical models of national economic systems based on the results of dynamic systems theory. In what follows, a computational algorithm for estimating the structural stability of mathematical models of national economic systems will be constructed that relies on Robinson’s theorem on weak structural stability (Theorem А) [71]. Theorem Let N 0 be some manifold and N be a compact subset in N 0 such that the closure of the interior of N coincides with N. Consider a given vector field in a neighborhood of the set N in N 0; this field defines the C 1-flow f in this neighborhood. Denote by Rð f , N Þ the chain-recurrent set of the flow f on N.

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Let Rð f , N Þ be contained in N. Assume it has the hyperbolic structure and, moreover, the flow f on Rð f , N Þ satisfies the transversality conditions of stable and instable manifolds. Then the flow f on N is weakly structurally stable. Particularly, if Rð f , N Þ is an empty set, then the flow f is weakly structurally stable in N. A similar result also holds for a discrete-time dynamic system (cascade) defined by the homeomorphism with image f : N ! N 0 . Therefore, the weak structural stability of the flow (or cascade) f can be estimated with numerical algorithms based on Theorem A: to this end, just perform a numerical estimation of the chain-recurrent set Rð f , N Þ for some compact domain N in the state space of the dynamic system under consideration. Now, introduce a localization algorithm of the chain-recurrent set for a compact subset in the state space of a dynamic system described by a system of ordinary differential (or difference) equations and algebraic equations. This algorithm is based on the symbolic image algorithm [14]. Note that the chain-recurrent subset is simulated using a directed graph (symbolic image) that represents a discretization of the shift mapping along the trajectories defined by this dynamic system. Suppose it is required to find an estimate of the chain-recurrent set Rð f , N Þ of some dynamic system in a compact set N of its state space. For a specific mathematical model of an economic system, as the compact set N consider, e.g., some parallelepiped of its state space that includes all possible system evolution trajectories on a given time interval. The localization algorithm for the chain-recurrent set consists of several steps as follows. Step 1. Define the mapping f defined on N and given by the shift along the trajectories of the dynamic system for a fixed time interval. Step 2. Construct a partition C of the compact set N into cells Ni. Assign a directed graph G in which vertexes correspond to the cells while edges between the cells Ni and Nj to the intersection of the image of one cell f(Ni) with another cell Nj. Step 3. Find all recurrent vertexes of the graph G (a recurrent vertex is a vertex that belongs to cycles). If the set of such vertexes is empty, then Rð f , N Þ is empty; in this case, terminate the localization process with the conclusion that the dynamic system is weakly structurally stable. Step 4. Partition the cells corresponding to the recurrent vertexes of the graph G into cells of lower dimension, and then construct a new directed graph G from them (see Step 2 of the algorithm). Step 5. Get back to Step 3. Note that Steps 3–5 are repeated until the diameters of the partition cells become less than a given threshold ε. The last collection of cells gives the desired estimate of the chain-recurrent set Rð f , N Þ. This estimation method allows to establish the weak structural stability of a dynamic system if the resulting chain-recurrent set Rð f , N Þ for a compact subset in its state space is empty.

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

7

However, if the discrete-time dynamic system is a priori the semi-cascade f, then verify the invertibility of the mapping f defined on N before estimating its weak structural stability using Robinson’s theorem (in this case, the semi-cascade defined by f is a cascade). Consider a numerical algorithm for estimating the invertibility of a differentiable mapping f : N ! N 0 , where N is some closed neighborhood of a discrete-time trajectory ff t ðx0 Þ, t ¼ 0, . . . , T g in the state space of a dynamic system. Suppose N contains a continuous curve L sequentially connecting the points ff t ðx0 Þ, t ¼ 0, . . . , T g . For example, as such a curve, it is possible to choose a piecewise linear curve with vertexes at the points of the above discrete-time trajectory of the semi-cascade. The invertibility of the mapping f : N ! N 0 can be verified in two stages as follows: Stage 1. Verify that the restriction of the mapping f : N ! N 0 to the curve L is invertible, i.e., f : L ! f ðLÞ. This is reduced to ascertaining that the curve f ðLÞ has no self-intersection points, i.e., ðx1 6¼ x2 Þ ) ð f ðx1 Þ 6¼ f ðx2 ÞÞ; x1 , x2 2 L. For the mapping f ðLÞ, the absence of self-intersection points can be determined, e.g., by checking the monotonic property of the limitation of the mapping f onto L along any coordinate of the state space of the semi-cascade f.   Choose a sufficiently large collection of points xi ¼ x1i , x2i , . . . , xni 2 L, yi ¼   f ðxi Þ, yi ¼ y1i , y2i , . . . , yni and consider their jth coordinate. If the inequality yji1 < yji2 (yji1 > yji2 ) holds for all values xji , i ¼ 1, . . . , n , such that xji1 < xji2 (xji1 < xji2 , respectively), then the mapping f : L ! f ðLÞ is estimated to be invertible. Stage 2. Verify that the mapping f is invertible in neighborhoods of the points of the curve L (local invertibility). In accordance with the inverse function theorem, this can be done in the following way. For a sufficiently large number of chosen points x 2 L, estimate

the Jacobians of the mapping f using difference derivatives: J ðxÞ ¼ det

∂f i ∂xj

ðxÞ , i, j ¼ 1, . . . , n: Here i and j are vector coordinates, while

n denotes the state space dimension of the dynamic system. If all the estimated Jacobians are nonzero and have the same sign, then J ðxÞ 6¼ 0 for all x 2 L, and hence the mapping f is invertible in some neighborhood of each point x 2 L. An aggregate algorithm for estimating the weak structural stability of a discretetime dynamic system (a semi-cascade defined by the mapping f ) in the state space N 0 2 Rn defined by the continuously differentiable mapping f can be written in the following form. Step 1. Find the discrete-time trajectory ff t ðx0 Þ, t ¼ 0, . . . , T g and also the curve L in a closed neighborhood N of which the weak structural stability of the dynamic system has to be estimated. Step 2. Estimate the invertibility of the mapping f in a neighborhood of the curve L using the algorithm described above.

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Step 3. Localize the chain-recurrent set Rð f , N Þ. In view of the obvious inclusion Rð f , N 1 Þ ⊆ Rð f , N 2 Þ for N 1 ⊂ N 2 ⊂ N 0 , as the compact set N choose any parallelepiped belonging to N 0 that contains L. Step 4. If Rð f , N Þ ¼ Ø, the dynamic system under study is weakly structurally stable in N. This aggregate algorithm can be also used for estimating the weak structural stability of a continuous-time dynamic system (the flow f ) with its trajectory L ¼ ff t ðx0 Þ, 0  t  T g considered as the curve L. In this case, Step 2 of the aggregate algorithm should be omitted. A mapping f t for some fixed t (t > 0) can be selected as the mapping f at Step 3.

1.3.2

Numerical Estimation Methods for Stability Indexes of Model Mappings

Following Orlov’s definition [12], a mathematical model of an economic system is a certain mapping F : D ! E that transforms values of initial (exogenous) data p 2 D into solutions (values of endogenous variables) y 2 E. Let a mathematical model of some real-life phenomena or process f be built and some actual values of the point p be determined, using available measurements or by solving the parametric identification problem. Then the natural question concerns the adequacy of this model. The condition of model stability against admissible disturbances of the initial data [12] is a prerequisite of its adequacy. If such stability is the case, small disturbances of the model’s initial data cause small changes of its solution. The monograph [12] defined a series of basic stability indexes of mathematical models (see below), yet without any algorithms for calculating them. Further exposition will describe the algorithms for estimating the stability indexes of a mathematical model that have been developed by the authors. These indexes characterize the stability of solutions of a mathematical model against initial data disturbances. Note that beforehand all model parameters and variables have to be reduced to dimensionless  form.  Designate as X ¼ X 1 , X 2 , . . . , X k the vector of some values of the model parameters for a time interval t 2 f0, . . . , T g: Denote by X 0 ¼ exogenous  X 10 , X 20 , . . . , X k0 the corresponding vector of their basis values for the same time interval. For dynamic models, the vector X consists of the values of all model parameters and also the initial values of the variables of associated differential (or difference) equations. For econometric models, the vector X consists of statistical data (measurements) used for finding equation coefficients.  the model i Designate as p ¼ p1 , p2 , . . . , pk , pi ¼ XXi , i ¼ 1, . . . , k, the vector of normalized 0

input data of the mathematical model, p0 ¼ ð1, 1, . . . , 1Þ.

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

9

Let D be the space of normalized input data vectors that includes all admissible collections of p, and also let D ⊂ Rk be a metric space with the Euclidean metric defined by the space Rk , p0 2 D.  Denote by Y ¼ Y ðpÞ ¼ Y 1 , Y 2 , . . . , Y n a chosen vector of the values of endogenous variables for a given interval (or time) calculated for the chosen values of p. For dynamic models, the vector Y consists of the values of a selected collection of the endogenous model variables for the above interval (or time). For econometric models, the vector Y consists of the coefficients of the model equations or the values of a selected collection of the endogenous model variables for the above interval (or time).   In particular, for p ¼ p notation Y 0 ¼ Y ðp0 Þ ¼ Y 10 , Y 20 , . . . , Y n0 . 0 , introduce the

Designate as y ¼ yðpÞ ¼

Y1 Y 10

,

Y2 Y 20

, ...,

Yn Y n0

the normalized vector of values of the

endogenous variables for the time T 1 ; y0 ¼ yðp0 Þ ¼ ð1, 1, . . . , 1Þ: Let E ⊂ Rn be a set that contains all possible output values y for p 2 D with the Euclidean metric of space Rn , y0 2 E . The model under consideration defines the mapping F of the set D (domain) into the set E (codomain). For a chosen point p 2 D and a value α > 0, denote by U α ðpÞ the intersection of an α-neighborhood of p with the set A: U α ðpÞ ¼ fp1 2 D : ρðp1 , pÞ  αg. Here and below, ρð∙,∙Þ indicates the Euclidean distance between two points of the Euclidean space. Designate as dðE 1 Þ the diameter of a subset E 1 ⊂ E, i.e., d ðE 1 Þ ¼ supðρðy1 , y2 Þ: y1 , y2 2 E 1 Þ. Definition 1.1 Given α > 0, the stability index of the model at a point p 2 D is the value βðp, αÞ ¼ dðF ðU α ðpÞÞ. Algorithm 1.1 For estimating the stability index βðp, αÞ of the model using Monte Carlo simulations includes several steps as follows. Step 1. Choose collections of input (X) and output (Y ) parameters and calculate their normalized values.   Step 2. Define the vector of normalized input data p ¼ p1 , p2 , . . . , pk , a value α > 0, and the set U α ðpÞ: Step 3. Generate a sufficiently large collection of M pseudorandom points ( p1, p2, . . ., pM), with the uniform distribution in U α ðpÞ: To this effect, sequentially apply a uniform randomizer to generate the coordinates pij ði ¼ 1, . . . , k; j ¼ 1, . . . , M Þ of the point pj within the ranges

2 P ½pi  α, pi þ α covering U α ðpÞ . If the inequality ki¼1 pij  pi  α2 holds (xj 2 U α ðpÞÞ, add this point to the collection. Step 4. For each point pj of the collection, determine the point yj ¼ F( pj), j ¼ 1, . . . , M, using simulations.     Step 5. Evaluate β ¼ max ρ yi , yj : i, j ¼ 1, . . . , M . Step 6. Stop.

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

For α ¼ 0.01, the resulting value β/2 characterizes the maximal change of the output variables (in percentage points) under a 1% disturbance of the input data. Definition 1.2 The absolute stability index of the model at a point x 2 D is the value βðxÞ ¼ inf βðp, αÞ. Here α0 is the maximum admissible relative deviation of the 0 0, find the values β pj , j ¼ 1, . . . , M, using Algorithm 1.2.   Step 3. Evaluate γ ¼ max β pj : Step 4. Stop.

j¼1, ..., M

If γ turns out to be smaller than some a priori given small value ε (i.e., γ is approximately 0), then the model mapping F is considered to be continuously dependent on the input values in the domain D. In particular, these algorithms were applied for estimating the econometric model of a small open economy and the computable general equilibrium model of economic sectors.

1.3.3

Numerical Estimation Methods for Stability of Differentiable Model Mappings

This subsection describes numerical methods (algorithms) for evaluating the stability of smooth mappings F: D ! E defined by a static or discrete-time dynamic model; stability will be understood in the sense of [2]. Such stability means that the qualitative properties of model mappings are preserved under small changes of an

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

11

associated model. If real economic phenomena have an adequate description using a mathematical model, the stability (or instability) of a model mapping may indicate of the stability (or instability) of corresponding dependencies of economic indexes on exogenous (controlled or uncontrolled) factors under small variations of these dependencies. Also the instability of a model mapping may testify that the model under consideration is inadequate. The numerical estimation methods for the stability of such mappings (with or without parametric control) rely on theoretical results connected with the stability of smooth mappings [6] for the cases of immersion, submersion, and submersion with fold. Necessary theoretical background is as follows. Consider a smooth mapping F on the closure D of an open set D0 given by F : D0 ! E,

ð1:1Þ

  where D0 and E are some manifolds, dim D0 ¼ n and dimðEÞ ¼ v. In the sequel, the role of D0 and E will be played by open bounded domains in the corresponding Euclidean spaces; the mapping F is assumed to be smooth on the closure D of the domain D0 . Under these conditions, mapping (1.1) is proper and hence satisfies the hypotheses of Mather’s theorem [6] and other propositions from [6] presented below, which require the compact property of the manifold D0 (or at least the properness of mapping (1.1)). Mapping (1.1) is said to be an immersion in D if n < v and rankðF Þ ¼ n at all points of D0 . Proposition 1.3.3 [6] If mapping (1.1) is a bijective immersion (with image), then it is stable. If v  2n þ 1 , then this mappings is stable in D0 if and only if F is a bijective immersion (with image). Mapping (1.1) is said to be submersion if n  v and rankðF Þ ¼ v at all points of D0 . Proposition 1.3.4 [6] If mapping (1.1) is a submersion in D0, then it is stable in D0. Now, study the case in which for n  v the domain D0 contains all singularities of the mapping F. Denote by S1 ðF Þ a non-empty set of points of the domain D0 at 0 which S1 ðF Þ ¼ p 2 D : rankF ¼ v  1 , i.e., the Jacobian matrix of the mapping F is of the maximum rank minus 1. 1 In addition, let the transversality condition j1 F ⋈ S1 hold, where  0 j F denotes the 1 1-stream of the mapping F, S1 is a subset in the space J D , E of 1-streams consisting of the streams of corank 1, and ⋈ indicates transversality. As is wellknown, in this case the set S1 ðF Þ is a submanifold in D0 of dimension   0 v  1 ; 1 0 1 moreover, S1 is a submanifold in J D , E , and its codimension in J D , E is   n  v þ 1. The space J 1 D0 , E is a manifold of dimension n þ v þ nv , and its tangent spaces have the structure of the space Rn  Rv  HomðRn , Rv Þ, where Rn is the tangent space to D0, Rv is the tangent space to E, HomðRn , Rv Þ denotes the set of

12

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

all linear mappings from Rn into Rv , i.e., the Jacobian matrices for all smooth mappings from D0 into E at a given point. Definition Assume a mapping F : D0 ! E satisfies the condition j1 F ⋈ S1. A point p 2 S1 ðF Þ is said to be a fold point if the sum of the tangent space to S1 ðF Þ and the kernel of the tangent mapping dF at this point has dimension n, i.e., if T p S1 ðF Þ þ KerðdF Þp ¼ T p D0 :

ð1:2Þ

The sum of dimensions in the left-hand side of this relationship obviously coincides with the dimension of the right-hand side, ðv  1Þ þ ðn  v þ 1Þ ¼ n . Hence, condition (1.2) is equivalent to the fact that the terms in the left-hand side have a unique common point (the origin) and

cos ∠ T p S1 ðF Þ, KerðdF Þp 6¼ 1:

ð1:3Þ

Definition A mapping F : D0 ! E is said to be a submersion with fold if the set of its singularities is non-empty and each of them is a fold point. In this case, the submanifold S1 ðF Þ in D0 is said to be a fold. As is well-known, if F : D0 ! E is a submersion with fold, then the restriction of the mapping F to the fold S1 ðF Þ is an immersion. The next result is the case. Theorem 1.3.5 [6] Let (1.1) be a submersion with fold. Then the restriction FjS1 ðFÞ of this mapping to the fold is an immersion. The mapping F is stable if and only if the restriction FjS1 ðFÞ has normal intersection. In particular, this leads to the following. Corollary 1.3.6 If F : D0 ! E is a submersion with fold and the restriction FjS1 ðF Þ is injective, then the mapping F : D0 ! E is stable. Really, any bijective immersion (with image) is an immersion with normal intersections, see Definition 3.1 in [6]. A series of algorithms suggested below are intended for estimating the stability of model mappings for the cases of immersion, submersion, and submersion with fold. Algorithm for Estimating the Set of Singularities of a Model Mapping Hereinafter, mapping (1.1) will be considered as a smooth mapping described by some mathematical model that transforms a certain parallelepiped D of its exogenous parameters into a value set E of its endogenous variables. Denote by p ¼ ðp1 , . . . , pn Þ 2 D ⊂ Rn the vector of all arguments of mapping (1.1) and by y ¼ yðpÞ ¼ ðy1 , . . . , yv Þ 2 E ⊂ Rv the corresponding image of the point

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

13

p , i.e., the solution vector of the model. In this case, the Jacobian matrix of dimensions v  n for mapping (1.1) at the point p can be written as  J ð pÞ ¼

∂yi ð pÞ ∂pj

 ð1:4Þ i¼1,...,v; j¼1,2,...,n

An estimate of the Jacobian matrix (1.4) at some point p 2 D obtained by numerical differentiation of (1) will be also designated as J ðpÞ. Let l be the total v! number of maximum order minors in J ðpÞ. Actually, l ¼ C nv ¼ n!ðvn Þ! if n < v and v n! l ¼ C n ¼ v!ðnvÞ! if n  v . Finally, denote by jM i ðpÞj, i ¼ 1, . . . , l, an estimated determinant of such a minor of order min ðv, nÞ in the Jacobian matrix J ðpÞ, p 2 D. The following algorithm estimates the set of singularities of a mapping F and, if this set is empty, also estimates the condition rankðJ ðpÞÞ ¼ min ðv, nÞ for all p 2 D:

ð1:5Þ

Under this condition the mapping F has no singularities in the domain D. Algorithm 1.4 An aggregate algorithm for estimating the set of singularities of mapping (1.1). Step 1. Decompose the parallelepiped D into a sufficiently large number of elementary parallelepipeds Dk of the same size, and define a grid P of N points being the vertexes of these parallelepipeds: P ¼ pj : j ¼ 1, . . . , N .   Step 2. Calculate all elements of the matrices J pj for j ¼ 1, . . . , N.   Step 3. For i ¼ 1, . . . , l, j ¼ 1, . . . , N, calculate the determinants j M i pj j. Step 4. For each i ¼ 1, . . . , l , define the set D(i) as the union of all closed parallelepipeds Dk with the following property: not all values j M i pj j at the vertexes of Dk have the same sign. T e ¼ l DðiÞ. Step 5. Find the set D i¼1 e is empty, condition (1.5) is estimated as true. Stop. Step 6. If the set D e instead of Step 7. Otherwise, perform Steps 1–6 of the algorithm with the domain D e D and smaller parallelepipeds in the partition of D. Remarks e (if non-empty) consists of the elementary parallelepipeds covering all 1. The set D singularities of the mapping F; so the error of Algorithm 1 (in terms of the e of these parallelepimaximum distance between the points from the vertex set P peds and the set of all singularities of the mapping F) does not exceed the e diameter of the parallelepipeds from D.  2. If, at Step 4, for some i all values j M i pj j have the same sign (i.e., a certain set D (i) turns out to be empty), then the subsequent steps of Algorithm 1.4 can be

14

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

e is empty and condition (1.5) is estimated as true. skipped. In this case, the set D For a sufficiently large number of iterations, Steps 1–7 allow estimating condition e (1.5) or the set of all singularities of the mapping F using the set D. e then 3. If, in the case n  v, Algorithm 1.4 yields the empty estimate for the set D, the mapping F is estimated as a submersion in the domain D0 ; hence, by Proposition 1.3.4, the mapping is stable in this domain. e 4. If, in the case v  2n þ 1, Algorithm 1.4 yields a non-empty estimate for the set D, 0 then the mapping F is estimated as instable in the domain D ; see Proposition 1.3.3. Algorithm for Estimating the Nonlocal Injectivity of Model Mappings Assume n < υ, i.e., the dimension of the domain D of mapping (1.1) is smaller than that of the codomain E. This paragraph will present an algorithm for estimating the nonlocal injectivity of mapping (1.1) defined by the model in the domain D. (Nonlocal injectivity is the absence of non-close points in D that have the same images in the mapping F.) Jointly with condition (1.5), which guarantees the local injectivity in a neighborhood of each point p 2 D, this property means the existence of an inverse mapping for F defined over the set F(D).   Fix a sufficiently small value ε > 0. For each point pj 2 P, denote by d pj the set of all points pk 2 P such that j pj  pk j> ε. Here |∙| indicates vector magnitude. Algorithm 1.5 For estimating the nonlocal injectivity of mapping (1.1). Step 1. For each point pj 2 P, calculate the value

  mj ¼ min F ðpk Þ  F pj : pk 2d ðpj Þ

ð1:6Þ

Step 2. Calculate m ¼ min mj , and determine the points pj , pk 2 P such that pj 2P

 

F ðpk Þ  F pj ¼ m. Step 3. Repeat Steps 1 and 2 of this algorithm with the grid P replaced by a new grid P1 as follows: P1 consists of the vertexes of smaller parallelepipeds and contains all points of P such that the distance to one of the points pj , pk does not exceed 2ε. Algorithm 1.5 with a sufficiently large number of iterations leads to one of two possible situations as follows. (a) The resulting sequence of values m is decreasing approximately in proportion to the step of the grid P1 . In this case, the mapping F is estimated as non-injective (in other words, the set F ðDÞ as self-intersecting). (b) The inequality m > ε holds for all grids with a sufficiently small step. In this case, the mapping F is estimated as injective (in other words, the set F ðDÞ as non-self-intersecting).

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

15

e then If, in the case n < v, Algorithm 1.4 yields the empty estimate for the set D, the mapping F is estimated as an immersion. If Algorithm 1.5 estimates this immersion as an injective mapping, then by Proposition 1.3.3 the mapping F is stable in the domain D0 . If, in the case v  2n þ 1, Algorithm 1.5 estimates F ðDÞ as a self-intersecting set, then by Proposition 1.3.3 the mapping F is instable in the domain D0 . Algorithms for Estimating Fold and Stability of Submersion with Fold The algorithms suggested in this paragraph are intended to estimate the following conditions in the case v  n : • rankðJ ðpÞÞ ¼ v  1 for a non-empty set of all singularities of the mapping F; • Transversality (j1 F ⋈ S1 ) • Conditions (1.2) for all points S1 ðF Þ, which estimate the set of all singularities of the mapping F as a fold (in combination with the two conditions mentioned earlier) • The injectivity of the mapping F on the fold, which estimates the mapping F from D0 to E as a stable submersion with fold Algorithm 1.6 Aggregate algorithm for estimating the condition rankðJ ðpÞÞ ¼ v  1 over the set of all singularities of the mapping F and the transversality j1 F ⋈ S1 in the space of 1-streams of mappings from D0 into E. Consider the case n  v, and assume Algorithm 1.4 has yielded an estimate of all e ⊂ D and a vertex set P e of the singularities of the mapping F as a non-empty subset D e elementary parallelepipeds that form D. e estimate the condition Step 1. For each elementary parallelepiped Dk ⊂ D, k rankðJ ðpÞÞ ¼ v  1, where p 2 D , in the following way. n o 2n Let pkj be the collection of all vertexes of a parallelepiped Dk and j¼1

n ol

be the collection of the ðv  1Þ th minors of the Jacobian matrix

M i pk i¼1 j n!n J pkj evaluated at a point pkj ; l ¼ Cv1 n n ¼ ðv1Þ!ðnvþ1Þ!. k If, for a chosen

parallelepiped D , there exists a number i such that all determi

nants M i pkj , j ¼ 1, . . . , 2n , have the same sign, then rankðJ ðpÞÞ is estimated by

the value v  1 in Dk . e then the set D e gives If rankðJ ðpÞÞ is estimated by the value v  1 for all Dk ⊂ D, an estimate of the submanifold S1 ðF Þ. k e Otherwise,

if there

exists a parallelepiped D ⊂ D such that for each i ¼ 1, . . . , l

n k the values M i pj , j ¼ 1, . . . , 2 , have different signs, then the subset of all singularities of the mapping F is estimated as S1 ðF Þ. In this case, the set of all singularities of the mapping F is not estimated as a fold. Stop. e perform Steps 3–5 as described below. Step 2. For each point p of the grid P,

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

e estimate the basis vectors of the tangent space to the submanifold Step 3. For p 2 P,   S1 in the manifold J 1 D0 , E of 1-streams in the following way. Substep 3.1. Estimate the basis (nondegenerate) ðv  1Þth minor of the matrix J ðpÞ. The mapping matrix ðdF Þp of theoretical rank ðv  1Þ is estimated using a numerically calculated Jacobian matrix J ðpÞ in which all vth minors have almost zero determinants. Therefore, first determine a row of the matrix J ðpÞ that is close to a linear combination of all other rows. If the rank of a matrix is the number of its rows minus 1, then by the basis minor theorem, one of the rows represents a linear combination of the others. Let {J 1 , J 2 , . . . , J v } be the collection of all normalized rows of the matrix J ðpÞ. (In the course of normalization, the elements of each row are divided by its magnitude; if the magnitude of some row is 0, the problem is solved.) Let Pi , (i ¼ 1, . . . , v), be a linear combination of all rows from this collection except for the row J i, which is considered as a plane in space Rn. Let mi be the distance between a point J i 2 Rn and the plane Pi: mi ¼ dðJ i , Pi Þ . The value mi can be found by minimizing a function of ðv  1Þ variables ðα1 , . . . , αi1 , αiþ1 , . . . , αv Þ of the form

2 Xv

j Di ðα1 , . . . , αi1 , αiþ1 , . . . , αv Þ ¼ J i  α J

: j j¼1,j6¼i

ð1:7Þ

Choose a number i corresponding to the minimum value mi . Rearrange the variables in the codomain E so that this row is last in the matrix J ðpÞ (i.e., v ¼ i). Using an exhaustive search of all ðv1Þ!ðn!nvþ1Þ! minors, choose a nondegenerate ðv  1Þth minor of the matrix J ðpÞ that does not contain the last row and corresponds to the maximum magnitude of its determinant divided by the magnitudes of all its columns. Rearrange the variables in the domain D so that this minor is among the first ðv  1Þ columns of the matrix J ðpÞ . As a result, the basis ðv  1Þ th minor appears in the left upper corner of the matrix J ðpÞ. Substep 3.2. Estimate the orthonormal basis vectors of the tangent space to the   e using Proposition 5.3 of the submanifold S1 in J 1 D0 , E at a point p 2 P book [6].   A B Write the matrix J ðpÞ in the block representation , where A denotes the C G basis ðv  1Þth minor calculated at Substep 3.1 (a matrix of dimensions ðv  1Þ  ðv  1Þ), B is a matrix of dimensions ðv  1Þ  ðn  v þ 1Þ, while С and G are row vectors of dimension ðv  1Þ and ðn  v þ 1Þ , respectively. In accordance with Proposition 5.3 in [6], the elements of S1 from a neighborhood of the matrix J ðpÞ e1 B, e ¼C eA e where sign “~ ” applies to the corresponding satisfy the relationship G blocks of matrices from S1; in particular, G CA1 B for the matrix J ðpÞ. Throughout this substep, replace the row G with the row G ¼ CA1 B.

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

17

Choose a sufficiently small value δ so that the matrix A has the determinant of the same sign (nonzero) after adding δ to any of! its elements. For i ¼ Ai Bi 1, . . . , ðvn  n þ v  1Þ, denote by J i ðpÞ ¼ the matrix obtained from C i Gi J ðpÞ by adding δ to a single element of the matrix A (or B or C). The other elements of  1 the matrices A, B, and C remain invariable; Gi ¼ C i Ai Bi . As the basis vector of  the tangent space T ðS1 Þ at a point p in J 1 D0 , E choose the orthonormal basis fei g of n þ v þ ðvn  n þ v  1Þ elements constructed by the Gram–Schmidt procedure for a linearly independent system consisting of ðvn  n þ v  1Þ matrices ðJ i ðpÞ  J ðpÞÞ, which are treated as vectors in space Rvn, and n standard orthonormal basis vectors of space Rn together with v standard orthonormal basis vectors of space Rv . e estimate n vectors generating the image of the tangent Step 4. For a point p 2 P, 1 space  0 at this  point with the mapping defined by the stream j F into the manifold 1 J D , E of 1-streams. e let pj be an adjacent point in which the jth coordinate For a point p of the grid P, (j ¼ 1, . . . , n) differs from the corresponding coordinate of the point p in a small value δ while the other coordinates of these points coincide. Choose n generating vectors f j of the image of the tangent space in the tangent space Rn  Rv    HomðRn , Rv Þ to the manifold J 1 D0 , E in the following way. For the subspace Rn , the corresponding coordinates of the vector f j are the coordinates of the vector pj  p. For the subspace Rv, the corresponding coordinates   of the vector f j are the coordinates of the vector F pj  F ðpÞ. For the subspace HomðRn , Rv Þ, the corresponding coordinates of the vector f j are the elements of the   matrices J pj  J ðpÞ.  Normalize the elements of the resulting collection f j , j ¼ 1, . . . , n, by dividing each vector by its magnitude in Rnþvþvn . In the original  notations, this procedure gives the normalized collection of the generating elements f j of the desired image of the tangent space to D0 with the mapping defined by the stream j1 F.   Step 5. Estimate the transversality of the subspaces Lðfei gÞ and L f j in Rnþvþvn  that are generated by the resulting vector collections fei g and f j in the following way. For j ¼ 1, . . . , n, sequentially find the projections ef j of the vectors f j on the plane  P Lðfei gÞ: ef j ¼ i f j , e i ei . Here ð∙,∙Þ denotes the scalar product in space Rnþvþvn . If

the inequality ef j  f j  ε holds for a sufficiently small value ε, then the collection ~f f

fei g is supplemented by the vector

j j

. So the number of orthonormal vectors in ef j f j the collection fei g increases by 1. Perform the same operations for a next number j.

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

If the set fei g contains n þ v þ vn vectors, then the spaces under study are treated as the ones with transversal intersection. Otherwise the set of all singularities of the mapping F is estimated as a fold. Stop.   e the subspaces Lðfei gÞ and L f j Step 6. If for all p 2 P have transversal intersection in Rvþnþvn , then the condition j1 F ⋈ S1 is estimated as true. Stop. Algorithm 1.7 Aggregate algorithm for estimating the mapping F : D0 ! E as a stable submersion with fold. Consider the case n  v, and assume Algorithm 1.6 has yielded the estimate v  1 for the rank of the Jacobian matrix of the mapping F over the non-empty set of all its singularities. Also assume the transversality condition j1 F ⋈ S1 has been estimated as true. First, using Steps 1–5 of this algorithm, estimate the fold condition (1.6) for the e of the set of all singularities of the mapping F. calculated estimate P e perform Steps 2–4 as described below. Step 1. For each point p of the grid P, e Step 2. For p 2 P , estimate the basis vectors e1 , . . . , ev1 of the tangent space T p S1 ðF Þ in the following way. e where M >> v  1, that are closest to the point p Choose M points of the grid P, except for the latter itself: {p1 , . . . , pM }. Define a collection of M vectors of the form {f i ¼ pi  p : i ¼ 1, . . . , M}, where the points pi and p are treated as radius vectors. Denote by T ¼ T ðe1 , . . . , ev1 Þ the linear hull of an arbitrary collection of ndimensional vectors (e1 , . . . , ev1 ). Designate as dð f i , T Þ the distance between the point f i and the plane T. The sum of squared distances from the points f i is Sðe1 , . . . , ev1 Þ ¼

XM i¼1

ðd ð f i , T ÞÞ2 :

ð1:8Þ

Find the coordinates of the desired vectors e1 , . . . , ev1 with the least squares method by minimizing the objective function Sðe1 , . . . , ev1 Þ. In fact, this function remains invariable in case of multiplying any vector ei by a nonzero value. So the vectors ei will be assumed to have unit length. e estimate the basis vectors (g1 , . . . , gnvþ1 ) of the core KerðdF Þ Step 3. For p 2 P, p in the following way. Substep 3.1. Recall that the mapping matrix ðdF Þp of theoretical rank ðv  1Þ is estimated using a numerically calculated Jacobian matrix J ðpÞ in which all vth minors have almost zero determinants. So first determine a row of the matrix J ðpÞ that is close to a linear combination of all other rows. If the rank of a matrix is the number of its rows minus 1, then by the basis minor theorem, one of the rows represents a linear combination of the others. So elimination of this matrix does not affect the kernel of the corresponding linear operator.

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

19

Let {J 1 , J 2 , . . . , J v } be the collection of all normalized rows of the matrix J ðpÞ. (In the course of normalization, the elements of each row are divided by its magnitude; if the magnitude of some row is 0, the problem is solved.) Let Pi , (i ¼ 1, . . . , v), be the linear hull of all rows from this collection except for the row J i, which is considered as a plane in space Rn. Let mi be the distance between a point J i 2 Rn and the plane Pi: mi ¼ dðJ i , Pi Þ. The value mi can be found by minimizing a function of ðv  1Þ variables ðα1 , . . . , αi1 , αiþ1 , . . . , αv Þ of the form

2 Xv

j Di ðα1 , . . . , αi1 , αiþ1 , . . . , αv Þ ¼ J i  α J

: j¼1,j6¼i j

ð1:9Þ

Choose a number i corresponding to the minimum value mi . Denote by e J ðpÞ the matrix obtained from J ðpÞ after elimination of row i. Substep 3.2. Using the Gauss method, solve the linear homogeneous system of ðv  1Þ equations with n unknowns and the system matrix e J ðpÞ. Find the basis of its solution space: {g1 , . . . , gnvþ1 }. e estimate the cosine of the angle between the planes T p S1 ðF Þ and Step 4. For p 2 P, KerðdF Þp in space Rn ( cos φp ) in the following way. Denote by (e1 , . . . , ev1 ) and (g1 , . . . , gnvþ1 ) the estimated  basis vectors  of these planes; see above. Also designate as ðα1 , . . . , αv1 Þ and β1 , . . . , βnvþ1 the P Pv1 corresponding collections of variables. Let e ¼ v1 i¼1 αi ei and g ¼ i¼1 β i gi be arbitrary vectors of the planes T p S1 ðF Þ and KerðdF Þp, respectively. Define a function   f of n variables α1 , . . . , αv1 , β1 , . . . , βnvþ1 that expresses the cosine of the angle between the vectors e and g in the form   fg y ¼ f α1 , . . . , αv1 , β1 , . . . , βnvþ1 ¼ : jf jjgj

ð1:10Þ

The maximum value of this function is treated as the desired value cos φp . e , then by Step 5. Choose a small value ε > 0 . If cos φp < 1  ε for all p 2 P 0 e is estimated as a fold and the mapping F : D ! E as a definition the set D submersion with fold. e is not e such that cos φp  1  ε, then the set D Otherwise, if there exists p 2 P estimated as a fold. Stop. Step 6. Estimate the injectivity (one-to-one relation with image) of the mapping F e over the fold D. In accordance with Theorem 1.3.5, the restriction of the mapping F to a fold is an immersion, which guarantees the local injectivity of this mapping. Nonlocal

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

e The value injectivity is estimated using Algorithm 1.5 with the grid P replaced by P. ε in this algorithm must exceed the double diameter of the elementary parallelepie peds of the set D. e must be After size reduction at Step 3 of Algorithm 1.5, the new cells of the set D checked for singularities using Algorithm 1.4. The cells containing no singularities e Algorithm 1.5 with a sufficiently large number of must be eliminated from the set D. iterations leads to one of two possible situations as follows. (a) The resulting sequence of values m is decreasing approximately in proportion to e is estimated as e1. In this case, the mapping F over the fold D the step of the grid P non-injective (in other words, the set F ðDÞ as self-intersecting). Here an additional study is required for estimating the normality of the self-intersection points   e . of F D (b) The inequality m > ε holds for all grids with a sufficiently small step. In this e is estimated as injective case, the restriction of the mapping F over the fold D (in other words, the set F ðDÞ as non-self-intersecting). On the strength of Corollary 1.3.6, then the mapping F : D0 ! E is estimated as stable. Stop. As demonstrated by computer simulations, the four algorithms proposed in this subsection have an exponential relationship between the amount of calculations to estimate the stability of model mappings and the dimension n of their domains. This result has a simple explanation: the total number of vertexes of each elementary parallelepiped (described by the algorithms) is 2n ; besides, after dividing each parallelepiped into k equal parts in each edge, the total number of smaller elementary parallelepipeds is k n . This computational complexity of the developed algorithms can be reduced by choosing an acceptable number of most significant factors (input parameters) for solving a specific problem of macroeconomic analysis or parametric control based on the model. In addition, it is easily verified that the amounts of calculations have at most a polynomial dependence on other basic parameters of the algorithms, namely, on the dimension v of the codomain Е (for Algorithms 1.4 and 1.6 with an infinite growth of v) and on the value 1/δ, where δ is the diameter of the minimal elementary parallelepiped. Testing of Algorithms The algorithms were tested on the stable Whitney mapping defined by 

y1 ¼ x31 þ x1 x2 , y2 ¼ x2 ,

ð1:11Þ

where ðx1 , x2 Þ 2 R2 . As is well-known, the singularities of this mapping form the parabola 3x21 þ x2 ¼ 0; all points of this parabola (except for the origin) are folds, while the origin is the assembly point [2]. Figure 1.1 shows the fold points of the e of the Whitney estimated using Algorithms 1.4, 1.6, and 1.7 as the vertex set P e to rectangles covering this fold. Note that the distance from any point of the set P

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

21

Fig. 1.1 Fold estimate for Whitney mapping

e this fold does not exceed pffiffiffi the diameter of an elementary square box from the set D (i.e., the value 0:05 2). The estimate cos φð0,0Þ yielded by Algorithm 1.7 was close to 1, and hence the origin was not estimated as a fold point. Algorithm 1.7 was applied to the estimated set of all fold points, indicating that the restriction of F to that fold was injective. Therefore, for the rectangles D without the point (0, 0), Algorithms 1.4, 1.6, and 1.7 estimated the Whitney mapping as a stable submersion (no intersection between D and the fold estimate) or as a stable submersion with fold (some intersections between D0 and the fold estimate).

1.3.4

Numerical Estimation Methods for Stability of Germs of Differential Model Mappings

Note that the algorithms suggested in subsection 1.3.3 are not enough for estimating the stability of differentiable mappings in general case (if they do not belong to the classes of immersions, submersions, or submersions with fold). Relying on the fact that a necessary condition for the stability of a differentiable mapping is the stability of its germs at all singularities, this subsection will introduce an algorithm for estimating the stability of such germs based on theoretical results from the book [2]. Also it will present three algorithms for estimating the type of a stable germ of corank 1 for all possible relationships between the dimensions of the domain and image (codomain) of a model mapping. These algorithms will be used within the V&V model for estimating the low sensitivity of the qualitative properties of the model to its small disturbances. Algorithm for Estimating the Stability of Germs of Differentiable Mappings First, consider some notations and theoretical background from [2] that will be required for developing such an algorithm, namely, the definitions of a stable and

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

infinitesimally V-stable germ and theorems to establish sufficient conditions for the stability of a germ using its infinitesimal V-stability. Then the conditions from the definition of an infinitesimally V-stable germ will be equivalently rewritten in form of Condition 6. Algorithm 1.8 below is a tool for estimating Condition 6, i.e., the infinitesimal V-stability and hence stability of a germ. Consider a germ of a differentiable mapping F : Rm ! Rn at some singularity x0 2 Rm . The same notation will be used for a mapping F : D ! Rn that describes this germ and is defined over some parallelepiped D ⊂ Rm with center at the point x0. Such a germ is said to be stable if, for any arbitrary close mapping F (in an appropriate topology), there exist such diffeomorphisms in some neighborhoods of points of the domain of x0 and image F ðx0 Þ that transform the mapping F into F. See the monograph [2] for a formal definition of (left-right, differentiable) stability of a germ F. Definition A germ F is said to be stable if, for any arbitrarily small neighborhood U of a point x0 , it is possible to find a neighborhood E of the mapping F with the property that, for any mapping F from E, there exists a point x 2 U such that the germ F at x is (left-right, differentiably) equivalent to the germ F at x0 . A topology over the set of all differentiable germs at a point x0 is defined [2] by a collection of neighborhoods of the form ( E¼

)

jαj jαj ∂

∂ i i F: sup

α F ðxÞ  α F 0 ðxÞ :

∂ x jαjK, x2U, i¼1, ..., n ∂ x

ð1:12Þ

  Here F i denotes the coordinate function of a germ F ¼ F 1 , . . . , F n (i ¼ 1, . . . , n ); F 0 is a fixed germ––the center of the neighborhood E; U means some neighborhood of the point x0; x ¼ ðx1 , . . . , xm Þ and α ¼ ðα1 , . . . , αm Þ; αj gives a nonnegative integer; jαj ¼j α1 þ . . . þ αm j; K is an arbitrarily great value; finally, jαj jαj ∂ ∂ . α ¼ α1 1 α ∂ x ∂ x ...∂ m xm Introduce the following additional notations. Designate as ∂F=∂xj a germ that is defined by of the coordinate functions of a mapping F:  the partial derivatives  ∂F=∂xj ¼ ∂F 1=∂xj , . . . , ∂F n=∂xj , where j ¼ 1, . . . , m . The basis germs of mappings Rm ! Rn are constant germs er , (r ¼ 1, . . . , n), in which the rth coordinate function is identically equal to 1, while the other coordinate functions are 0: e1 ¼ ð1, 0, . . . , 0Þ,. . ., en ¼ ð0, . . . , 0, 1Þ. Let Ax0 be an algebra of all germs of differentiable functions Rm ! R at a point x0 , with ðAx0 Þn as the Ax0 -module of all germs of differentiable mappings Rm ! Rn at the point x0. Denote by В a submodule in ðAx0 Þn that is generated by m þ n2 germs of the form ∂F=∂xj , where j ¼ 1, . . . , m, and F i er , where i, r ¼ 1, . . . , n. Definition [2] A mapping germ F at the point x0 is said to be infinitesimally V-stable if the factor module ðAx0 Þn =B is generated over R images of the basis germs e1 , e2 , . . . , en .

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

23

Note that infinitesimal V-stability implies stability. This fact follows from Theorems 1.3.5 and 1.3.6, which were established in [2]. Theorem 1.3.5 The infinitesimal V-stability of a germ is equivalent to its infinitesimal stability. Theorem 1.3.6 (Local case of Mather’s theorem). An infinitesimally stable germ is stable. The following facts are well-known: • The submodule В is the set of all linear combinations of mapping germs ∂F=∂xj and F i er with coefficients from Ax0 . • The definition of V-stability is equivalent to the property that any element of the factor module ðAx0 Þn =B of the form G þ B , where G means some germ from ðAx0 Þn , can be written as the sum of a certain linear combination of the germs e1 , e2 , . . . , en with numerical coefficients and the submodule B. Using these facts, the infinitesimal V-stability conditions of a germ F (see the definition above) will be reformulated in an equivalent way. Condition 1.3.7 Any germ G from ðAx0 Þn can be represented as the sum of a linear combination of the constant germs e1 , e2 , . . . , en with numerical coefficients and a linear combination of the germs ∂F=∂xj and F i er with numerical coefficients from Ax0 . In the case x0 ¼ 0, any germ G from ðAx0 Þn can be written as the sum of a constant germ (a linear combination of the germs e1 , e2 , . . . , en with numerical coefficients) and a germ that takes value 0 at the point x0 ; in addition, the latter germ can be represented as the sum of linear combinations of the functions xj er with numerical coefficients from Ax0 , where xj denotes the jth coordinate of a vector x. Therefore, in Condition 1.3.7 it is possible to replace the words “any germ G” with “any germ of the form xj er , where j ¼ 1, . . . , m, r ¼ 1, . . . , n.” Moreover, for a sufficiently small neighborhood of x0 ¼ 0, the coefficients at the germs generating the submodule В are approximately the polynomial germs QK ðxÞ from Ax0 of an order not exceeding K, where K is a sufficiently large fixed number. In other words, Q K ð xÞ ¼

X

c x α:jαjK α

α

:

ð1:13Þ

α

where xα ¼ ðx1 Þ 1 . . . ðxm Þαm and cα are the coefficients of this polynomial. (This approximation holds up to infinitesimal terms of higher orders.). Hence, in the case x0 ¼ 0 , Condition 1.3.7 of an infinitesimally V-stable germ F has the following equivalent form. Condition 1.3.8 Any mapping germ of the form xj er , where j ¼ 1, . . . , m and ~ r ¼ 1, . . . , n, can be represented as a linear combination of the germs ∂F=∂x~j and F i e~r with the polynomial coefficients (1.13) up to infinitesimal terms of higher orders.

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

In accordance with Theorem 1.3.6, Condition 1.3.8 is sufficient for the stability of a germ F at its singularity––the origin. The aggregate algorithm below employs Condition 1.3.8 for estimating the stability of a germ at a singularity x0 of a model mapping F defined over a parallelepiped Dε with center at the point x0 , where ε denotes its diameter. A singularity x0 can be estimated using Algorithm 1.4 with a e subsequent choice of one point from the set P. Algorithm 1.8 Step 1. Perform an appropriate parallel shift and normalization of the coordinate system in Rm for placing the singularity x0 to the origin and making the parallelepiped Dε close to a cube. Here ε is a sufficiently small given value. Step 2. Partition each edge of Dε into a sufficiently large even number N of equal segments and construct a grid Pε from N m points. Step 3. Define grid functions F i, i ¼ 1, . . . , n, at the nodes p 2 Pε that correspond to the coordinate functions of the mapping F. Step 4. Using numerical differentiation (with a step less than that of the grid Pε ), define the grid functions F ij , i ¼ 1, . . . , n, j ¼ 1, . . . , m, at the nodes p 2 Pε that are the estimates of the partial derivatives ∂Fi=∂xj at the nodes p 2 Pε . Step 5. Specify a grid mapping (a linear combination of germs) using its coordinate grid functions defined on Pε :

   yi ¼ Ri x, αj : j ¼ 1, . . . , m; αj  K , fαi,r : r ¼ 1, . . . , n; jαi,r j  K g Xm Xn ð1:14Þ ¼ Q ðxÞF ij þ Q ðxÞF r , i ¼ 1, . . . , n: j¼1 j r¼1 i,r Here Qj and Qi,r are some polynomials of degrees not exceeding K with arbitrary coefficients αj and αi,r , respectively; K is a sufficiently large fixed number. Step 6. Using numerical differentiation (with a step less than that of the grid Pε ), define the grid functions F iβ , i ¼ 1, . . . , n , β ¼ ðβ1 , . . . , βm Þ , j β j K , at the jβ j

nodes p 2 Pε that are the estimates of the partial derivatives ∂β F i at the nodes ∂ x p 2 Pε . (For the cases jβj ¼ 0 and jβj ¼ 1, these functions have been defined at Steps 3 and 4 of the algorithm). Step 7. Define the grid functions Riβ, i ¼ 1, . . . , n, β ¼ ðβ1 , . . . , βm Þ, j β j K, at the nodes p 2 Pε that are the estimates of the partial derivatives β

jβ j

i ∂ β R ∂ x

∂j j β Q ∂ x j

using the jβ j

calculated values and the values of the partial derivatives and ∂β Qi,r of ∂ x the polynomials at the nodes p 2 Pε . Step 8. For each r ¼ 1, . . . , n and j ¼ 1, . . . , m, execute Steps 9–11. Step 9. For i ¼ 1, . . . , n, define the functions F iβ

M ir,j

 αj : j ¼ 1, . . . , m; αj  K , fαi,r : r ¼ 1, . . . , n; jαi,r j  K g



jβj ∂

i ¼ sup Rβ ðxÞ  β xj eir :

∂ x x2Pε ; β:jβjK



ð1:15Þ

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

25





jβ j jβ j Here, for β ¼ 0, ∂β xj eir ¼ xj if i ¼ r, and ∂β xj eir ¼ 0 if i 6¼ r. In the case ∂ x ∂ x

∂ j i ∂ j β j¼ 1, ∂xj x er ¼ 1 if i ¼ r and j ¼ j ; otherwise, ∂x xj eir ¼ 0 . If jβj > 1 , j

jβ j ∂ xj eir 0. β

∂ x

Step 10. Estimate the deviation of the germ xj er from its linear approximation: m¼



1 inf supi2f1,...,ng M ir,j : ε fαj :j¼1, ..., m; jαj jK g, fαi,r :r, i¼1, ..., n; jαi,r jK g

ð1:16Þ

Step 11. If m > δ, where δ is a sufficiently small given value, then move to Step 13. Step 12. The germ F is estimated as stable. Stop. Step 13. Reduction of ε and all edges of the parallelepiped Dε by 2 times. Step 14. If ε > ε0 , where ε0 is a sufficiently small given value, then get back to Step 2. Step 15. The stability of the germ F remains an open issue. Step 16. Stop. Algorithms for Estimating the Type of a Stable Germ of Corank 1 of Differential Mapping After a germ F is estimated as stable at a singularity x0 using Algorithm 1.8, it should be assigned a type in accordance with the classification of stable germs by genotype suggested in [2]. The book [2] presented three Moren’s theorems on classification of stable germs of corank 1; unfortunately, classification of stable germs of higher coranks still remains an open problem. The corank of a germ F at a singularity x0 is defined as the difference between the value min ðm, nÞ and the rank of the Jacobian matrix of this germ at x0 . This paragraph recalls Moren’s theorems as well as develops algorithms for estimating the type and some characteristics of a germ F. Consider a differentiable mapping F : Rm ! Rn defined in some neighborhood of its singularity 0 2 Rm , F ð0Þ ¼ 0 2 Rn . Study all possible relationships between the dimensions m and n. 1. Let the dimensions of the image and full domain of a germ F coincide: n ¼ m. Moren’s theorem [2] describes such stable germs in the following way. Theorem 1.3.9 A stable germ F : ðRn , 0Þ ! ðRn , 0Þ of corank 1 is right-left equivalent to the germ

26

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

8  1 k   1 2 1 k2 > þ . . . þ ~xk1~x1 , > > ey ¼ ~x þ ~x ~x > < ~y2 ¼ ~x2 , > > ... > > : n ~y ¼ ~xn ,

ð1:17Þ

    where ey ¼ ey1 , ey2 , . . . , eyn and ex ¼ ex1 , ex2 , . . . , exn are new coordinates in the image and preimage spaces of the germ F; a natural number k, 2  k  n þ 1, is the genotype order. (Equivalence of two germs means that one germ is reducible to the other with diffeomorphic changes of coordinates in the preimage and image spaces.) The aggregate algorithm below estimates the axis Oey1 for mapping (1.17) and also the genotype order k of the mapping F mentioned in this theorem. It relies on an obvious fact that there is a unique direction in the image space Rn (the axis Oey1) with the following property: the coordinate of the mapping F that corresponds to ey1 has zero derivative at the origin O in any direction in the preimage space. For any other axis in the image space, the corresponding derivative in some direction is nonzero at the point O. Algorithm 1.9 Step 1. Using Algorithm 1.4, estimate the set P of all singularities of the mapping F : D ! Rn , D ⊂ Rn . Step 2. If P is empty, then stop. The mapping F has no singularities. Step 3. Choose a singularity x0 2 P and, using Algorithm 1.8, estimate the stability of the germ F at x0 . Step 4. If the germ F is estimated as stable, then stop. Step 5. Estimate the Jacobian matrix of the mapping F at the singularity x0 and its rank r ¼ rankðJ ðx0 ÞÞ. Step 6. If r 6¼ n  1, then stop. The corank of F at the singularity x0 exceeds 1. Step 7. Define the shifted mapping y ¼ F ðx  x0 Þ þ F ðx0 Þ ¼ F ðxÞ, for which x ¼ 0 is the singularity under study and F ð0Þ ¼ 0. Step 8. Estimate the axis Oey1 . Substep 8.1. For an arbitrary unit vector ey 2 Rn (jeyj ¼ 1), define a coordinate function of n variables of the form a ¼ F~y ðxÞ ¼ pr~y F ðxÞ, which is the projection of the vector F ðxÞ to the vector ey. Substep 8.2. Define a function of n variables of the form s ¼ Gð~yÞ ¼ Pn ∂F~y ðxÞ 2 ∂F ðxÞ , where ∂x~y i is the corresponding difference derivative. i¼1 ∂xi x¼0

Substep 8.3. Estimate the constrained minimum s0 of the function s ¼ GðeyÞ subject to jeyj ¼ 1.

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

27

Substep 8.4. If s0 0, then the value ey is an estimate of the unit vector of the axis Oey1 ; otherwise stop. In this case, repeat Steps 1–8 of the algorithm with smaller steps of the grids and increments. Step 9. Estimate the genotype order k of the germ F. Substep 9.1. For an axis defined by an arbitrary unit vector x (jxj ¼ 1), specify the constraint c ¼ f x ðt Þ ¼ F~y1 ðtxÞ on the coordinate function F~y1 ðxÞ. Substep 9.2. Assign k ¼ 2. Substep 9.3. Estimate the maximum magnitude of the kth

derivative of the

ðkÞ

function c ¼ f x ðt Þ at the origin: m ¼ max f ðt Þ ; here approximate x, jxj¼1

x

t¼0

calculations involve difference derivatives. Substep 9.4. If m > ε, where ε is a sufficiently small value, then return the order k and stop. Substep 9.5. Assign k ≔ k þ 1. Substep 9.6. If k  n þ 1, then get back to Substep 9.3. Substep 9.7. The genotype order k of the point x0 is not determined. Substep 9.8. Stop. 2. Now, consider the case in which the image of the germ F has a strictly greater dimension than that of its domain: n > m. Here is another Moren’s theorem [2] that describes stable germs in this case. Theorem 1.3.10 If n > m, a stable germ F : ðRm , 0Þ ! ðRn , 0Þ of corank 1 in the preimage space is right-left equivalent to the germ 8  k  k2 > ey1 ¼ ex1 þ ex2 ex1 þ . . . þ exk1ex1 , > > >     > k1 k2 > > ey2 ¼ exk ex1 þ exkþ1 ex1 þ . . . þ ex2k2ex1 , > > > > > ... <    k2 t ðt1Þkt 1 k1 ey ¼ ex ex þ exkþ1 ex1 þ . . . þ extktex1 > > > > > eytþ1 ¼ ex2 , > > > > > ... > > : eyn ¼ exm ,

ð1:18Þ

where a natural number t ¼ n  m þ 1; t ðk  1Þ  m;  k  2 is  the genotype order;  ey ¼ ey1 , ey2 , . . . , eyn and ex ¼ ex1 , ex2 , . . . , exm are new coordinates in the image and preimage spaces of the germ F. The aggregate algorithm below estimates the plane П spanned over the axes ey1 , ey2 , . . . , eyt for mapping (1.18) as well as the genotype order k of the mapping F mentioned in Theorem 1.3.10. This algorithm is based on the following properties. For any axis from the plane П, the coordinate of the mapping F that corresponds to

28

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

this axis has zero derivative at the origin O in any direction of the preimage space. For any axis outside П, the derivative at the origin is nonzero in some direction in the image space. Algorithm 1.10 Step 1. Using Algorithm 1.4, estimate the set P of all singularities of the mapping F : D ! Rn , D ⊂ Rm . Step 2. If P is empty, then stop. The mapping F has no singularities. Step 3. Choose a singularity x0 2 P, and, using Algorithm 1.8, estimate the stability of the germ F at x0 . Step 4. If the germ F is estimated as stable, then stop. Step 5. Estimate the Jacobian matrix of the mapping F at the singularity x0 and its rank r ¼ rankðJ ðx0 ÞÞ. Step 6. If r 6¼ m  1 , then stop. The corank of F in the preimage space at the singularity x0 exceeds 1. Step 7. Define the shifted mapping y ¼ F ðx  x0 Þ þ F ðx0 Þ ¼ F ðxÞ, for which x ¼ 0 is the singularity under study and F ð0Þ ¼ 0. Step 8. Estimate plane П spanned over the axes ey1 , ey2 , . . . , eyt (where t ¼ n  m þ 1) for mapping (1.18). Substep 8.1. For an arbitrary unit vector ey 2 Rn (jeyj ¼ 1), define a coordinate function of n variables of the form a ¼ Fey ðxÞ ¼ prey F ðxÞ, which is the projection of the vector F ðxÞ to the vector ey.

Substep 8.2. Define a function of n variables of the form s ¼ Gð~yÞ ¼ Pn ∂F~y ðxÞ 2 ∂F ðxÞ , where ∂x~y i is the corresponding difference derivative. i¼1 ∂xi x¼0

Substep 8.3. Assign k ¼ 1, where k is the counter for the coordinate axes of the desired plane П. Substep 8.4. Estimate the constrained minimum s0 of the function s ¼ GðeyÞ subject to jeyj ¼ 1 and ey ∙ eyi ¼ 0, where i ¼ 1, . . . , k  1 are the numbers of the coordinate axes of plane П calculated at the previous steps and “” denotes scalar product (for k ¼ 1, this constraint disappears). Substep 8.5. If s0 0, then the value ey is an estimate of the unit vector of the axis Oeyk ; otherwise stop. In this case, repeat Steps 1–8 of the algorithm with smaller steps of the grids and increments. Substep 8.6. Assign k ≔ k þ 1. Substep 8.7. If k  t, then get back to Substep 8.4. Step 9. Estimate the genotype order k of the germ F. Substep 9.1. For an axis defined by arbitrary unit vectors ex 2 Rm (jexj ¼ 1) and ey 2 Rn (jeyj ¼ 1), specify the constraint c ¼ f x,y ðt Þ ¼ F y ðtxÞ on the coordinate function Fey ðxÞ. Substep 9.2. Assign k ¼ 2.

1.3 Numerical Estimation Methods for some Types of Stability of Mathematical. . .

29

Substep 9.3. Estimate the maximum magnitude of the of the

kth derivative

ðk Þ function c ¼ f x,y ðt Þ at the origin: m ¼ max max f x,y ðt Þ ; here approxy2П, jyj¼1x, jxj¼1

t¼0

imate calculations involve difference derivatives. Substep 9.4. If m > ε, where ε is a sufficiently small value, then return the order k and stop. Substep 9.5. Assign k ≔ k þ 1. Substep 9.6. If t ðk  1Þ  m, then get back to Substep 9.3. Substep 9.7. The genotype order k of the point x0 is not determined. Substep 9.8. Stop. 3. Finally, consider the case in which the image of the germ F has a strictly smaller dimension than that of its domain: n < m. Here is the third Moren’s theorem [2] that characterizes stable germs of corank 1 in this case. Theorem 1.3.11 If n < m , a stable germ F : ðRm , 0Þ ! ðRn , 0Þ of corank 1 for which the corank of the second differential of genotype does not exceed 1 at the origin is right-left equivalent to the germ 8  k  k2  2  2 > ey1 ¼ ex1 þ ex2 ex1 þ . . . þ exk1ex1 exnþ1 exnþ2 . . . ðexm Þ2 , > > > < ey2 ¼ ex2 , > > ... > > : n ey ¼ exn , ð1:19Þ     where ey ¼ ey1 , ey2 , . . . , eyn and ex ¼ ex1 , ex2 , . . . , exn are new coordinates in the image and preimage spaces of the germ F; a natural number k, 2  k  n þ 1, is the genotype order. The last aggregate algorithm presented in this subsection estimates the axis Oey1 for mapping (1.19) and also the genotype order k of the mapping F described in Theorem 1.3.11. The first part of this algorithm (Steps 1–8) is similar to the corresponding steps of Algorithm 1.9. Algorithm 1.11 Step 1. Using Algorithm 1.4, estimate the set P of all singularities of the mapping F : D ! Rm , D ⊂ Rn . Step 2. If P is empty, then stop. The mapping F has no singularities. Step 3. Choose a singularity x0 2 P, and, using Algorithm 1.8, estimate the stability of the germ F at x0 . Step 4. If the germ F is estimated as stable, then stop. Step 5. Estimate the Jacobian matrix of the mapping F at the singularity x0 and its rank r ¼ rankðJ ðx0 ÞÞ.

30

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Step 6. If r 6¼ m  1, then stop. The corank of F at the singularity x0 exceeds 1. Step 7. Define the shifted mapping y ¼ F ðx  x0 Þ þ F ðx0 Þ ¼ F ðxÞ, for which x ¼ 0 is the singularity under study and F ð0Þ ¼ 0. Step 8. Estimate the axis Oey1 . Substep 8.1. For an arbitrary unit vector ey 2 Rn (jeyj ¼ 1), define a coordinate function of n variables of the form a ¼ Fey ðxÞ ¼ prey F ðxÞ, which is the projection of the vector F ðxÞ to the vector ey.

Substep 8.2. Define a function of m variables of the form s ¼ GðeyÞ ¼ Pn i¼1



∂F ðxÞ

ey

∂xi

2 , where x¼0

∂F ðxÞ

ey

∂xi

is the corresponding difference derivative.

Substep 8.3. Estimate the constrained minimum s0 of the function s ¼ GðeyÞ subject to jeyj ¼ 1. Substep 8.4. If s0 0, then the value ey is an estimate of the unit vector of the axis Oey1; otherwise stop. In this case, repeat Steps 1–8 of the algorithm with smaller steps of the grids and increments. Step 9. Estimate the genotype order k of the germ F. Substep 9.1. For an axis defined by arbitrary unit vector x (jxj ¼ 1), specify the constraint c ¼ f x ðt Þ ¼ F 1 ðtxÞ on the coordinate function F 1 ðxÞ. ey ey Substep 9.2. Assign i ¼ 1, where i is the counter of the axes corresponding to the Morse singularity of the function F 1 ðxÞ. ey Substep 9.3. Estimate the maximum magnitude of the second derivative of the

ð2Þ

function c ¼ f x ðt Þ at the origin: m ¼ max

f ðt Þ ; here approxx, jxj¼1; x~xj ¼0, j m) in the image space of a studied mapping that has the following property. The coordinate of the mapping F that corresponds to ey1 (or to any axis from plane П in the case n > m) has zero derivative at the origin O in any direction in the preimage space. (The origin O corresponds to a singularity x0 .) Revealing such an axis (or plane) can be crucial for analyzing the behavior of a model germ at a singularity. The coordinate function corresponding to this axis gives an example of a local invariant of the model in the following sense: in some neighborhood of the singularity, this function is constant (up to infinitesimal terms of higher orders) in all exogenous model parameters of the mapping. If a singularity x0 is estimated by Algorithm 1.4 as a regular point of a given mapping, then the image space does not contain axis with the above properties; so this mapping has no local invariants in a neighborhood of the regular point under consideration.

1.4

Some Properties of Solutions of Parametric Control Problems

For developing Components 4 and 6 of the theory of parametric control, this section formulates and proves the following theorems: • Sufficient conditions for the existence of solutions of variational calculus problems on the design and choice of optimal parametric control laws (from a given finite collection of Algorithms) • Sufficient conditions for a continuous dependence of optimal criterion values on uncontrolled parameters (functions) in the parametric control problems • Sufficient conditions for the existence of all bifurcation points for extremals of optimal parametric control choice problems All the results established for nonautonomous systems remain in force for their autonomous counterparts with a constant exogenous vector function a(∙).

1.4.1

Sufficient Conditions for the Existence of Solutions of Optimal Parametric Control Design and Choice Problems

Sufficient Conditions in Dynamic Optimization Problem on Design of Optimal Parametric Control Laws for Continuous-Time Dynamic Systems Consider a continuous-time controlled system of the form

32

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

x_ ðt Þ ¼ f ðxðt Þ, uðt Þ, aðt ÞÞ,

t 2 ½0, T ,

xð0Þ ¼ x0 ,

ð1:20Þ ð1:21Þ

with the following notations: t as time; x ¼ xðt Þ ¼ ðx1 ðt Þ, . . . , xm ðt ÞÞ as a vector function of system state; u ¼ uðt Þ ¼ ðu1 ðt Þ, . . . , uq ðt ÞÞ as a vector function of a ¼ aðt Þ ¼ ða1 ðt Þ, . . . , as ðt ÞÞ as a given vector function; x0 ¼ control; 1 m x0 , . . . , x0 as a given initial state of the system; and finally, f as a given vector function of appropriate arguments. The design problem of optimal economic tools consists in the minimization or maximization of a criterion: ðT K ¼ F ðt, xðt ÞÞdt,

ð1:22Þ

0

where F is a given function, subject to state-space constraints xðt Þ 2 X ðt Þ,

t 2 ½0, T ,

ð1:23Þ

where X(t) is a given set and also subject to explicit control constraints uðt Þ 2 U ðt Þ,

t 2 ½0, T ,

ð1:24Þ

where U(t) is a given set. A variational calculus problem on the design of optimal parametric control laws in the continuous-time dynamic system has the following statement. Problem 1.4.1 Given a function a (), find a control vector u() that satisfies condition (1.24) so that the corresponding solution of the dynamic system (1.20) and (1.21) satisfies condition (1.23) and also maximizes (minimizes) criterion (1.22). For proving the solvability of Problem 1.4.1, first the existence of a unique solution of the Cauchy problems (1.20) and (1.21) is required. To obtain this property, employ a well-known result from theory of ordinary differential equations. Let a value T > 0, a metrizable compact set U and a continuous function φ : ½0, T   Rm  U ! Rm be such that, for any ρ  0, there exists a value σ  0 for which jφðt, y, uÞ  φðt, y0 , uÞj  σjy  y0 j

8t 2 ½0, T, y, y0 2 ρB, u 2 U,

ð1:25Þ

where В denotes a unit ball in Rm , and also there exists a constant η  0 for which

1.4 Some Properties of Solutions of Parametric Control Problems

jyφðt, y, uÞj  ηð1 þ jyj2 Þ 8t 2 ½0, T, y 2 Rm , u 2 U:

33

ð1:26Þ

Consider the Cauchy problem: _ ¼ φðt, yðtÞ, uðtÞÞ yðtÞ

8t 2 ð0, TÞ,

yð0Þ ¼ y0 ,

ð1:27Þ ð1:28Þ

where y0 2 Rm : The next result was established in [19]. Lemma 1.4.1 Under the above assumptions, for any measurable mapping u : ½0, T  ! U, Problems (1.27) and (1.28) has a unique solution y : ½0, T  ! Rm that satisfies the bound

1=2 eηT 8t 2 ½0, T : jyðt Þj  jy0 j2 þ 2ηT

ð1:29Þ

The following is the case. Theorem 1.4.2 Let the function а(∙) be continuous on the interval ½0, T , U be a compact set in Rq, and the function f be continuous. Also suppose for any ρ  0 there exist constants σ  0 and η  0 such that jf ðx, u, aðt ÞÞ  f ðx0 , u, aðt ÞÞj  σ jx  x0 j8t 2 ½0, T , x, x0 2 ρB, u 2 U,

ð1:30Þ

and

jxf ðx, u, aðt ÞÞj  η 1 þ jxj2 8t 2 ½0, T , x 2 Rm , u 2 U:

ð1:31Þ

Then for any measurable mapping u: [0, T] ! U, the Cauchy problems (1.20) and (1.21) has a unique solution x: [0, T] ! Rm that satisfies the bound 1=2 ηT

jxðtÞj  ðjx0 j2 þ 2ηTÞ

e

8t 2 ½0, T:

ð1:32Þ

Proof Introduce the notation φðt, х, иÞ ¼ f ðх, и, aðt ÞÞ: Since the functions а and f are continuous, the same property holds for the function φ. Inequalities (1.25) and (1.26) follow from relationships (1.30) and (1.31). And the desired result is immediate from Lemma 1.4.1. This completes the proof of Theorem 1.4.2. For showing the solvability of Problem 1.4.1, use a well-known result from optimal control of dynamic systems described by differential equations. Specify a closed subset E ⊂ ½0, T   Rm  U . Consider the system described by the Cauchy problems (1.27) and (1.28) under the hypotheses of Lemma 1.4.1; let a control law

34

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

be a measurable mapping u : ½0, T  ! U . For this system, a “state–control” pair is said to be admissible if it satisfies relationships (1.27) and (1.28) and also the inclusion: ðt, xðt Þ, uðt ÞÞ 2 E:

ð1:33Þ

Define a Caratheodory function Ф with nonnegative values over the set ½0, T   ÐT ðR  U Þ. Define the functional I ¼ Φðt, xðt Þ, uðt ÞÞdt: m

0

Problem 1.4.2 Minimize the functional I over the set of admissible pairs of system (1.27) and (1.28). For any point ðt, xÞ 2 ½0, T   Rm, define the section E t,x ¼ fu 2 U : ðt, x, uÞ2 Eg. Define the set Γt,x ¼ fφðt, x, uÞ : ðt, x, uÞ 2 E t,x Þg and also the function g : ½0, T  Rm  Rm ! R by the equality gðt, x, yÞ ¼ min fΦðt, x, uÞj ðt, x, uÞ 2 E, φðt, x, uÞ ¼ yg: The following result on the solvability of the optimization problem is wellknown; see Proposition 4.2 in [19]. (Here R denotes the set of all real values supplemented by 1 and +1.) Lemma 1.4.3 Under the hypotheses of Lemma 1.4.1, let the set Γt,x be convex for all t 2 ½0, T , x 2 Rm and the function gðt, x, ∙ Þ : Rm ! R be convex for all ðt, xÞ 2 ½0, T   Rm . Then Problem 1.4.2 has a solution. S Assume Х is the closure of the union XðtÞ. t2½0, T

Lemma 1.4.4 Let Х be a compact set and the function F be continuous over ½0, T   X . Then the function Ф defined by the equality Φðt, x, uÞ ¼ F 0  F ðt, xÞ, where F 0 is the maximum of the function F over the set ½0, T   X, satisfies the hypotheses of Lemma 1.4.3. Proof The maximum F 0 exists in accordance with the Weierstrass extreme value theorem. Obviously, the function Ф defined above takes nonnegative values only. It is a Caratheodory function on the strength of continuity of F. Finally, the function gðt, x, ∙ Þ is convex for all ðt, xÞ 2 ½0, T   Rm because the function F does not depend on the control u. This completes the proof of Lemma 1.4.4. S Assume U is the closure of the union t2½0,T  U ðt Þ. For reducing constraints (1.23) and (1.24) to form (1.33), define the set E ⊂ ½0, T   Rm  U so that Et0 ¼ fðt, x, uÞ 2 Ejt ¼ t 0 g ¼ X ðt 0 Þ  U ðt 0 Þ, t 0 2 ½0, T : Lemma 1.4.5 Let the mappings t ! X ðt Þ and t ! U ðt Þ be continuous at each point t 2 ½0, T  in the following sense: if xk 2 X ðt k Þ and uk 2 U ðt k Þ, where t k 2 ½0, T and k ¼ 1, 2, . . . и and also t k ! t, xk ! x, and uk ! u as k ! 1, then x 2 X ðt Þ, u 2 U ðt Þ: Then the set Е is closed.

1.4 Some Properties of Solutions of Parametric Control Problems

35

Proof Consider a sequence fðt k , uk , xk Þg of elements from the set Е such that ðt k , uk , xk Þ ! ðt, u, xÞ. By the definition of the set Е, the inclusion ðt k , uk , хk Þ 2 E implies the inclusions t k 2 ½0, T, xk 2 X ðt k Þ, and uk 2 U ðt k Þ: Because the interval ½0, T  is closed, the inclusion t 2 ½0, T  also holds. By the hypotheses of this lemma, x 2 X ðt Þ and u 2 U ðt Þ: Therefore, the inclusion ðt, u, xÞ 2 E is the case, indicating that the set Е is closed. This completes the proof of Lemma 1.4.5. Now, verify that the hypotheses of Lemma 1.4.5 are not very restricting. For example, in the scalar case, consider a typical situation with XðtÞ ¼ fxj aðtÞ  x  bðtÞg, t 2 ½0, T, where functions а and b are continuous. Assume xk 2 X ðt k Þ, where t k 2 ½0, T, k ¼ 1, 2, . . ., and also t k ! t and xk ! x: In other words, the inequalities aðt k Þ  xk  bðt k Þ, k ¼ 1, 2, . . . hold. Passing to the limit as k ! 1 gives the inequality aðt Þ  x  bðt Þ by the continuity of the functions а and b, and hence x 2 X ðt Þ: Thus, the mapping t ! X ðt Þ is continuous on the interval [0, T]. A similar approach can be adopted for checking the hypotheses of Lemma 1.4.5 in more general cases where the boundaries of the admissible sets of the control and state vectors are specified by continuous time-varying functions. For reducing Eq. (1.20) to form (1.27), it suffices to define φðt, х, иÞ ¼ f ðх, и, aðt ÞÞ: Then the set Γt,x figuring in the hypotheses of Lemma 1.4.3 can be written as Γt,x ¼ ff ðx, w, aðt ÞÞj w 2 U ðt Þg:

ð1:34Þ

Denote by Vа the set of admissible “state–control” pairs of system (1.20) and (1.21) with a given function а. Actually, these are pairs of the vector functions ðx, uÞ that satisfy relationships (1.20), (1.21), (1.23), and (1.24). Lemma 1.4.3 directly leads to the following. Theorem 1.4.6 Let the function а() be continuous on the interval ½0, T , U be a compact set in Rq, and the function f be continuous over X  U  A. Suppose for any ρ  0 there exist constants σ  0 and η  0 such that jf ðx, u, aðt ÞÞ  f ðx0 , u, aðt ÞÞj  σjx  x0 j, x, x0 2 ρB, u 2 U

ð1:35Þ

and



2

xf ðx, u, aðt ÞÞj  η 1 þ jxj 8t 2 ½0, T , x 2 Rm , u 2 U

ð1:36Þ

Let X be a compact set and the function F be continuous over [0, T]  X. In addition, suppose the mappings t ! X(t) and t ! U(t) are continuous for t 2[0, T] in the following sense: if xk2 X(tk) and uk2 U(tk), where tk2[0, T] and k ¼ 1, 2,. . . and also tk ! t, xk ! x, and uk ! u as k ! 1, then x 2X(t) and u 2 U(t). Then Problem

36

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

1.4.1 has a solution in the class of measurable functions provided that the set Va is non-empty, while the set Γt,x is convex for all t 2[0, T] and x 2 X(t). Sufficient Conditions in Dynamic Optimization Problem on Choice of Optimal Parametric Control Laws for Continuous-Time Dynamic Systems (from a Given Finite Collection of Algorithms) Consider the continuous-time controlled system (1.20) and (1.21). Let the control vector и be chosen from a collection of given control laws of the form uj ðt Þ ¼ Gj ðv, xðt ÞÞ, t 2 ð0, T Þ,

j ¼ 1, . . . , r:

ð1:37Þ

  Here Gj denotes a given vector function of its arguments, while v ¼ v1 , . . . , vl is the vector of adjustable parameters. The adjustable parameters are subjected to the constraints: v 2 V,

ð1:38Þ

where V indicates some subset of space Rl . Moreover, by assumption the adjustable parameters must be such that the corresponding control law (1.37) satisfies condition (1.24), i.e., Gj ðv, xðt ÞÞ 2 U ðt Þ, t 2 ð0, T Þ,

ð1:39Þ

where U ðt Þ is a given set. The state-space constraints have the form xðt Þ 2 X ðt Þ, t 2 ð0, T Þ,

ð1:40Þ

where X ðt Þ is a given set. Introduce the optimality criteria ðT

  K j ¼ K j ða, vÞ ¼ F t, xj ðt Þ dt,

ð1:41Þ

0





where xj ¼ xj ðt Þ ¼ x1j ðt Þ, . . . , xm j ðt Þ is the solution of the Cauchy problems (1.20) and (1.21) with a given function а() and control vector u ¼ uj ðt Þ ¼

u1j ðt Þ, . . . , uqj ðt Þ , i.e., with the jth control law (1.37). Consider an auxiliary optimization problem as follows. Problem 1.4.3 Given a function а(), for each of r control laws, find the vector of adjustable parameters v so that the corresponding solution x ¼ xj of problems (1.20) and (1.21) with the control law u ¼ uj defined by (1.37) satisfies conditions (1.38), (1.39), and (1.40) and also maximizes functional (1.41).

1.4 Some Properties of Solutions of Parametric Control Problems

37

Formulate the following variational calculus problem on the choice of an optimal parametric control law for the nonautonomous continuous-time system from a given finite collection of algorithms. Problem 1.4.3 Given a function а(), among all optimal control laws in the sense of Problem 1.4.3 choose the one maximizing the optimality criterion (1.41). Substituting the control law (1.37) into Eq. (1.20) yields   x_ ðt Þ ¼ f xðt Þ, Gj ðv, xðt ÞÞ, aðt Þ , t 2 ð0, T Þ:

ð1:42Þ

(For the sake of compactness, here the subscript j is omitted for the component xj that corresponds to this control law.) Denote by W ja the set of admissible “state–control parameter” pairs of the system, i.e., the pairs ðx, vÞ that satisfy relationships (1.42) and (1.21), and also inclusions (1.38), (1.39) and Ð(1.40). Therefore, Problem 1.4.3 is T reduced to maximization of the functional K ¼ 0 F ½t, xðt Þdt over the set W ja . First, establish the solvability of the Cauchy problems (1.42) and (1.21). Lemma 1.4.1 directly leads to the following. Theorem 1.4.7 Let the function а() be continuous on the interval ½0, T , U and V be compact sets, and the functions f and Gj be continuous. Suppose for any ρ  0 there exist constants σ  0, χ  0, and η  0 such that jf ðx, u, aðtÞÞ  f ðx0 , u0 , aðtÞÞj  σðjx  x0 j þ ju  u0 jÞ 2 ρB, u, u0 2 U, jGj ðv, xÞ  Gj ðv, x0 Þj  χjx  x0 j

8t 2 ½0, T, x, x0

8x, x0 2 ρB, v 2 V,

ð1:43Þ ð1:44Þ

and

 

xf x, Gj ðv, xÞ, aðt Þ  η 1 þ jyj2 8t 2 ½0, T , x 2 Rm , v 2 V:

ð1:45Þ

Then for any v 2 V, problems (1.31), (1.10) has a unique solution x: [0, T] ! Rm that satisfies the bound 1=2 ηT

jxðtÞj  ðjx0 j2 þ 2ηTÞ

e

8t 2 ½0, T:

ð1:46Þ

Now, prove the solvability of Problem 1.4.3. The following result is the case. Theorem 1.4.8 Under the hypotheses of Theorem 1.4.7, let the set W ja be non-empty for a given function а() and a given value j 2 f1, . . . , r g. In addition, let the sets V, U, and X be compact, the sets X ðt Þ and U ðt Þ be compact for all t 2 ½0, T , and the function F be continuous. Then Problem 1.4.3 has a solution.

38

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Proof Condition (1.40) together with the continuity of F implies that this function is bounded. Then, by the non-emptiness of the set W ja , there exists the least upper bound (sup K) of the function K over the set W ja of all admissible pairs of the system. Then there exists a sequence fðxk , vk Þg of elements from W ja that converges to this bound K k ! sup K,

ð1:47Þ

ÐT where K k ¼ F ½t, xk ðt Þdt: The inclusion ðxk , vk Þ 2 W ja means that 0

  x_ k ðt Þ ¼ f xk ðt Þ, Gj ðvk , xk ðt ÞÞ, aðt Þ , t 2 ð0, T Þ,

ð1:48Þ

x k ð 0Þ ¼ x 0 ,

ð1:49Þ

xk ðt Þ 2 X ðt Þ, t 2 ð0, T Þ,

ð1:50Þ

vk 2 V,

ð1:51Þ

Gj ðvk , xk ðt ÞÞ 2 U ðt Þ, t 2 ð0, T Þ,

ð1:52Þ

Theorem 1.4.7 gives the bound

12 jxk ðt Þj  jx0 j2 þ 2ηT eηT

8t 2 ½0, T :

ð1:53Þ

Since the set V is bounded, the Bolzano–Weierstrass theorem allows extracting a convergent subsequence of the sequence fðxk , vk Þg so that xk ðtÞ ! xðtÞ 8t 2 ½0, T,

ð1:54Þ

vk ! v:

ð1:55Þ

On the strength of the closedness of the set V and condition (1.51), the limiting value v satisfies inclusion (1.38). Then conditions (1.54) and (1.55) in combination with the continuity of the function Gj guarantee the convergence Gj ðvk , xk ðt ÞÞ ! Gj ðv, xðt ÞÞ, t 2 ð0, T Þ:

ð1:56Þ

Because the set U ðt Þ is closed, condition (1.52) leads to inclusion (1.39). On the other hand, since the function f is continuous, conditions (1.54) and (1.56) imply the convergence     f xk ðt Þ, Gj ðvk , xk ðt ÞÞ, aðt Þ ! f xðt Þ, Gj ðv, xðt ÞÞ, aðt Þ , t 2 ð0, T Þ: The function хk satisfies the integral relationship

ð1:57Þ

1.4 Some Properties of Solutions of Parametric Control Problems

39

ðt xk ðtÞ ¼ x0 þ f ðxk ðτÞ, Gj ðvk , xk ðτÞÞ, aðτÞÞdτ, t 2 ð0, TÞ:

ð1:58Þ

0

Passing to the limit on the strength of conditions (1.54) and (1.57) yields the equality ðt xðtÞ ¼ x0 þ f ðxðτÞ, Gj ðv, xðτÞÞ, aðτÞÞdτ, t 2 ð0, TÞ:

ð1:59Þ

0

Now, differentiate both sides of (1.59) with respect to t for showing (1.20). The initial condition (1.21) obviously holds under passing to the limit in equality (1.49) due to (1.54). Thus, the limiting pair ðx, vÞ is admissible. As the function F is continuous, condition (1.54) leads to the convergence F ½t, xk ðt Þ ! F ½t, xðt Þ t 2 ½0, T ,

ð1:60Þ

and consequently ðT

ðT F ½t, xk ðt Þdt ! F ½t, xðt Þdt:

0

ð1:61Þ

0

ÐT Expressions (1.61) and (1.47) give the equality F ½t, xðt Þdt ¼ supK: So the least 0

upper bound of the functional K is achieved at the pair ðx, uÞ 2 W ja over the set of all admissible pairs of the system. This completes the proof of Theorem 1.4.8. Clearly, Problem 1.4.3 is reduced to the maximization of the function Φa ð jÞ ¼ max K j over the finite set f1, . . . , r g with a given function а(). The right-hand

ðx, vÞ2W ja

side of this equality represents the maximum value of the functional with a given function а() and the jth control law. This value exists in accordance with Theorem 1.4.8. Recall that any function achieves maximum over a finite set. Thus the following result is immediate. Theorem 1.4.9 Under the hypotheses of Theorem 1.4.8, Problem 1.4.3 has a solution. Sufficient Conditions in Dynamic Optimization Problem on Design of Optimal Parametric Control Laws for Discrete-Time Dynamic Systems Consider a discrete-time controlled system of the form

40

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

xðt þ 1Þ ¼ f ðxðt Þ, uðt Þ, aðt ÞÞ,

t ¼ 0, 1, . . . , n  1,

xð0Þ ¼ x0 ,

ð1:62Þ ð1:63Þ

with the following notations: t as time; x ¼ xðt Þ ¼ ðx1 ðt Þ, . . . , xm ðt ÞÞ as a vector function of system state of discrete argument; u ¼ uðt Þ ¼ ðu1 ðt Þ, . . . , uq ðt ÞÞ as a vector function of control of discrete argument; a ¼ aðt Þ ¼ða1 ðt Þ, . . . , as ðt ÞÞ as a given vector function of discrete argument; x0 ¼ x10 , . . . , xm 0 as a given initial state of the system; and finally, f as a given vector function of appropriate arguments. The design problem of optimal economic tools consists in the minimization or maximization of a criterion K¼

n X

F ½t, xðt Þ ! max ð min Þ,

ð1:64Þ

t¼1

where F is a given function, subject to state-space constraints xðt Þ 2 X ðt Þ,

t ¼ 1, . . . , n,

ð1:65Þ

where X(t) is a given set and also subject to explicit control constraints uðt Þ 2 U ðt Þ, t ¼ 0, 1, . . . , n  1,

ð1:66Þ

where U(t) is a given set. A variational calculus problem on the design of optimal parametric control laws in the discrete-time dynamic system has the following statement. Problem 1.4.4 Given a function a (), find a control vector u() that satisfies condition (1.66) so that the corresponding solution of the dynamic system (1.62) and (1.63) satisfies condition (1.65) and also maximizes (minimizes) criterion (1.54). Denote by Vа the set of admissible “state–control” pairs of this system with a given function а. Actually, these are pairs of the vector functions (x, u) that satisfy relationships (1.62), (1.63), (1.65), and (1.66). Introduce the notations n n[ 1 [ U ðt Þ. X ¼ X ðt Þ and U ¼ t¼1

t¼0

The following result is true. Theorem 1.4.10 Let the set Va be non-empty for a given function а(). In addition, let the sets X(t) and U(t – 1) be closed and bounded for all t ¼ 1, . . ., n and the function f be continuous in the first two arguments over the set X  U, while the function F be continuous in the second argument over the set X. Then Problem 1.4.4 has a solution.

1.4 Some Properties of Solutions of Parametric Control Problems

41

The proof of this theorem is based on the property that any continuous function achieves minimum and maximum over a compact set. Sufficient Conditions in Dynamic Optimization Problem on Choice of Optimal Parametric Control Laws for Discrete-Time Dynamic Systems (from a Given Finite Collection of Algorithms) Consider the discrete-time controlled system (1.62) and (1.63). The state-space constraints have the form xðt Þ 2 X ðt Þ, t ¼ 1, . . . , n,

ð1:67Þ

where X ðt Þ is a given set. Let the control vector и be chosen from a collection of given control laws of the form uj ðt Þ ¼ Gj ðv, xðt ÞÞ, t ¼ 1, . . . , n, j ¼ 1, . . . , r,

ð1:68Þ

  Here Gj denotes a given vector function of its arguments, while v ¼ v1 , . . . , vl is the vector of adjustable parameters. The adjustable parameters are subjected to the constraints v 2 V,

ð1:69Þ

where V indicates some subset of space Rl . Moreover, by assumption the adjustable parameters must be such that the corresponding control law (1.68) satisfies the condition Gj ðv, xðt ÞÞ 2 U ðt Þ, t ¼ 0, . . . , n  1,

ð1:70Þ

where U ðt Þ is a given set. Introduce the optimality criteria K j ¼ K j ða, vÞ ¼

n X   F t, xj ðt Þ ,

ð1:71Þ

t¼1



where xj ¼ xj ðt Þ ¼ x1j ðt Þ, . . . , xm j ðt Þ is the solution of the Cauchy problems (1.62) and (1.63) with a given function а() and control vector u ¼ uj ðt Þ ¼

u1j ðt Þ, . . . , uqj ðt Þ , i.e., with the jth control law (1.68). Consider an auxiliary optimization problem as follows. Problem 1.4.5 Given a function а(), for each of r control laws, find the vector of adjustable parameters v so that the corresponding solution x ¼ xj of problem (1.62) and (1.63) with the control law u ¼ uj defined by (1.68) satisfies conditions (1.67), (1.69), and (1.70) and also maximizes functional (1.71).

42

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Formulate the following variational calculus problem on the choice of optimal parametric control laws for the nonautonomous discrete-time system from a given finite collection of algorithms. Problem 1.4.5 Given a function а(), among all optimal control laws in the sense of Problem 1.4.5, choose the one maximizing the optimality criterion (1.71). First, establish the solvability of Problem 1.4.5 with a fixed control law. Substituting the control law (1.68) into Eq. (1.62) yields   xðt þ 1Þ ¼ f xðt Þ, Gj ðv, xðt ÞÞ, aðt Þ , t ¼ 0, 1, . . . , n  1:

ð1:72Þ

(For the sake of compactness, here the subscript j is omitted for the component xj that corresponds to this control law.) Denote by W ja the set of admissible “state– control parameter” pairs of the system, i.e., the pairs ðx, vÞ that satisfy relationships (1.63) and (1.72) and also inclusions (1.67), (1.69), and (1.70).P Therefore, Problem   1.4.5 is reduced to the maximization of the functional K j ¼ nt¼1 F t, xj ðt Þ over the set W ja . n n[ 1 [ Let the sets Х and U be given by X ¼ X ðt Þ and U ¼ U ðt Þ. The following t¼1

t¼0

result is the case. Theorem 1.4.11 Let the set W ja be non-empty for a given function а() and a given value j 2 f1, . . . , r g. In addition, let the sets V, X ðt Þ, and U ðt  1Þ be compact and bounded for all t ¼ 1, . . . , n, the function f be continuous in the first two arguments over the set X  U, the function Gj be continuous over the set V  X, and the function F be continuous in the second argument over the set X. Then Problem 1.4.5 has a solution. The proof of this theorem is based on the property that any continuous function achieves minimum and maximum over a compact set. As mentioned earlier, any function achieves maximum over a finite set. Thus the following result is immediate. Theorem 1.4.12 Under the hypotheses of Theorem 1.4.11, Problem 1.4.5 has a solution. Sufficient Conditions in Dynamic Optimization Problem on Design of Optimal Parametric Control Laws for Discrete-Time Stochastic Dynamic Systems Consider a discrete-time stochastic controlled system of the form xðt þ 1Þ ¼ f ðxðt Þ, uðt Þ, aÞ þ ξðt Þ, t ¼ 0, 1, . . . , n  1,

ð1:73Þ

x ð 0Þ ¼ x 0 :

ð1:74Þ

with the following notations: t as time; x ¼ xðt Þ 2 Rm as the state of system (1.73) and (1.74), a random   vector function of discrete argument (a vector random process); x0 ¼ x10 , . . . , xm 0 as a given initial state of the system, a deterministic vector; u ¼ uðt Þ 2 Rq as the vector of controlled parameters, a vector function of discrete

1.4 Some Properties of Solutions of Parametric Control Problems

43

s argument; a 2 Rs as the vector  1 of uncontrolled  parameters, where a 2 A and A ⊂ R m are a given set; ξ ¼ ξðt Þ ¼ ξ ðt Þ, . . . , ξ ðt Þ as a given vector random process that describes disturbances (e.g., Gaussian noise); and finally, f as a given vector function of appropriate arguments.

An optimality criterion to be maximized has the form K¼E

nXn

o F ð x ð t Þ Þ , t t¼1

ð1:75Þ

where F t are given functions, subject to state-space constraints E½xðt Þ 2 X ðt Þ,

t ¼ 1, . . . , n,

ð1:76Þ

where X ðt Þ is a given set. The problems below also include explicit control constraints uðt Þ 2 U ðt Þ,

t ¼ 0, . . . , n  1,

ð1:77Þ

where U ðt Þ ⊂ Rq is a given set. A variational calculus problem on the design of optimal parametric control laws in the discrete-time stochastic dynamic system has the following statement. Problem 1.4.6 Given the vector of uncontrolled parameter a 2 A, find a parametric control law u() that satisfies condition (1.77) so that the corresponding solution of the dynamic system (1.73) and (1.74) satisfies condition (1.76) and also maximizes criterion (1.75). Denote by U ad the set of admissible control laws of this system––the aggregate of all control laws uðt Þ satisfying constraint (1.77) for which the expectation E½xðt Þ of the corresponding solution of the system satisfies inclusion (1.75). The following result is the case. Theorem 1.4.13 Let the random variables ξðt Þ be absolutely continuous and have zero means for each a 2 A and any t ¼ 1, . . . , n, and let the sets X ðt Þ and U ðt Þ be closed and bounded for all t: In addition, let the function f satisfy the Lipschitz condition and the functions F t be Lipschitz continuous, and let the absolute values of f (for u 2 U and a 2 A) and F t not exceed some linear functions in |x|. Then, if the set U ad is non-empty, Problem 1.4.6 has a solution. Proof By the Weierstrass extreme value theorem, a continuous function has maximum on a non-empty, close, and bounded set. Thus, it suffices to show that the multivariate function K ¼ K ðuÞ defined by (1.75) is continuous and the set U ad is close and bounded. Recall that this set is non-empty in accordance with the hypotheses of Theorem 1.4.13. Demonstrate that there exist the expectations of the values appearing in the statespace constraint (1.73). Indeed, from Eq. (1.73) it follows that

44

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

E½xðt þ 1Þ ¼ E½f ðxðt Þ, uðt Þ, λÞ þ E½ξðt Þ:

ð1:78Þ

The second term in the right-hand side of this equation makes sense by the hypotheses of the theorem, while the first one is calculated by the formula ð E½f ðxðt Þ, uðt Þ, λÞ ¼

Rn

f ðω, uðt Þ, λÞpxðtÞ ðωÞdω,

ð1:79Þ

where pxðtÞ denotes the probability density function of the random variable xðt Þ. This formula can be used if the integral has absolute convergence. Such convergence is guaranteed by the growth constraints imposed on the function f and the existence of the expectation of xðt Þ for any t ¼ 1, . . . , n (this fact is easily established by induction). The expectation in the right-hand side of equality (1.75) is well-defined on the strength of the growth constraints on the function F t and the existence of the expectation of xðt Þ . Let the convergence uk ! u , uk 2 U ad take place. From Eq. (1.73) it follows that jxk ðt þ 1Þ  xðt þ 1Þj ¼ jf ðxk ðt Þ, uk ðt Þ, λÞ  f ðxðt Þ, uðt Þ, λÞj,

ð1:80Þ

where xk and x are the solutions of problem (1.73) and (1.74) with the control laws uk and u, respectively. Then the following relationship holds: jxk ðt þ 1Þ  xðt þ 1Þj  Lf ½jxk ðt Þ  xðt Þj þ juk ðt Þ  uðt Þj,

ð1:81Þ

where Lf is the Lipschitz constant of the function f . The same line of reasoning together with xk ð0Þ ¼ xð0Þ (see the initial condition (1.74)) yields  2 jxk ðt þ 1Þ  xðt þ 1Þj  Lf jxk ðt  1Þ  xðt  1Þj  2 þ Lf juk ðt  1Þ  uðt  1Þj þ Lf juk ðt Þ  uðt Þj Xt  sþ1 L  juk ðt  sÞ  uðt  sÞj  εk , s¼0 f

ð1:82Þ

where εk ! 0 as k ! 1. Denote by LF the maximum of the Lipschitz constants of the functions F t ; for t ¼ 1, . . . , n, j F t ½xk ðtÞ  F t ½xðtÞ j LF εk :

ð1:83Þ

Taking the expectations of both parts of this inequality gives EfjF t ½xk ðt Þ  F t ½xðt Þjg  LF εk :

ð1:84Þ

1.4 Some Properties of Solutions of Parametric Control Problems

45

Consequently, EfjF t ½xk ðt Þ  F t ½xðt Þjg ! 0 , and the sequence under study is convergent, EfF t ½xk ðt Þg ! EfF t ½xðt Þg. On the strength of (1.75), this means that the function K ¼ K ðuÞ is continuous. The boundedness of the set U ad follows from the boundedness U(t). Next, the closedness of the set U ad follows from the continuity of the mapping U ad ! X and the compactness of the set X (by a well-known theorem, the complete preimage of a compact set under a continuous mapping is closed). Then the solution of Problem 1.4.6 follows from the Weierstrass extreme value theorem. This completes the proof of Theorem 1.4.13. Sufficient Conditions in Dynamic Optimization Problem on Choice of Optimal Parametric Control Laws for Discrete-Time Stochastic Dynamic Systems (from a Given Finite Collection of Algorithms) The next parametric control problem also involves the discrete-time stochastic dynamic control system described by Eqs. (1.73) and (1.74) under the state-space constraints (1.76). In this problem, the control vector is chosen from a collection of given control laws of the form uj ðtÞ ¼ Gj ðv, xðtÞÞ, t ¼ 0, . . . , n  1; j ¼ 1, . . . , r,

ð1:85Þ

  where Gj is a given vector function of its arguments and v ¼ v1 , . . . , vl denotes the vector of adjustable parameters of the control law Gj . The adjustable parameters v are subjected to constraints v 2 V,

ð1:86Þ

where V indicates some subset of space Rl . Moreover, by assumption the adjustable parameters must be such that the corresponding control law (1.85) satisfies condition (1.77), i.e., h

i E Gj v, xvj ðt Þ 2 U ðt Þ,

t ¼ 0, . . . , n  1:

ð1:87Þ

Here xvj is the solution of problems (1.73) and (1.74) with the chosen values of the adjustable parameter v, uncontrolled parameter a, and the jth control law (1.85). Introduce the optimality criteria nXn

o v K j ¼ K j ðv, aÞ ¼ E F x ð t Þ t j t¼1

ð1:88Þ

Formulate the following variational calculus problem on the choice of optimal parametric control laws for the discrete-time stochastic dynamic system from a given finite collection of algorithms.

46

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Problem 1.4.7 Given the vector of uncontrolled parameter a 2 A, for each of r control laws, find the vector of adjustable parameters v so that the corresponding solution x ¼ xj of problems (1.73) and (1.74) with the control law u ¼ uj defined by 1.85) satisfies conditions (1.86) and (1.87) and also maximizes functional (1.88), with further choice of the best control law in terms of the optimality criterion (1.88). Now, establish sufficient conditions for the existence of a solution of Problem 1.4.7. Denote by xvj the solution of system (1.73) and (1.74) with the chosen j-th parametric control law (1.85), its adjustable parameter v, and the uncontrolled parameter α:



xvj ðt þ 1Þ ¼ f xvj ðt Þ, Gj v, xvj ðt Þ , a þ ξðt Þ,

t ¼ 0, 1, . . . , n  1,

xvj ð0Þ ¼ x0 :

ð1:89Þ ð1:90Þ

Define the admissible set V jad of the adjustable parameters as the set of all values v 2 V satisfying condition (1.86) for which the corresponding solution of problems (1.89) and (1.90) satisfies the inclusions: h

i E Gj v, xvj ðt Þ 2 U ðt Þ; h i E xvj ðt Þ 2 X ðt Þ;

t ¼ 0, 1, . . . , n  1;

ð1:91Þ

t ¼ 1, . . . , n:

ð1:92Þ

Problem 1.4.7 is said to be nontrivial if, for each j ¼ 1, . . . , r, the corresponding set V jad is non-empty and contains some open set. Theorem 1.4.14 Let the random variables ξðt Þ be absolutely continuous and have zero means for each a 2 A and any t ¼ 1, . . . , n, and let the sets U ðt Þ, X ðt Þ, and V be compact for all t: In addition, let the functions f , Gj , and F t satisfy the Lipschitz   condition, and let the functions j f x, Gj ðv, xÞ, λ j and j F t ðxÞ j not exceed some linear functions in |x| uniformly in v 2 V . Then, if the sets V jad are non-empty, Problem 1.4.7 has a solution. Proof It suffices to show that all functions K j (1.88) are continuous, while all sets V jad are closed and bounded, where j ¼ 1, . . . , r. The existence of all expectations below can be argued similar to the proof of Theorem 1.4.13. Taking into account the additivity of expectation, find the values K j ¼ K j ð vÞ ¼ Hence, for any v, w 2 V jad ,

n h io v E F x ð t Þ : t j t¼1

Xn

ð1:93Þ

1.4 Some Properties of Solutions of Parametric Control Problems

47

Xn Xn jK j ðvÞ  K j ðwÞj ¼ j t¼1 EfF t ½xvj ðtÞg  EfF t ½xwj ðtÞgj t¼1 Xn  jEfF t ½xvj ðtÞg  EfF t ½xwj ðtÞgj: t¼1

ð1:94Þ

On the other hand, relationships Eqs. (1.89) and (1.90) imply









v



xj ðt þ 1Þ  xwj ðt þ 1Þ ¼ f xvj ðt Þ, Gj v, xvj ðt Þ , λ  f xwj ðt Þ, Gj w, xwj ðt Þ , λ

h



i



 Lf xvj ðt Þ  xwj ðt Þ þ Gj v, xvj ðt Þ  Gj w, xwj ðt Þ , t ¼ 0, 1, . . . , n  1 ð1:95Þ where Lf is the Lipschitz constant of the function f . Denote by LG the maximum of the Lipschitz constants of the function Gj . Then it is possible to write



v



xj ðt þ 1Þ  xwj ðt þ 1Þ  Lf ð1 þ LG Þ xvj ðt Þ  xwj ðt Þ þ Lf LG jv  wj, t ¼ 0, 1, . . . , n  1:

ð1:96Þ



On the strength of the inequality xvj ð0Þ  xwj ð0Þ ¼ 0, this leads to the bound

Xt  l

v

xj ðt þ 1Þ  xwj ðt þ 1Þ  Lf LG l¼0 Lf ð1 þ LG Þ jv  wj  βjv  wj8v, w 2 V jad ,

ð1:97Þ

where β ¼ Lf LG

Xt  l L ð 1 þ LG Þ : l¼0 f

ð1:98Þ

Denote by LF the maximum of the Lipschitz constants of the function F t , and by analogy write jF t ½xvj ðtÞ  F t ½xwj ðtÞj  LF jxvj ðtÞ  xwj ðtÞj  LF βjv  wj

8v, w 2 V jad :

ð1:99Þ

Thus, for a sufficiently small difference of h i the adjustable h i parameters v and w, the v w v w values xj ðt Þ and xj ðt Þ (as well as F t xj ðt Þ and F t xj ðt Þ ) are arbitrarily close to each other. Define the convergent sequence w ¼ vk ! v. Then, taking the expectations of the left- and right-hand sides of inequality (1.99), obtain the inequality

48

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

h h i h i i

E F t xvj ðt Þ  F t xvj k ðt Þ  LF βjv  vk j:

ð1:100Þ

This proves the convergence n h io n h io E F t xvj k ðt Þ ! E F t xvj ðt Þ ,

ð1:101Þ

and in turn the continuity of the function K j ðvÞ. Since V jad ⊂ V, all sets V jad are bounded. The closedness of the sets V jad follows from the h closeness i of the sets h U ðt Þ and Xiðt þ 1Þ, the continuity of the mappings v v ! E xj ðt þ 1Þ and v ! E Gj v, xvj ðt ÞÞ (see above), and the definition of the set V jad as the complete preimage of these sets under continuous mappings. The desired result is immediate on using the Weierstrass extreme value theorem.

1.4.2

Sufficient Conditions for a Continuous Dependence of Optimal Criteria Values on Uncontrolled Functions in Optimal Parametric Control Design and Choice Problems

For developing Component 6 of the theory of parametric control, this subsection establishes sufficient conditions for a continuous dependence of optimal criterion values on uncontrolled functions а() in the parametric control problems for the nonautonomous deterministic dynamic systems and also sufficient conditions for a continuous dependence of optimal criterion values on uncontrolled parameters in the parametric control problems for the autonomous stochastic dynamic systems. All the results obtained for the nonautonomous systems remain in force for their autonomous counterparts with a constant exogenous vector function a(∙). The next theorem provides sufficient conditions for a continuous dependence of optimal criterion values in Problem 1.4.4. Theorem 1.4.15 Under the hypotheses of Theorem 1.4.10, in a neighborhood (Euclidean topology) of the function а(), let the function f be continuous in the third argument and satisfy the Lipschitz condition in the first argument over the set Х uniformly in the second and third arguments. Then the optimal criterion value in Problem 1.4.4 has a continuous dependence on the uncontrolled function at the point а(). Proof Assume the convergence ak ! a

ð1:102Þ

1.4 Some Properties of Solutions of Parametric Control Problems

49

holds in space Rsn. In accordance with Theorem 1.4.10, Problem 1.4.4 with uncontrolled functions а and ak has some solutions и and иk, respectively. Denote by x½b, w the solution of the state-space Eqs. (1.62) and (1.63) that correspond to an uncontrolled function b and a control parameter w. So the function x½b, w satisfies the relationships x½b, wðt þ 1Þ ¼ f ðx½b, wðt Þ, wðt Þ, bðt ÞÞ, t ¼ 0, 1, . . . , n  1, x½b, wð0Þ ¼ x0 :

ð1:103Þ ð1:104Þ

Consequently, 0  К ðx½а, uk Þ  К ðx½а, uÞ, К ðx½аk , uk Þ  К ðx½аk , uÞ  0,

ð1:105Þ

where K ðyÞ is the value of the optimality criterion K with a given system state у. As a result, 0  К ðx½а, uÞ  К ðx½а, uk Þ  fК ðx½а, uÞ  К ðx½аk , uÞg þ fК ðx½аk , uÞ  К ðx½аk , uk Þg þfК ðx½аk , uk Þ  К ðx½а, uk Þg  2sup jК ðx½а, wÞ  К ðx½аk , wÞj: w2U

ð1:106Þ Using formulas (1.103), derive the following chain of upper bounds: jx½ak , wðt þ 1Þ  x½a, wðt þ 1Þj ¼ jf ðx½ak , wðtÞ, wðtÞ, ak ðtÞÞ  f ðx½a, wðtÞ, wðtÞ, aðtÞÞj  jf ðx½ak , wðtÞ, wðtÞ, ak ðtÞÞ  f ðx½ak , wðtÞ, wðtÞ, aðtÞÞj 

þ jf ðx½ak , wðtÞ, wðtÞ, aðtÞÞ  f ðx½a, wðtÞ, wðtÞ, aðtÞÞj sup jf ðy, φ, ak ðtÞÞ  f ðy, φ, aðtÞÞj þ Ljx½ak , wðtÞ  x½a, wðtÞj, t ¼ 0, 1, . . . , n  1,

y2X, φ2U

ð1:107Þ where L is the Lipschitz constant of the function f in the first argument that does not depend on w. This gives the inequality jψ k ðt þ 1Þj  ηk þ Ljψ k ðt Þj, t ¼ 0, 1, . . . , n  1,

ð1:108Þ

where ψ k ðtÞ ¼ x½ak , wðtÞ  x½a, wðtÞ, t ¼ 0, 1, . . . , n  1, ηk ¼ max sup jf ðy, φ, ak ðtÞÞ  f ðy, φ, aðtÞÞj: t¼0, 1, ..., n1y2X, φ2U

Next, establish the relationships

ð1:109Þ

50

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

jψ k ðt þ 1Þj  ηk þ Ljψ k ðtÞj  ηk þ Lðηk þ Ljψ k ðt  1ÞjÞ ¼ ð1 þ LÞηk þ L2 jψ k ðt  1Þj    ð1 þ LÞηk þ L2 ðηk þ Ljψ k ðt  2ÞjÞ ¼ 1 þ L þ L2 ηk þ L3 jψ k ðt  2Þj  . . . t X  Lr ηk þ Lrþ1 jψ k ð0Þj, t ¼ 0, 1, . . . , n  1: r¼0

ð1:110Þ From the initial condition (1.104), it follows that jψ k ð0Þj ¼ 0: Then j ψ k ð t þ 1Þ j  η k

n X

Lr , t ¼ 0, 1, . . . , n  1:

ð1:111Þ

r¼0

Hence, jx½ak , wðt þ 1Þ  x½a, wðt þ 1Þj  ηk

n X

Lr , t ¼ 0, 1, . . . , n  1,

ð1:112Þ

r¼0

and the right-hand side of this inequality is independent of w. Because the function f is continuous in the third argument, while the sets X and U are closed and bounded, condition (1.102) implies the convergence ηk ! 0: Then inequality (1.112) guarantees the convergence x½ak , wðtÞ ! x½a, wðtÞ uniformly in w, t ¼ 1, . . . , n

ð1:113Þ

Since the function F is continuous, F½t, x½ak , wðtÞ ! F½t, x½a, wðtÞ uniformly in w, t ¼ 1, . . . , n

ð1:114Þ

This means that К ðx½аk , wÞ ! К ðx½а, wÞ uniformly in w,

ð1:115Þ

sup jК ðx½а, wÞ  К ðx½аk , wÞj ! 0:

ð1:116Þ

and consequently

w2U

Passing to the limit in inequality (1.106) together with condition (1.116) yields the convergence К ðx½а, uk Þ ! К ðx½а, uÞ: The following bound is the case:

ð1:117Þ

1.4 Some Properties of Solutions of Parametric Control Problems

51

jК ðx½аk , uk Þ  К ðx½а, uÞj  jК ðx½аk , uk Þ  К ðx½а, uk Þj þ jК ðx½а, uk Þ  К ðx½а, uÞj:

ð1:118Þ Passing to the limit in this inequality together with conditions (1.116) and (1.117) proves the convergence К ðx½аk , uk Þ ! К ðx½а, uÞ:

ð1:119Þ

By the definitions of the control parameters и and иk, the maximum value of the criterion K corresponding to the uncontrolled function ak converges to its maximum corresponding to the limiting value а of the uncontrolled function. This completes the proof of Theorem 1.4.15. To proceed, establish sufficient conditions for a continuous dependence of optimal criterion values on uncontrolled functions in Problem 1.4.5 (choice of optimal parametric control laws from a given finite collection of algorithms) for the nonautonomous discrete-time dynamic system. First, prove a corresponding result for Problem 1.4.5. Theorem 1.4.16 Under the hypotheses of Theorem 1.4.11, in a neighborhood of a point а let the function f be continuous in the third argument and satisfy the Lipschitz condition in the first two arguments over the set X  U uniformly in the third argument. In addition, let the function Gj satisfy the Lipschitz condition in the second argument over the set Х uniformly in the first argument. Then the optimal criterion value in Problem 1.4.5 has a continuous dependence on the uncontrolled function at the point а. Proof Assume the convergence ak ! a

ð1:120Þ

holds in space Rsn. In accordance with Theorem 1.4.11, Problem 1.4.5 with uncontrolled functions а and ak has some solutions v and vk, respectively. Denote by x½b, w the solution of the state-space Eqs. (1.72) and (1.73) that correspond to an uncontrolled function b(∙) and a control parameter w. So the function x½b, w satisfies the relationships   x½b, wðt þ 1Þ ¼ f x½b, wðt Þ, Gj ðv, x½b, wðt ÞÞ, bðt Þ , t ¼ 0, 1, . . . , n  1, ð1:121Þ x½b, wð0Þ ¼ x0 :

ð1:122Þ

By analogy with inequality (1.106), 0  К ðx½а, vÞ  К ðx½а, vk Þ  2supjК ðx½а, wÞ  К ðx½аk , wÞj: w2V

Denoting

ð1:123Þ

52

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

x ¼ x½a, w, xk ¼ x½ak , w,

ð1:124Þ

from relationship (1.121), derive the following chain of upper bounds: jxk ðt þ 1Þ  xðt þ 1Þj ¼ jf ðxk ðtÞ, Gj ðw, xk ðtÞÞ, ak ðtÞÞ  f ðxðtÞ, Gj ðw, xðtÞÞ, aðtÞÞj  jf ðxk ðtÞ, Gj ðw, xk ðtÞÞ, ak ðtÞÞ  f ðxk ðtÞ, Gj ðw, xk ðtÞÞ, aðtÞÞj þ jf ðxk ðtÞ, Gj ðw, xk ðtÞÞ, aðtÞÞ  f ðxðtÞ, Gj ðw, xðtÞÞ, aðtÞÞj 

sup jf ðy, φ, ak ðtÞÞ  f ðy, φ, aðtÞÞj þ L½jxk ðtÞ  xðtÞj þ jGj ðw, xk ðtÞÞ  Gj ðw, xðtÞÞj

y2X, φ2U



sup jf ðy, φ, ak ðtÞÞ  f ðy, φ, aðtÞÞj þ Lð1 þ MÞjxk ðtÞ  xðtÞj, t ¼ 0, 1, . . . , n  1,

y2X, φ2U

ð1:125Þ where L is the Lipschitz constant of the function f in the first two arguments, while М the Lipschitz constant of the function Gj in the second argument. This gives the inequality jψ k ðt þ 1Þj ηk þ N jψ k ðt Þj, t ¼ 0, 1, . . . , n  1,

ð1:126Þ

N ¼ Lð1 þ MÞ, ψ k ðtÞ ¼ xk ðtÞ  xðtÞ, t ¼ 0, 1, . . . , n  1,

ð1:127Þ

where

ηk ¼

max

sup jf ðy, φ, ak ðtÞÞ  f ðy, φ, aðtÞÞj:

t¼0, 1, ..., n1y2X, φ2U

The same technique as in the proof of Theorem 1.4.13 allows to obtain the upper bound jψ k ðt þ 1Þj  ηk

n X

N r , t ¼ 0, 1, . . . , n  1,

ð1:128Þ

r¼0

and also the convergence ηk ! 0: Therefore, jψ k ðt Þj ! 0, t ¼ 1, . . . , n, and hence x½ak , wðtÞ ! x½a, wðtÞ uniformly in w, t ¼ 1, . . . , n

ð1:129Þ

Since the function F is continuous, F½t, x½ak , wðtÞ ! F½t, x½a, wðtÞ uniformly in w, t ¼ 1, . . . , n This means that

ð1:130Þ

1.4 Some Properties of Solutions of Parametric Control Problems

К ðx½аk , wÞ ! К ðx½а, wÞ uniformly in w:

53

ð1:131Þ

Because the control space is finite-dimensional and the set U is bounded, supjК ðx½а, wÞ  К ðx½аk , wÞj ! 0:

ð1:132Þ

w2V

Passing to the limit in inequality (1.123) together with condition (1.132) yields the convergence К ðx½а, vk Þ ! К ðx½а, vÞ:

ð1:133Þ

The following upper bound is the case: jК ðx½аk , vk Þ  К ðx½а, vÞj  jК ðx½аk , vk Þ  К ðx½а, vk Þj þ jК ðx½а, vk Þ  К ðx½а, vÞj:

ð1:134Þ Passing to the limit in this inequality together with conditions (1.132) and (1.133) proves the convergence К ðx½аk , vk Þ ! К ðx½а, vÞ:

ð1:135Þ

By the definitions of the control parameters v and vk, the maximum value of the criterion K corresponding to the uncontrolled function ak converges to its maximum corresponding to the limiting value а of the uncontrolled function. This completes the proof of Theorem 1.4.16. Recall that the maximum of two (and hence any finite number of) continuous functions is continuous. Thus Theorem 1.4.16 produces a similar result for Problem 1.4.5. Theorem 1.4.17 Under the hypotheses of Theorem 1.4.11, in a neighborhood of a point а, let the function f be continuous in the third argument and satisfy the Lipschitz condition in the first two arguments over the set X  U uniformly in the third argument. In addition, let the function Gj satisfy the Lipschitz condition in the second argument over the set Х uniformly in the first argument. Then the optimal criterion value in Problem 1.4.5 has a continuous dependence on the uncontrolled function at the point а. Next, establish sufficient conditions for a continuous dependence of optimal criterion values on uncontrolled functions in Problem 1.4.1 for the nonautonomous continuous-time dynamic system. Theorem 1.4.18 Under the hypotheses of Theorem 1.4.6, in a neighborhood of a point а, let the function f be continuous in the second argument and satisfy the Lipschitz condition in the first and third arguments over the set X  A uniformly in

54

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

the second argument. Then the optimal criterion value in Problem 1.4.1 has a continuous dependence on the uncontrolled function at the point а. Proof Assume the convergence ak ! a

ð1:136Þ

holds in space (C[0, T])s. In accordance with Theorem 1.4.6, Problem 1.4.1 with uncontrolled functions а and ak has some solutions и and иk, respectively. Denote by x½b, w the solution of the state-space Eqs. (1.20) and (1.21) that correspond to an uncontrolled function b and a control parameter w. So the function x½b, w satisfies the relationships x_ ½b, wðtÞ ¼ f ðx½b, w, wðtÞ, bðtÞÞ, t 2 ð0, TÞ,

ð1:137Þ

x½b, wð0Þ ¼ x0 :

ð1:138Þ

By analogy with (1.106), the same considerations as in the proof of Theorem 1.4.13 give the inequality 0  К ðx½а, uÞ  К ðx½а, uk Þ  2sup jК ðx½а, wÞ  К ðx½аk , wÞj:

ð1:139Þ

w2U

Denoting x ¼ x½a, w, xk ¼ x½ak , w, from conditions (1.137), obtain x_ k ðtÞ  x_ ðtÞ ¼ f ðxk ðtÞ, wðtÞ, ak ðtÞÞ  f ðxðtÞ, wðtÞ, aðtÞÞ, t 2 ð0, TÞ:

ð1:140Þ

Integrate this relationship over t with (1.138) to get the following chain of upper bounds: ðt jxk ðtÞ  xðtÞj  jf ðxk ðτÞ, wðτÞ, ak ðτÞÞ  f ðxðτÞ, wðτÞ, aðτÞÞjdτ 0

ðt

ðt

ðt

 L jxk ðτÞ  xðτÞjdτ þ L jak ðτÞ  aðτÞjdτ  L jxk ðτÞ  xðτÞjdτ 0

0

ð1:141Þ

0

þ LTkak  akΘ , t 2 ð0, TÞ, where L is the Lipschitz constant of the function f that does not depend on w. Using Grönwall’s inequality, it follows that jxk ðtÞ  xðtÞj  ckak  akΘ , t 2 ð0, TÞ, where a positive constant с depends on L and Т only. Due to condition (1.136), this leads to the convergence

ð1:142Þ

1.4 Some Properties of Solutions of Parametric Control Problems

xk ðt Þ ! xðt Þ, t 2 ð0, T Þ,

55

ð1:143Þ

and hence x½ak , wðtÞ ! x½a, wðtÞ uniformly in w, t 2 ð0, TÞ:

ð1:144Þ

Since the function F is continuous, F½t, x½ak , wðtÞ ! F½t, x½a, wðtÞ uniformly in w, t 2 ð0, TÞ:

ð1:145Þ

This means that К ðx½аk , wÞ ! К ðx½а, wÞ uniformly in w:

ð1:146Þ

sup jК ðx½а, wÞ  К ðx½аk , wÞj ! 0:

ð1:147Þ

As a result,

w2U

Passing to the limit in inequality (1.139) together with condition (1.147) yields the convergence К ðx½а, uk Þ ! К ðx½а, uÞ:

ð1:148Þ

The following upper bound is the case: jК ðx½аk , uk Þ  К ðx½а, uÞj  jК ðx½аk , uk Þ  К ðx½а, uk Þj þ jК ðx½а, uk Þ  К ðx½а, uÞj:

ð1:149Þ Passing to the limit in inequality (1.149) together with conditions (1.147) and (1.148) yields the convergence К ðx½аk , uk Þ ! К ðx½а, uÞ:

ð1:150Þ

By the definitions of the control parameters и and иk, the maximum value of the criterion K corresponding to the uncontrolled function ak converges to its maximum corresponding to the limiting value а of the uncontrolled function. This completes the proof of Theorem 1.4.18. Finally, present sufficient conditions for a continuous dependence of optimal criterion values on uncontrolled functions in Problem 1.4.3 (choice of optimal parametric control laws from a given finite collection of algorithms) for the nonautonomous continuous-time dynamic system. Theorem 1.4.19 Under the hypotheses of Theorem 1.4.8, in a neighborhood of a point а, let the function f satisfy the Lipschitz condition over the set X  U  A. In

56

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

addition, let the function Gj satisfy the Lipschitz condition in the second argument over Х uniformly in the first argument. Then the optimal criterion value in Problem 1.4.3 has a continuous dependence on the uncontrolled function at the point а. Proof Assume the convergence ak ! a

ð1:151Þ

holds in space (C[0, T])s. In accordance with Theorem 1.4.8, Problem 1.4.3 with uncontrolled functions а and ak has some solutions v and vk, respectively. Denote by x½b, w the solution of the state-space Eqs. (1.42) and (1.21) that correspond to an uncontrolled function b and a control parameter w. So the function x½b, w satisfies the relationships   x_ ½b, wðt Þ ¼ f x½b, wðt Þ, Gj ðw, xðt ÞÞ, bðt Þ , t 2 ð0, T Þ, x½b, wð0Þ ¼ x0 :

ð1:152Þ ð1:153Þ

By analogy with (1.106), the same considerations as in the proof of Theorem 1.4.15 give the inequality 0  К ðx½а, uÞ  К ðx½а, uk Þ  2supjК ðx½а, wÞ  К ðx½аk , wÞj:

ð1:154Þ

w2V

Denoting x ¼ x½a, w, xk ¼ x½ak , w, from condition (1.15), obtain x_ k ðtÞ  x_ ðtÞ ¼ f ðxk ðtÞ, Gj ðw, xk ðtÞÞ, ak ðtÞÞ  f ðxðtÞ, Gj ðw, xðtÞÞ, aðtÞÞ, t 2 ð0, TÞ:

ð1:155Þ

Integrate this relationship over t with (1.153) to get the following chain of upper bounds: ðt jxk ðtÞ  xðtÞj  j f ðxk ðτÞ, Gj ðw, xk ðτÞÞ, ak ðτÞÞ  f ðxðτÞ, Gj ðw, xðτÞÞ, aðτÞÞjdτ 0

ðt  L ½jxk ðτÞ  xðτÞj þ jGj ðw, xk ðτÞÞ  Gj ðw, xðτÞÞj þ jak ðτÞ  aðτÞjdτ 0

ðt  Lð1 þ MÞ jxk ðτÞ  xðτÞjdτ þ LTkak  akΘ , t 2 ð0, TÞ, 0

ð1:156Þ

1.4 Some Properties of Solutions of Parametric Control Problems

57

where L is the Lipschitz constant of the function f, while М the Lipschitz constant of the function Gj in the second argument. Using Grönwall’s inequality, it follows that jxk ðtÞ  xðtÞj  ckak  akΘ , t 2 ð0, TÞ,

ð1:157Þ

where a positive constant с depends on L and Т only. Due to condition (1.151), this leads to the convergence x½ak , wðtÞ ! x½a, wðtÞ uniformly in w, t 2 ð0, TÞ:

ð1:158Þ

Since the function F is continuous, F½t, x½ak , wðtÞ ! F½t, x½a, wðtÞ uniformly in w, t 2 ð0, TÞ,

ð1:159Þ

and hence К ðx½аk , wÞ ! К ðx½а, wÞ uniformly in w:

ð1:160Þ

supjК ðx½а, wÞ  К ðx½аk , wÞj ! 0:

ð1:161Þ

As a result,

w2V

Passing to the limit in inequality (1.154) together with condition (1.161) yields the convergence К ðx½а, uk Þ ! К ðx½а, uÞ:

ð1:162Þ

The following upper bound is the case: jК ðx½аk , uk Þ  К ðx½а, uÞj  jК ðx½аk , uk Þ  К ðx½а, uk Þj þ jК ðx½а, uk Þ  К ðx½а, uÞj:

ð1:163Þ

Converging to the limit in inequality (1.163) together with conditions (1.161) and (1.162) yields the convergence К ðx½аk , uk Þ ! К ðx½а, uÞ:

ð1:164Þ

By the definitions of the control parameters и and иk, the maximum value of the criterion K corresponding to the uncontrolled function ak converges to its maximum corresponding to the limiting value а of the uncontrolled function. This completes the proof of Theorem 1.4.19. By analogy with Theorems 1.4.17 and 1.4.19 can be used for obtaining the following result. (Note that here system continuity becomes insignificant.)

58

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Theorem 1.4.20 Let the hypotheses of Theorem 1.4.19 hold for all j ¼ 1, . . . , r . Then the optimal criterion value in Problem 1.4.3 has a continuous dependence on the uncontrolled function at the point а. Concluding this subsection, consider theorems on a continuous dependence of optimal criterion values on uncontrolled parameters in parametric control problems for stochastic dynamic systems. The proofs of these theorems (with appropriate simplifications dictated by constant functions a) are similar to those of Theorems 1.4.15, 1.4.16, and 1.4.17, respectively. Theorem 1.4.21 Let the hypotheses of Theorem 1.4.13 hold for any a 2 A: Then the optimal criterion value in Problem 1.4.6 is a continuous function of the parameter a 2 A. Theorem 1.4.22 Let the hypotheses of Theorem 1.4.14 hold for any a 2 A: Then, for a chosen number j of the parametric control law, the optimal criterion value K j in Problem 1.4.7 is a continuous function of the parameter a 2 A. Corollary 1.4.23 Let the hypotheses of Theorem 1.4.22 hold for all j ¼ 1, . . . , r. Then the optimal criterion value K ¼ max K j in Problem 1.4.7 is a continuous function of the parameter a 2 A.

j¼1, ..., r

The results of this subsection will be employed for proving the existence of bifurcation points for the extremals of dynamic optimization problems; see the next subsection.

1.4.3

Sufficient Conditions for the Existence of Bifurcation Points for Extremals of Optimal Parametric Control Choice Problems

Introduce the concept of a bifurcation point for the extremals of the variational calculus problem on the choice of optimal parametric control laws from a given finite collection of algorithms. If there exists a bifurcation point for the extremals of this problem with some uncontrolled function а(), then in its neighborhood one optimal parametric control law is replaced by another. Consider an abstract variational calculus problem on the choice of optimal parametric control laws from a given finite collection of algorithms that generalizes Problems 1.4.3, 1.4.5, and 1.4.7. The given data are the following: А as the set of uncontrolled functions (or parameters); V ja , a 2 A, j ¼ 1, . . . , r, as the family of admissible control laws (values of adjustable parameters); K j ¼ K j ða, vÞ, where v 2 V ja , a 2 A, j ¼ 1, . . . , r, as the family of goal functionals (optimality criteria). Define a bifurcation point for the extremals of a family of maximization problems with given functionals over the corresponding sets of admissible control laws, i.e., the abstract variational

1.4 Some Properties of Solutions of Parametric Control Problems

59

calculus problem on the choice of optimal parametric control laws from a given finite collection of algorithms. Definition A value a 2 A is said to be a bifurcation point for the extremals of the maximization problem for mappings v ! K k ða, vÞ over the sets V ka , k ¼ 1, . . . , r, if there exist two different numbers i, j 2 f1, . . . , r g such that maxi K i ða, vÞ ¼ maxj K j ða, vÞ ¼ max max K k ða, vÞ v2V a

v2V a

k¼1, ..., r v2V ka

ð1:165Þ

and moreover in any neighborhood of the point а, there exists a point b 2 A such that the value max max K j ðb, vÞ

ð1:166Þ

k¼1, ..., r v2V j b

is achieved for a unique number k. Theorem 1.4.24 Under the hypotheses of Theorem 1.4.17 (Theorem 1.4.20 or Corollary 1.4.23), let А be a connected set, and let there exist two different values a0 , a1 2 A such that the maximums of the functions v ! K j ða, vÞ over the sets V ja are maximized over all j ¼ 1, . . . , r at two different numbers j0 , j1 , i.e., max max K j ða0 , vÞ < max K j0 ða0 , vÞ, j v2V a00 j ¼ 1, . . . , r v2V ja0 j 6¼ j0 max max K j ða1 , vÞ < max K j1 ða1 , vÞ: j v2V a11 j ¼ 1, . . . , r v2V ja1

ð1:167Þ

j 6¼ j1 Then there exists a bifurcation point a 2 A for the extremals of Problem 1.4.3 (Problem 1.4.5 or Problem 1.4.7, respectively) on the choice of optimal parametric control laws from a given finite collection of algorithms. Proof Since the set А is connected, draw a continuous curve a ¼ aðsÞ and s 2 ½0, 1, between the points a0 , a1 that belong to the set А so that að 0Þ ¼ a0 , að 1Þ ¼ a1 :

ð1:168Þ

Denote K j ðsÞ ¼ max K j ðaðsÞ, vÞ, s 2 ½0, 1: In accordance with Theorem 1.4.17 j v2V aðsÞ

(Theorem 1.4.20 or Corollary 1.4.23), the function s ! K j ðsÞ is continuous on the interval ½0, 1. Hence, the function K  ¼ K  ðsÞ, where K  ðsÞ ¼ max K j ðsÞ, has the j¼1, ..., r

same property on this interval.

60

1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Introduce the set  Δð jÞ ¼ s 2 ½0, 1j K j ðsÞ ¼ K  ðsÞ , j ¼ 1, . . . , r:

ð1:169Þ

This set is closed as the complete preimage of a closed set that consists of a single point (zero) for the continuous function y ¼ yðsÞ, where yðsÞ ¼ K j ðsÞ  K  ðsÞ: Therefore, the interval ½0, 1 can be represented as the union ½0, 1 ¼

[

Δð jÞ:

ð1:170Þ

j¼1, ..., r

(By the hypotheses of this theorem, the union comprises at least two non-empty closed sets.) Also the hypotheses of this theorem imply the relationships 0 2 Δð j0 Þ, 1= 2Δð j0 Þ:

ð1:171Þ

Then the boundary points of the set Δð j0 Þ that lie within the interval (0,1) form a non-empty set. Hence, there exists a lower bound s0 for them. This value is also a boundary point of some other set Δð j2 Þ and also belongs to it. Consequently, for a ¼ aðs0 Þ the value max K j ðaðsÞ, vÞ achieves maximum over all j ¼ 1, . . . , r at least j v2V aðsÞ

at two different numbers j0 and j2. At the same time, for 0  s < s0, this maximum is achieved at j0 only. Thus, a(s0) is the desired bifurcation point. This completes the proof of Theorem 1.4.24. The next result is a direct consequence of Theorem 1.4.24. Corollary 1.4.25 Under the hypotheses of Theorem 1.4.17 (Theorem 1.4.20 or Corollary 1.4.23), let А be a connected set, and let the parametric control law Gj0 yield a solution of Problem 1.4.3 (Problem 1.4.5 or Problem 1.4.7, respectively) for a value a ¼ a0 2 A and no solution for a value a ¼ a1 2 A, where a0 6¼ a1 : In other words, let max max K j ða0 , vÞ < max K j0 ða0 , vÞ, j v2V a00 j ¼ 1, . . . , r v2V ja0 j 6¼ j0 max max K j ða1 , vÞ > max K j0 ða1 , vÞ: j v2V a01 j ¼ 1, . . . , r v2V ja1

ð1:172Þ

j 6¼ j0 Then there exists at least one bifurcation point a 2 A for the extremals of Problem 1.4.3 (Problem 1.4.5 or Problem 1.4.7, respectively). In conclusion, consider a numerical algorithm for calculating the bifurcation point of an uncontrolled function or parameter a in Problems 1.4.3, 1.4.5, or 1.4.7 (the

1.5 Example: Parametric Control of Cyclic Dynamics Based on Kondratiev’s Cycle

61

choice of optimal parametric control laws from a given finite collection of algorithms) under the hypotheses of Theorem 1.4.24. Connect the points a0 and a1 by a smooth curve S ⊂ A. Divide this curve into n equal parts with a sufficiently small step. For the resulting partition {βi 2 S, i ¼ 0, . . . , n, β0 ¼ a0 , βn ¼ a1 g , determine the numbers ji of the optimal parametric control laws that solve Problems 1.4.3, 1.4.5, or 1.4.7 with a ¼ βi . Then find the least number i for which the corresponding number of the parametric control law differs from j0 . In this case, the bifurcation point of the parameter a lies on the segment ðβi1 , βi Þ of the curve S. For this segment of the curve, the idea is to calculate the bifurcation point with a given accuracy ε using the bisection method. This method yields a point γ 2 ðβi1 , βi Þ on one side of which the parametric control law Gj0 is optimal and on the other is not (within an admissible deviation ε from the value γ). By Corollary 1.4.25, there exists a bifurcation point of the problem on ðβi1 , βi Þ, and it can be estimated by any point of this segment. The next section gives an example of optimal parametric control choice and estimation of bifurcation points in Kondratiev’s cycle model.

1.5

Example: Parametric Control of Cyclic Dynamics Based on Kondratiev’s Cycle Model

Model Description This model [7] combines non-equilibrium economic growth and nonuniform scientific and technological advancement. The model is described by the following system of equations, including two differential and one algebraic equations: 8 nðt Þ ¼ Ayðt Þa , > > > > > < dx=dt ¼ xðt Þðxðt Þ  1Þðy0 n0  yðt Þnðt ÞÞ,   μ þ l0 2 > dy=dt ¼ nðt Þð1  nðt ÞÞyðt Þ xðt Þ  2 þ , > > n0 y 0 > > : n0 ¼ Ay0 α :

ð1:173Þ

The notations are the following: t as time (in months); x as the efficiency of innovations; y as the capital productivity ratio; y0 as the capital productivity ratio corresponding to the equilibrium trajectory; n as the rate of savings; n0 as the rate of saving corresponding to the equilibrium trajectory; μ as the disposal coefficient of capital assets (per month); l0 as the employment growth rate corresponding to the equilibrium trajectory; and finally, A and a as some model constants. A preliminary estimation of the model parameters was performed using statistical information on the Republic of Kazakhstan for the years 2001–2005 [15]. The deviations between the observed statistical and calculated data for that period did not exceed 1.9%.

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Fig. 1.2 Cyclic phase trajectory of Kondratiev’s cycle model

The parameter identification procedure yielded the following values of the exogenous parameters of the model: α ¼ 0.0046235; y0 ¼ 0.081173; n0 ¼ 0.29317; μ ¼ 0.00070886; l0 ¼ 0.00032161; and x(0) ¼ 1.91114. Retrospective forecasting for the years 2006–2007 produced the errors of 6.1% and 12.1% for the capital productivity ratio and 2.3% and 11% for the rate of savings. The corresponding cyclic phase trajectory of Kondratiev’s cycle model is shown in Fig. 1.2. The period of cyclic trajectory corresponding to the statistical information on the Republic of Kazakhstan for the given years was estimated as 232 months. Estimating the Robustness of Kondratiev’s Cycle Model without Parametric Control The structural stability (robustness) of the mathematical model was estimated in accordance with Component 4 of the theory of parametric control theory (see Sect. 1.3.1) over a chosen compact set of its state space. Figure 1.3 presents an estimate of the chain-recurrent set R( f,N ) calculated by the corresponding algorithm over the domain N ¼ ½1:7, 2:3  ½0:066, 0:098 of the phase plane Oxy of system (1.173). Since the set R( f,N ) was non-empty, the weak structural stability of Kondratiev’s cycle model over N could not be estimated using Robinson’s theorem. However, the domain N contained a non-hyperbolic

μþl0 singularity––the center x0 ¼ 2  n0 y , y0 ; therefore system (1.173) was not 0

weakly structurally stable over N. Parametric Control of the Evolution of Economic Systems Based on Kondratiev’s Cycle Model Choose an optimal parametric control law from the following collection of algorithms:

1.5 Example: Parametric Control of Cyclic Dynamics Based on Kondratiev’s Cycle

63

Fig. 1.3 Chain-recurrent set for Kondratiev’s cycle model

y ð t Þ  y ð 0Þ ; y ð 0Þ y ð t Þ  y ð 0Þ 2Þ n0 ðt Þ ¼ n0   k2 ; y ð 0Þ x ð t Þ  x ð 0Þ ; 3Þ n0 ðt Þ ¼ n0  þ k3 x ð 0Þ x ð t Þ  x ð 0Þ : 4Þ n0 ðt Þ ¼ n0   k4 x ð 0Þ 1Þ n0 ðt Þ ¼ n0  þ k1

ð1:174Þ

Here ki denotes the scenario coefficient; n0 is the value of the exogenous parameter n0 obtained by preliminary estimation of parameters. The choice of an optimal parametric control law at the level of the econometric parameter n0 can be formulated as the following problem. Within the framework of the mathematical model (1.173), find an optimal parametric control law from the collection of algorithms (1.174) under which  2  2 ! T xð t Þ  x0 yð t Þ  y0 1 X K¼ þ ! min T t¼1 x0 y0 subject to the constraints

ð1:175Þ

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

0  yðt Þ  1, 0  nðt Þ  1, 0  xðt Þ:

ð1:176Þ

Here T ¼ 232 is the length of one cycle, also called period. The basic value of this criterion without parametric control is K ¼ 0.0307. The formulated parametric control problem with criterion (1.175) was solved using control law (4) from collection (1.174); the optimal value of this criterion was calculated as K ¼ 0.007273, with the corresponding value k 4 ¼ 0:300519 of the adjustable coefficient. The graphs of the endogeneous model variables, without parametric control and with the constructed optimal parametric control law in terms of the criterion K, are demonstrated in Figs. 1.4 and 1.5 below. Estimating the Structural Stability of Kondratiev’s Cycle Model with Parametric Control For stability analysis, the optimal parametric control laws (1.174) with the obtained values of the adjustable coefficients were substituted into the right-hand side of the second and third equations of system (1.173) for the parameter n0. Then, using the numerical algorithm for estimating the weak structural stability of a discrete-time dynamic system, the chain-recurrent set Rð f , N Þ was estimated as an empty set (or singleton) for the compact set N determined by the inequalities 1:7  x  2:3 and 0:066  y  0:098 in the state space of the variables (x, y). This means that Kondratiev’s cycle model with the optimal parametric control law was estimated as weakly structurally stable over the compact set N.

0.12

y

0.1

y0

0.08 0.06 0.04 0.02 0

2001

2003

2005

2007

2009

2011

2013

2015

2017

2019

2021

Years Without parametric control

With optimal parametric control

Fig. 1.4 Capital productivity ratio, without parametric control and with optimal parametric control law (4) in terms of criterion K

1.6 Example: Parametric Control of Nonlinear Dynamic System Based on Lorenz Model. . . 65

x 2.5

x0

2 1.5 1 0.5 0

2001

2003

2005

2007

2009

2011

2013

2015

2017

2019

2021

Years Without parametric control

With optimal parametric control

Fig. 1.5 Efficiency of innovations, without parametric control and with optimal parametric control law (4) in terms of criterion K

Dependence of the Optimal Value of Criterion K on Exogenous Parameter in Variational Calculus Problem for Kondratiev’s Cycle Model Now, analyze how the optimal value of the criterion K depends on the exogenous parameters μ (the disposal coefficient of capital assets) and a for the parametric control laws (1.174) with the obtained optimal values of the adjustable coefficients ki, where the values of the parameters (μ, a) belong to the rectangle A ¼ ½0:00063, 0:00147  ½0:01, 0:71 in the plane. The graphs of the optimal values of the criterion K as functions of the uncontrolled parameters (for parametric control laws (0) and (2), yielding the maximum criterion values) that were obtained by computer simulations can be seen in Fig. 1.6. The projection of the intersection line of the two surfaces into the plane (μ, α) consists of the bifurcation points for the extremals of the variational calculus problem under study.

1.6

Example: Parametric Control of Nonlinear Dynamic System Based on Lorenz Model and Estimation of Bifurcation Points for Extremals

The parametric control approach to nonlinear dynamic systems suggested in Sect. 1.4.1 was applied to the classical Lorenz model. Note that this model was intensively studied by researchers, using analytical tools and also computer simulations. The Lorenz model has the form

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Fig. 1.6 Optimal values of criterion K as functions of exogenous parameters μ and a

8 > < x_ ¼ σx þ σy, y_ ¼ xz þ rx  y, > : z_ ¼ xy  bz:

ð1:177Þ

Computer simulations identified several domains of parameter values in which this model revealed different dynamic properties. For example [10], computer simulations in the domain fr 2 ð24:06, 30:1, σ ¼ 10, b ¼ 8=3g established that there existed three chaotic trajectories (singularities) of the form Oð0; 0; 0Þ, O1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

bðr  1Þ; bðr  1Þ; r  1 , O2  bðr  1Þ;  bðr  1Þ; r  1

ð1:178Þ and also a chaotic Lorenz attractor; moreover, for r 2 ð24:06, 24:7368Þ the singularities O1 and O2 were stable while for r 2 ½24:7368, 30:1 instable. Note that in this domain the singularity О is instable [10]. In addition, computer simulations [18] demonstrated that system (1.177) had a strange attractor in the domain fr 2 ½300, 332:5, σ ¼ 16, b ¼ 4g. It is required to reduce the attractor’s dimensions or (at best) to replace a chaotic trajectory with a trajectory converging to a stable singularity or a limit cycle using parametric control methods.

1.6 Example: Parametric Control of Nonlinear Dynamic System Based on Lorenz Model. . . 67

To this effect, the optimal parametric control choice problem with one of the parameters (r, σ, b) can be formulated as follows. For the Lorenz model, choose an optimal parametric control law with one of the three parameters (r, σ, b) from the collection of algorithms xðtÞ  xðt 0 Þ þ const j , xðt 0 Þ xðtÞ  xðt 0 Þ þ const j , U 2j ðtÞ ¼ k2j xðt 0 Þ yðtÞ  yðt 0 Þ þ const j , U 3j ðtÞ ¼ þk3j yðt 0 Þ yðtÞ  yðt 0 Þ U 4j ðtÞ ¼ k4j þ const j , yðt 0 Þ zðtÞ  zðt 0 Þ U 5j ðtÞ ¼ þk5j þ const j , zðt 0 Þ zðtÞ  zðt 0 Þ U 6j ðtÞ ¼ k6j þ const j , zðt 0 Þ

1Þ U 1j ðtÞ ¼ þk1j 2Þ 3Þ 4Þ 5Þ 6Þ

ð1:179Þ

i.e., find an optimal law from the set {Uij} and its coefficient kij > 0 so that vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðT " 2  2  2 # u xð t Þ  x0 yð t Þ  y0 zðt Þ  z0 1u þ þ K¼ t dt ! min : T x y z fU ij , kij g 0

ð1:180Þ Here Uij denotes the i-th control law for the j-th parameter (i ¼ 1, . . . , 6; j ¼ 1, 2, 3); numbers j ¼ 1, 2, and 3 correspond to the parameters r, σ, and b, respectively; ðx0 ; y0 ; z0 Þ is the desired stationary behavior of the system solution defined as the middle point of the state space of a given strange attractor; finally, ðx ¼ 1; y ¼ 1; z ¼ 1Þ are numerical values characterizing approximate proportions of the attractor in three coordinate axes. The parametric choice problem was solved under state-space constraints determined by the attractor’s dimensions in each coordinate axis. System (1.174) was solved numerically using the standard Runge–Kutta method of order 4 with step 0.001. The value of criterion (1.180) without parametric control is K ¼ 101.0337. Figure 1.7 shows the chaotic attractor for the parameter values fr ¼ 332, σ ¼ 16, b ¼ 4g. In all figures of this subsection, axis Ox is pointing leftward, axis Oy rightward, and axis Oz upward.

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Fig. 1.7 Chaotic attractor of Lorenz model with fr ¼ 332, σ ¼ 16, b ¼ 4g

For the mathematical model (1.177) with the parameters fr ¼ 332, σ ¼ 16, b ¼ 4g, the problem is to choose an optimal parametric control law with one of the parameters (r, σ, b) from the collection of algorithms (1.179) that reduces the attractor’s dimensions in terms of the minimum value of functional (1.180), where ðx0 ¼ 10; y0 ¼ 25; z0 ¼ 350Þ: As indicated by the solution of this optimal parametric control choice problem, the best result K ¼ 87.6959 can be achieved using the law σ ¼ 0:0280ðyðt Þ  yðt 0 ÞÞ þ 16 (see Fig. 1.8). Also, the following problem can be formulated for the strange attractor in Fig. 1.7. For the mathematical model (1.177) with the parameters fr ¼ 332, σ ¼ 16, b ¼ 4g, choose an optimal parametric control law with one of the parameters (r, σ, b) from the collection of algorithms (1.179) that transforms the chaotic trajectory of the system to a trajectory converging to a limiting cycle in terms of the minimum value of functional (1.180), where ðx0 ¼ 10; y0 ¼ 25; z0 ¼ 350Þ: The solution of this optimal parametric control choice problem is presented in Fig. 1.9. Note that the chaotic behavior of the dynamic system was suppressed and the limiting cycle was obtained using the optimal parametric control law r ¼ 0:0530ðxðt Þ  xðt 0 ÞÞ þ 332. The corresponding minimum value of the criterion was K ¼ 96.8293. Now, consider parametric control of the chaotic attractor in Fig. 1.10 for the Lorenz model (1.177) with the parameters fr ¼ 24:74, σ ¼ 10, b ¼ 8=3g: For the mathematical model (1.177) with the parameters fr ¼ 24:74, σ ¼ 10, b ¼ 8=3g, the problem is to choose an optimal parametric control law with one of the parameters (r, σ, b) from the collection of algorithms

1.6 Example: Parametric Control of Nonlinear Dynamic System Based on Lorenz Model. . . 69 Fig. 1.8 Strange attractor of smaller dimension obtained by parametric control law σ

(1.179) that reduces the attractor’s dimensions in terms of the minimum value of functional (1.180), where ðx0 ¼ 0; y0 ¼ 0; z0 ¼ 22:5Þ. The value of criterion (1.180) without parametric control is K ¼ 12.2129. As indicated by the solution of this optimal parametric control choice problem, the best result K ¼ 6.1392 can be achieved using the law b ¼ 0:13ðyðt Þ  yðt 0 ÞÞ þ 8=3 (see Fig. 1.11). Also, the following problem was formulated for the strange attractor in Fig. 1.10. For the mathematical model (1.177) with the parameters fr ¼ 24:74, σ ¼ 10, b ¼ 8=3g , choose an optimal parametric control law with one of the parameters (r, σ, b) from the collection of algorithms (1.179) that transforms the chaotic trajectory of the system to a trajectory converging to an asymptotically stable singularity in terms of the minimum value of functional (1.180), where ðx0 ¼ 0; y0 ¼ 0; z0 ¼ 22:5Þ: The solution of this optimal parametric control choice problem is illustrated in Figs. 1.12 and 1.13. Note that the chaotic behavior of the dynamic system was suppressed and the trajectory converging to the asymptotically stable singularity (7.2279; 7.2279; 19.5749) was constructed using the optimal parametric control law r ¼ 0:17ðyðt Þ  yðt 0 ÞÞ þ 24:44 . The corresponding minimum value of the criterion was K ¼ 10.7236. In addition, the chaotic behavior of the dynamic system was suppressed, and the trajectory converging to the asymptotically stable singularity (8.1170; 8.1170; 24.7072) was constructed using the optimal parametric control law r ¼ 0:35ðzðt Þ  zðt 0 ÞÞ þ 24:44. The corresponding minimum value of the criterion was K ¼ 12.0533; see Fig. 1.13.

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Fig. 1.9 Limit cycle obtained by suppression of chaotic attractor of Lorenz model

Fig. 1.10 Chaotic attractor of Lorenz model with fr ¼ 24:74, σ ¼ 10, b ¼ 8=3g

1.6 Example: Parametric Control of Nonlinear Dynamic System Based on Lorenz Model. . . 71 Fig. 1.11 Strange attractor of smaller dimension obtained by parametric control law b

Fig. 1.12 Trajectory converging to asymptotically stable singularity (7.2279; 7.2279; 19.5749) obtained by suppression of chaotic attractor of Lorenz model

Interestingly, if a parametric control law from collection (1.179) with the parameter r yielded an asymptotically stable singularity, then the latter was one of points (1.178). The singularity in Fig. 1.12 corresponded to the point O1 for r ¼ 20.5749 and the singularity in Fig. 1.13 to the point O2 for r ¼ 25.7072. This point O2 was

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1 Parametric Control of Macroeconomic Systems: Basic Components of Theory

Fig. 1.13 Trajectory converging to asymptotically stable singularity (8.1170; 8.1170; 24.7072) obtained by suppression of chaotic attractor of Lorenz model

asymptotically stable for system (1.177) with the parametric control law r ¼ 0:35ðzðt Þ  zðt 0 ÞÞ þ 24:44 for σ ¼ 10, b ¼ 8=3 yet instable for system (1.177) without parametric control. In other words, using parametric control laws, it was possible to localize and stabilize the singularity of the original system.

Chapter 2

Macroeconomic Analysis and Parametric Control Based on Global Multi-country Dynamic Computable General Equilibrium Model (Model 1)

This chapter illustrates the efficiency of parametric control based on the dynamic computable general equilibrium (CGE) model that was developed from the static model Globe 1 [43]. The class of CGE models is a widespread tool to analyze macroeconomic systems and also to assess an efficient economic policy [22, 31, 34, 35, 77]. As noted by Dixon and Parmenter in [34], this class of mathematical models will be probably dominating in government’s economic policy and business strategies. Among others this class of models includes the static model Globe 1; see [43]. Chapter 2 presents the following results: • The global multi-country dynamic computable general equilibrium model (Model 1) that was developed from the static model Globe 1 under accepted hypotheses and a conceptual description of the global economy • The adaptation of Model 1 to the goals of research that was performed in GAMS [41]; more specifically, the economic dynamics of five countries of the Eurasian Economic Union (EAEU), the European Union (as a single country), the USA, China, and the Rest of World (as a single country) were studied by choosing a finite collection of economic sectors for each region, generating initial data in the form of social accounting matrices (SAMs), and calibrating Model 1 • The applicability of computer simulations with Model 1 to the real macroeconomic systems that was tested using the methods of Chap. 1 • Macroeconomic analysis based on Model 1 • A series of parametric control problems based on Model 1 (statements and solutions) • The applicability of parametric control laws that was tested using the methods of Chap. 1 • The dependence of optimal criteria values on uncontrolled parameters in parametric control problems

© Springer Nature Switzerland AG 2020 A. A. Ashimov et al., Macroeconomic Analysis and Parametric Control of a Regional Economic Union, https://doi.org/10.1007/978-3-030-32205-2_2

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2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

74

2.1 2.1.1

Modeling Procedure Conceptual Description of the Global Economy

In contrast to [43], this chapter introduces a conceptual description of interacting economies of different Regions within the framework of the global multi-country dynamic computable general equilibrium model. In particular, this description contains: • Hypotheses accepted in the course of modeling • Formal rules of agents’ behavior • Formal conditions under which the static model Globe 1 was further developed into the dynamic one • Formal conditions of equilibrium in the markets of all types of Products and Factors, as well as formal conditions of price balances, internal balances for government accounts, and external balances for trading accounts. The conceptual description yields a rigorous mathematical model for the economic interaction of different Regions of the global economy. Hypotheses of Conceptual Description Model 1 describes the global economy under a series of main hypotheses as follows. The global economy is represented by several types of economic agents in different Regions that interact with each other, namely: • • • •

Sectors of a Region Households of a Region Government of a Region Agent Globe

Each Sector supplies a single type of Product in accordance with its name. To this end, each Sector uses two Factors of production belonging to the Households of its Region, namely, Labor and Capital. Agent Globe imports transportation services from other Regions and exports them in form of transportation margins for delivering Products between other Regions. By assumption, in each Region all Sectors, Households, and Government (in terms of consumption of final Products) consist of very many small and identical Elementary agents (firms, separate households, and government institutions). Elementary agents are acting under perfect competition: any individual behavior of elementary agents does not affect prices in markets, i.e., they are price takers. By assumption, Elementary agents have perfect rationality in the sense that they know the values of all exogenous and endogenous variables for a current year. For choosing their behavior, Elementary agents solve a series of optimization problems under perfect competition with fixed prices. Other hypotheses include the following: (a) annual equilibria are achieved in the markets of all types of Products and Factors, and (b) formal conditions are satisfied

2.1 Modeling Procedure

75

for price balances, internal balances for government accounts, and external balances for trading accounts. The hypotheses on the behavioral rules of agents as well as associated balance and auxiliary equations will be given in Sect. 2.1.1. As a result, the global economy is described by the interaction of Elementary agents, Governments and Banking sectors of Regions, and agent Globe. The optimization problems of Agents considered below are obtained by aggregating the corresponding problems solved by each Elementary agent independently. Other behavioral rules of Households and Sectors (see Sect. 2.1.1) are also obtained by aggregating the behavioral rules of corresponding collections of Elementary agents. In final analysis, the global economy is completely characterized by the (optimal) behavioral rules of all Agents mentioned (see Sect. 2.1.1) under several balance relationships. The static model Globe 1 was further developed into the dynamic Model 1 by formalizing the dynamics of several variables as follows: the technological coefficients of the production functions for the gross value added (GVA) of all Sectors of Regions and the supply of Factors by the Households of Regions. All equations of the global multi-country dynamic computable general equilibrium model below containing no new variables were imported from [43]. The Basic Setup of Model 1: List of Main Endogenous and Exogenous Variables and Parameters All variables of Model 1 are denoted by upper- or lowercase Latin letters and digits and may have subscripts (r, w, a, c, cp, f, h, t). The Regions of Model 1 are indicated by subscripts r (for domestic Region) or w (foreign Regions), which take values from 1 to 9 (see the numbering system of Regions below). The Sectors of Model 1 are indicated by subscript a, which takes values from 1 to 16 (see the numbering system of Sectors below). Product indicated by subscript c (or cp) has a unique correspondence to Sector a producing it. The Factors of a Region are indicated by subscript f, which takes values 1 (Labor) and 2 (Capital). The subscript h ¼ 1 means Households. The subscript t (year under consideration) applies to all variables; in the sequel, it will be explicitly written only in the expressions with variables for 2 consecutive years (t and t + 1). The notations and names of the variables and parameters imported to Model 1 from the static model Globe 1 will be given in original form [43]. Exogenous Variables of Model 1: tadvaa,r,t tfs1,r,t cxfs2,r,t cfs2,r,t acc,r,t

Rate of change for ADVAa,r,t (shift parameter of CES production functions for GVA by Sector a in Region r for year t; see below) Rate of change for Labor supply in Region r for year t Disposal coefficient of Capital in Region r for year t Influence coefficient of investment demand on Capital in Region r for year t Shift parameter of Armington CES consumption function for Product c in Region r for year t

76

acr c,r,t at c,r,t at rc,r,t adxba,r,t dabadxa,r,t adx01a,r,t adfdbf ,a,r,t dadfd f ,a,r,t deprecf ,r,t qcdconst c,h,r,t qgdconst c,r,t tmbw,c,r,t tsbc,r,t txba,r,t tyfbf ,r,t tyhbh,r,t tfbf ,a,r,t TSc,r,t TX a,r,t TYF f ,r,t TYH h,r,t TF f ,a,r,t TM w,c,r,t ADX a,r,t ADFDf ,a,r,t DSHH r,t QGDADJ r,t QINVDc,r,t

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Shift parameter of Armington CES import function for Product c to Region r for year t Shift parameter of CET production function for Product c in Region r for year t Shift parameter of CET export distribution function for Product c from Region r for year t Basic shift parameter of CES production function for Sector a in Region r for year t Variation of basic shift parameter of CES production function for Sector a in Region r for year t Inflection of shift parameter of CES production function for Sector a in Region r for year t Basic shift parameter for efficiency of Sector a on Factor f in Region r for year t Variation of shift parameter for efficiency of Sector a on Factor f in Region r for year t Depreciation coefficient of Factor f in Region r for year t Minimum consumption of Product c by Households h in Region r for year t Government demand for Product c in Region r for year t Base tariff rate on Product c imported from Region w to Region r for year t Base sales tax rate on Product c in Region r for year t Base indirect tax rate on Sector a in Region r for year t Base direct tax rate on income from Factor f in Region r for year t Base direct tax rate on Household h in Region r for year t Base tax rate on the use of Factor f by Sector a in Region r for year t Sales tax rate on Product c in Region r for year t Indirect tax rate on Sector a in Region r for year t Direct tax rate on income from Factor f in Region r for year t Direct tax rate on Household h in Region r for year t Tax rate on the use of Factor f by Sector a in Region r Tariff rates on Product c imported from Region w to Region r for year t Shift parameter for CES production functions for QX in Region r for year t Shift parameter for specific efficiency of Sector a on Factor f in Region r for year t (in basic calculation, 1) Partial household savings rate scaling factor of Households in Region r for year t Government consumption demand scaling factor in Region r for year t (1, used in scenarios only) Investment demand by Product c in Region r for year t

2.1 Modeling Procedure

DTF r,t WALRASr,t WFDIST f ,a,r,t

77

Uniform adjustment to factor use tax by Sectors in Region r for year t Slack variable for Walras’ Law in Region r for year t (exogenous variable equal to 0) Sectoral proportion for the price of Factor f for Sector a in Region r for year t (in basic calculation, 1)

Exogenous Parameters of Model 1: δc,r ρcc,r ρm c,r δrw,c,r δxa,r ρxa,r δva f ,a,r ρva a,r γ c,r ρtc,r γ rc,w,r ρec,r βc,h,r

Share parameter of Armington CES consumption function for Product c in Region r Elasticity of Armington CES consumption function for Product c in Region r Elasticity of Armington CES import function for Product c from Region r Share parameter of Armington CES import function for Product c from Region w to Region r Share parameter of CES production function for Sector a in Region r Elasticity of CES production function for Sector a in Region r Share parameter of CES production function for GVA on Factor f of Sector a in Region r Elasticity of CES production function for GVA on Factor f of Sector a in Region r Share parameter of CET distribution function for Product c in Region r Elasticity of CET distribution function for Product c in Region r Share parameter of CET export distribution function for Product c from Region r to Region w Elasticity of CET export distribution function for Product c from Region r Minimum budget share of Households h on purchase of Product c in Region r

Initial Conditions of Dynamic Equations (Parameters of Model 1): ADVAa,r,2001 FSf ,r,2001

Shift parameter of CES production functions for GVA of Sector a in Region r for year 2001 Supply of Factor f in Region r for year 2001

Endogenous Variables of Model 1: ADVAa,r,t FSf ,r,t KAPGOV r,t QM c,r,t

Shift parameter of CES production functions for GVA of Sector a in Region r for year t Supply of Factor f in Region r for year t Government savings in Region r for year t Imports of Product c to Region r for year t

78

QDc,r,t PDc,r,t PM c,r,t QQc,r,t QMRw,c,r,t QMLc,r,t PMRw,c,r,t PMLc,r,t QVAa,r,t QINT a,r,t PINT a,r,t PVAa,r,t QX a,r,t WF f ,r,t MTAX r,t ETAX r,t STAX r,t ITAX r,t FYTAX r,t HTAX r,t FTAX r,t FDf ,a,r,t QE c,r,t PE c,r,t QXC c,r,t QERc,w,r,t PERc,w,r,t YF f ,r,t YFDIST f ,r,t YH h,r,t PQDc,r,t QCDc,h,r,t HEXPh,r,t SHH h,r,t PWM w,c,r,t

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Domestic demand for Product c in Region r for year t Consumer price of domestic supply of Product c in Region r for year t Domestic price of competitive imports of Product c to Region r for year t Supply of composite Product c in Region r for year t Imports of Product c from Region w to Region r for year t Supply of composite import c from large share regions to Region r for year t Domestic price of imports of Product c from Region w to Region r for year t Domestic price of imports of Product c from Region r for year t Quantity of aggregate value added (QVA) of Sector a in Region r for year t Aggregate quantity of intermediates used by Sector a in Region r for year t Price of aggregate intermediate input of Sector a in Region r for year t Value-added price for Sector a in Region r for year t Domestic output by Sector a in Region r for year t Price of Factor f in Region r for year t Import tax revenue in Region r for year t Export tax revenue in Region r for year t Sales tax revenue in Region r for year t Indirect tax revenue in Region r for year t Factor income tax revenue in Region r for year t Household income tax revenue in Region r for year t (exclusive of Factors) Factor use tax revenue in Region r for year t Demand for Factor f by Sector a in Region r for year t Domestic output exported by Product c from Region r for year t Domestic price of exports by Product c in Region r for year t Domestic output by Product c in Region r for year t Exports of Product c to Region w from Region r for year t Domestic price of exports of Product c from Region r to Region w Income from Factor f in Region r for year t Income from Factor f in Region r for distribution after depreciation and taxation for year t Income of Household h in Region r for year t Consumer price of composite Product c in Region r for year t Consumption of Product c by Household h in Region r for year t Consumption expenditures of Household h in Region r for year t Savings of Household h in Region r for year t

2.1 Modeling Procedure

ERr,t PWE c,w,r,t PQSc,r,t QINTDc,r,t QGDc,r,t PX a,r,t YGr,t EGr,t QT w,c,r,t TOTSAV r,t INVEST r,t KAPREGw,r,t PWMFOBw,c,r,t PT c,r,t VFDOMDr,t KAPWORr,t PXC c,r,t

79

CIF price of competitive imports of Product c from Region w to Region r for year t Exchange rate in Region r for year t (domestic per world unit (USD)) World price of exports of Product c from Region r to Region w for year t (USD) Supply price of composite Product c in Region r for year t Demand for intermediate inputs by Product c in Region r for year t Government consumption demand by Product c in Region r for year t Composite price of output by Sector a in Region r for year t Government income in Region r for year t Government expenditure in Region r for year t Quantity of margin services for aggregate imports of Product c from Region w to Region r for year t Total (aggregate) savings in Region r for year t Total (aggregate) investment expenditures in Region r for year t Bilateral current account balance of Regions r and w for year t FOB price of competitive imports of Product c from Region w to Region r for year t Price of imported transportation services of Product c to Region r for year t (same price to imports from all regions, PWE c,r,}glo} ) Value of final domestic demand in Region r for year t (in buyer’s prices) Current account balance in Region r for year t Producer price of composite domestic output of Product c in Region r for year t (coincides with PX a,r )

Conceptual Description of Sector’s Behavior Each Sector a of Region r is represented by a sufficiently large number N Br,a,r of identical Elementary agents (firms). In the course of operation, each Sector annually implements several functions in the aggregate of all corresponding actions of its elementary agents as follows: 1. Each Sector a supplies a unique type c of Products using Factors (hired Labor and leased Capital) that belong to the Households of its Region r and intermediate Products (domestic and imported) of each type cp. The behavior of each Sector satisfies aggregate first-order conditions for the following nested optimization problems that are solved by its Elementary agents: • Br1ða, r, iÞ, calculation of the optimal shares of intermediate consumption and QVA in the output of Elementary agent i • Br2ða, r, iÞ, calculation of the optimal quantities of Factors used by Elementary agent i in the production of final Products (VA)

80

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

2. Each Sector a sells made Product c to other Sectors, Government, and Households of domestic Region as well as exports a part of its output to foreign Regions. This behavior of each Sector also satisfies aggregate first-order conditions for the following nested optimization problems that are solved by its Elementary agents: • Br3ðc, r, iÞ , calculation of the share of exports in the aggregate output of Elementary agent i • Br4ðc, r, iÞ, calculation of the shares of exports to each foreign Region w in the aggregate exports of its Product c 3. Each Sector a pays the Government of Region r net tax charges (taxes minus subsidies); see the problems G1ðr Þ, G2ðr Þ, G4ðr Þ, and G5ðr Þ in the conceptual description of Government’s behavior below. These taxes participate in pricing (including the price coefficients) for the Products and Factors purchased by this Sector as well as for the exported Product. 4. Within export and import operations (jointly with other Sectors and consumers), each Sector redistributes the investments of Households and Governments among the Regions of Model 1 (see formulas (2.61) and (2.72) below). 5. The Elementary agents of each Sector a, like the Elementary agents of other Sectors, Households, and Government of Region r, solve the two nested optimization problems U1ðc, r, iÞ and U2ðc, r, iÞ for calculating the consumption shares of the domestic and imported Product c in Region r, with specification by Regions (see the optimization problems solved by the Elementary agents of Sectors and Households below). Now, consider formal statements of the four listed optimization problems as well as first-order conditions to find their solutions, including the optimal behavior of Sector a. First-level problem Br1ða, r, iÞ Calculate the optimal shares of intermediate consumption and QVA in the output of Elementary agent i of Sector a in Region r (r 6¼ Globe).
 Given the domestic output QX a,r,i , the value-added price ðPVAa,r Þ , and the price of aggregate intermediate input
 ðPINT a,r Þ, the problem is to find the quantity of and the aggregate quantity of intermediates aggregate value added QVA a,r,i
 QINT a,r,i that minimize the value of intermediates purchase and QVA of Elementary agent i, min

QVAa,r,i , QINT a,r,i




 PVAa,r QVAa,r,i þ PINT a,r QINT a,r,i ,

ð2:1Þ

subject to the following constraint on the CES production function of agent i: 1

 x x  x QX a,r,i  ADX a,r δxa,r QVAa,r,i ρa,r þ 1  δxa,r QINT a,r,i ρa,r ρa,r ¼ 0:

ð2:2Þ

2.1 Modeling Procedure

81

The parameters of equation (2.2) satisfy the natural restrictions 0 < δxa,r < 1 , ρxa,r > 0, and ADX a,r > 0. Solving the problem Br1ða, r, iÞ based on the first-order conditions yields the following optimal relationship between the QVA of Elementary agent i and the aggregate quantity of intermediates used by this agent in Region r:

QVAa,r,i

" # 1 ð1þρxa,r Þ δxa,r PINT a,r
 ¼ QINT a,r,i : x PVAa,r 1  δa,r

ð2:3Þ

Now, study the aggregate first-order conditions determining the behavior of Sector a. Recall that all Elementary agents are identical by the accepted hypothesis. Hence, Sector a in Region r has the following characteristics: • The domestic output by Sector a in Region r is the sum of all such outputs by its Elementary agents: QX a,r ¼ N Br,a,r QX a,r,i , or QX a,r,i ¼ QX a,r =N Br,a,r . • The quantity of aggregate value added of Sector a in Region r is the sum of all such quantities of its Elementary agents: QVAa,r ¼ N Br,a,r QVAa,r,i, or QVAa,r,i ¼ QVAa,r =N Br,a,r . • The aggregate quantity of intermediates of Sector a in Region r is the sum of all such quantities of its Elementary agents: QINT a,r ¼ N Br,a,r QINT a,r,i, or QINT a,r,i ¼ QINT a,r =N Br,a,r . Substitute these expressions for QX a,r,i , QVAa,r,i , and QINT a,r,i into relationships (2.2), (2.3) and perform some trivial transformations to derive the following aggregate first-order conditions. The aggregate CES production function of Sector a in Region r is given by 1

 x x  x QX a,r  ADX a,r δxa,r QVAa,r ρa,r þ 1  δxa,r QINT a,r ρa,r ρa,r ¼ 0:

ð2:4Þ

Here the parameters of Equation (2.4) satisfy the natural restrictions 0 < δxa,r < 1, ρxa,r > 0, and ADX a,r > 0. Finally, using (2.3) write the following relationship between the QVA of Sector a and the aggregate quantity of intermediates used by this Sector in Region r: " QVAa,r ¼ QINT a,r

# 1 ð1þρxa,r Þ δxa,r PINT a,r
 : PVAa,r 1  δxa,r

ð2:5Þ

Second-level problem Br2ða, r, iÞ Calculate the optimal demand structure for the Factors of production for Elementary agent i of Sector a in Region r (r 6¼ Globe).
 Given the quantity of aggregate value added QVAa,r,i , the prices of all Factors f in Region r (WF f ,r ), the corresponding sectoral proportions (WFDIST f ,a,r , which

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

82

are 1 in the basic calculation), the tax rates on the use of Factor f by Sector a in Region r (TF f ,a,r ), the problem is to find the demands for Factor f by Elementary agent i (FDf 1,a,r,i and FDf 2,a,r,i) that minimize the value of the Factors of production, min

FDf 1,a,r,i , FDf 2,a,r,i

X

  FD , WF WFDIST 1 þ TF f ,r f ,a,r f ,a,r f ,a,r,i f

ð2:6Þ

subject to the following constraint on the CES production function for the QVA of agent i: QVAa,r,i  ADVAa,r

hX  f


ρvaa,r i1=ρvaa,r δva ¼ 0: f ,a,r ADFDf ,a,r FDf ,a,r,i

ð2:7Þ

The parameters of Equation (2.7) satisfy the natural restrictions P va δf ,a,r ¼ 1 and ρva δva a,r > 0. In addition, ADFDf ,a,r is the shift parameter f ,a,r > 0, f

for the specific efficiency of Sector a on Factor f in Region r (in basic calculation, 1). The technological coefficient (the shift parameter in the terminology of [43]) ADVAa,r,t of this production function is calculated from the dynamic equation
 ADVAa,r,t ¼ ADVAa,r,t1 1 þ tempadvaa,r,t ,

ð2:8Þ

where tempadvaa,r,t denotes the rate of change for ADVAa,r,t1 . The aggregate solution of the problem Br2ða, r, iÞ is the following equation that implicitly defines the desired values of the demands for Factors f ¼ 1, 2 by Sector a in Region r (FDf ,a,r ):
 WF f ,r WFDIST f ,a,r 1 þ TF f ,a,r hX 
ρvaa,r i1 va δ ADFD FD ¼ PVAa,r QVAa,r f ,a,r fp,a,r fp,a,r fp  δva f ,a,r ADFDf ,a,r

ρva a,r

FDf ,a,r

ρva a,r 1

ð2:9Þ

:

The aggregate CES production function for QVA of Sector a in Region r (QVAa,r ) obtained from (2.7) has the form QVAa,r  ADVAa,r

hX  f


ρvaa,r i1=ρvaa,r δva ADFD FD ¼ 0: f ,a,r f ,a,r f ,a,r

ð2:10Þ

In accordance with behavioral rule 1, the domestic output by Sector a in Region r (QX a,r ) coincides with the output (QXC c,r ) by the corresponding Product с(a) in this Region, i.e., QX a,r ¼ QXC cðaÞ,r :

ð2:11Þ

2.1 Modeling Procedure

83

First-level problem Br3ðc, r, iÞ Calculate the optimal share of exports in the output of Product c ¼ cðaÞ for Elementary agent i of Sector a in Region r (r 6¼ Globe). Given the domestic output of Elementary agent i by Product c in Region r (QXC c,r,i ), the domestic price of exports by Product c in Region r (PE c,r ), and the consumer price of domestic supply of Product c in Region r (PDc,r), the problem is to find the optimal values of the domestic output of Elementary agent i by Product c exported from Region r (QE
c,r,i ) and the domestic demand of Elementary agent i for Product c in Region r QDc,r,i ) that maximize sales revenue, max




QE c,r,i , QDc,r,i

 PE c,r QE c,r,i þ PDc,r QDc,r,i ,

ð2:12Þ

subject to the following constraint on the CET distribution function for the output of Product с:  1=ρt
 t t c,r QXC c,r,i  at c,r γ c,r QE c,r,i ρc,r þ 1  γ c,r QDc,r,i ρc,r ¼ 0:

ð2:13Þ

The parameters of Equation (2.13) satisfy the natural restrictions 0 < γ c,r < 1 , ρtc,r > 0 and at c,r > 0. Solving the problem Br3ðc, r, iÞ based on the first-order conditions yields the following optimal relationship (aggregate solution) between the domestic output of Sector a by Product c exported
 from Region r (QE c,r ) and the domestic demand for Product c in Region r QDc,r :
 1 PE c,r 1  γ c,r ðρtc,r 1Þ : γ c,r PDc,r

 QE c,r ¼ QDc,r

ð2:14Þ

The aggregate CET distribution function for the output of Product c by Sector a in
 Region r QXC c,r obtained from (2.13) has the form  1=ρt
 t t c,r ¼ 0: QXC c,r  at c,r γ c,r QE c,r ρc,r þ 1  γ c,r QDc,r ρc,r

ð2:15Þ

If this Sector supplies Product c for export only (QDc,r ¼ 0 ) or for domestic consumption within Region r only (QE c,r ¼ 0), then its Elementary agents do not solve the problem Br3. In this case, relationships (2.13) and (2.14) are replaced by QXC c,r ¼ QE c,r þ QDc,r

ð2:16Þ

Second-level problem Br4ðc, r, iÞ Calculate the optimal regional structure of exports by Product c for Elementary agent i of Sector a in Region r (r 6¼ Globe).

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

84

Given the domestic output of Elementary agent i by Product c exported from Region r (QE c,r,i 6¼ 0) and the domestic prices of exports of Product c from Region r to each Region w (PERc,w,r Þ, find the optimal values of exports of Product c from Region r to other Regions w (QERc,w,r,i ) that maximize the revenue from export sales, max

X w6¼r

QERc,w,r,i , w6¼r

 PERc,w,r QERc,w,r,i ,

ð2:17Þ

subject to the following constraint on the CET distribution function for the exports of Product с from Region r: QE c,r,i  at rc,r

hX


w6¼r

γ rc,w,r QERc,w,r,i ρc,r e

The parameters of Equation (2.18) satisfy P γ rc,w,r > 0, γ rc,w,r ¼ 1, ρec,r > 0 and at rc,r > 0.

i1=ρec,r

the

¼ 0: natural

ð2:18Þ restrictions

w

Solving the problem Ar4ðc, r, iÞ based on the first-order conditions yields the following optimal relationship (aggregate solution) between the exports of Product
 c from Region r to each foreign Region w QERc,w,r and the corresponding output of exported Product c (QE c,r ):

QERc,w,r ¼ QE c,r

PERc,w,r e PE c,r γ rc,w,r at rc,r ρc,r

!1= ρe 1 ð c,r Þ

:

ð2:19Þ

The aggregate CET distribution function for the exports of Product c by Sector a in Region r obtained from (2.18) has the form QE c,r  at rc,r

hX


w6¼r

γ rc,w,r QERc,w,r ρc,r e

i1=ρec,r

¼ 0:

ð2:20Þ

If this Sector supplies Product c for domestic consumption within Region r only (QE c,r ¼ 0), then Equations (2.18) and (2.18) are replaced by QERc,w,r ¼ 0. Conceptual Description of Household’s Behavior The Households of each Region r (r 6¼ Globe) are represented by a sufficiently large number N H,r of identical Elementary agents (separate households). In the course of operation, each Household annually implements several functions in the aggregate of all corresponding actions of its Elementary agents as follows: 1. Each Household hires out Labor and leases Capital to the Sectors of its Region (dynamic behavioral rules H1ðr Þ and H2ðr Þ). 2. Each Household gains income from its two Factors (Labor and Capital) used by the Sectors of Region r (behavioral rule H3ð f , r Þ), after deduction of tax charges

2.1 Modeling Procedure

3.

4. 5.

6. 7.

85

and depreciation (behavioral rule H4ð f , r Þ); each Household forms its income based on Factors (behavioral rule H5ðr Þ). Each Household pays the Government of Region r net tax charges (taxes minus subsidies); see the problems G1ðr Þ, G3ðr Þ, G6ðr Þ, and G7ðr Þ in the conceptual description of Government’s behavior below. Each Household determines its expenditures (on the consumption of all types of composite Products) and savings (behavioral rule H6ðr Þ). Each Household maximizes its utility function subject to a budget constraint and determines the consumption of each Product c; see the optimization problem H7ðr Þ below. Each Household invests in Capital jointly with all Governments and Households of other Regions; see formulas (2.61), (2.72), and (2.22). Elementary agents (separate households), like the Elementary agents of Sectors and Governments of Region r, solve the two nested optimization problems U1ðc,r,iÞ and U2ðc,r,iÞ for calculating the consumption shares of the domestic and imported Product c in Region r, with specification by Regions (see the optimization problems solved by the Elementary agents of Sectors and Households below).

Now, consider formal descriptions of the behavioral rules H1 –H6 and also a formal statement of the optimization problem H7ðr Þ together with its first-order conditions determining the optimal behavioral rules of Households. Behavioral rule H1ðr Þ Calculate the aggregate Labor supply to the Sectors of Region r (r 6¼ Globe) by domestic Households. For each year t, the aggregate Labor supply (FS1,r,t ) to the Sectors of Region r by domestic Households is calculated using the exogenous rate of change for Labor supply (tfs1,r,t ) and the Labor supply in a previous year from the dynamic equation
 FS1,r,t ¼ FS1,r,t1 1 þ tfs1,r,t :

ð2:21Þ

Behavioral rule H2ðr Þ Calculate the aggregate Capital supply to the Sectors of Region r (r 6¼ Globe) by domestic Households. For each year t, the aggregate Capital supply (FS2,r,t) to the Sectors of Region r by domestic Households is calculated using the Labor supply in a previous year with the corresponding exogenous disposal coefficient of Capital (cxfs2,r,t ) and the aggregate P investment demand by Product c in domestic Region ( c QINVDc,r,t1 ) with the corresponding exogenous influence coefficient of investment demand (cfs2,r,t ). The aggregate Capital supply is calculated from the dynamic equation X
 QINVDc,r,t1 : FS2,r,t ¼ FS2,r,t1 1  cxfs2,r,t þ cfs2,r,t c

ð2:22Þ

86

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

In Model 1, domestic Households gain income from a single source—by selling the services of their Factors of production of domestic Region. Behavioral rule H3ð f ,r Þ Calculate the income from each Factor f in Region r (r 6¼ Globe). Households gain income (YF f ,r) from the demand for Factor f by each Sector a in domestic Region (FDf ,a,r ) multiplied by the average price of Factor f in domestic Region (WF f ,r) and the sectoral proportion for the price of this Factor (WFDIST f ,a,r), i.e., YF f ,r ¼

X a

WF f ,r WFDIST f ,a,r FDf ,a,r :

ð2:23Þ

Behavioral rule H4ð f ,r Þ Calculate the income from each Factor f in Region r ðr 6¼ GlobeÞ after depreciation and taxation. This income (YFDIST f ,r) is the income from Factor f (YF f ,r) after deduction of its depreciation with the corresponding coefficient (deprecf ,r ) and tax charges on the income from Factor f with the corresponding direct income tax rate (TYF f ,r ), i.e.,

 YFDIST f ,r ¼ YF f ,r 1  deprecf ,r 1  TYF f ,r :

ð2:24Þ

The depreciation coefficient of Labor is deprec1,r ¼ 0. Behavioral rule H5ðr Þ Calculate the income of domestic Households of Region r (r 6¼ Globe) from all Factors after deduction of depreciation and tax charges on the income from all Factors. This income (YH h,r ) is the sum of the corresponding incomes from Factors that belong to the Households of Region r, i.e., YH h,r ¼

X f

hvashh,f ,r YFDIST f ,r

ð2:25Þ

Note that hvashh,f ,r ¼ 1 denotes an auxiliary parameter. Behavioral rule H6ðr Þ Calculate the consumption expenditures of domestic Households of Region r (r 6¼ Globe), known as the budget constraints. The consumption expenditures of domestic Households of Region r (HEXPh,r ) are their incomes (YH h,r ) minus their net tax charges with the corresponding direct tax rate (TYH h,r ) and their savings defined by the corresponding exogenous share (SHH h,r ), i.e., HEXPh,r ¼ YH h,r ð1  TYH h,r Þð1  SHH h,r Þ:

ð2:26Þ

2.1 Modeling Procedure

87

Problem H7ðr, iÞ Calculate the optimal quantity of each composite Product for an Elementary agent i—a Household of Region r (r 6¼ Globe). Given the budget constraint of agent i (consumption expenditures HEXPh,r,i), the exogenous minimum consumption of Product c by Household i (qcdconst c,h,r,i), and the consumer price of composite Product c in Region r (PQDc,r ), the problem is to find the consumptions (QCDc,h,r,i ) of each Product c by this Household that maximize the Stone–Geary utility function, max

fQCDc,h,r,i g

Y
βc,h,r QCD  qcdconst , c,h,r,i c,h,r,i c

ð2:27Þ

subject to the budget constraint X c

PQDc,r QCDc,h,r,i ¼ HEXPh,r,i

ð2:28Þ

(In (2.27), 0 < βc,h,r < 1 are given parameters.) The aggregate first-order conditions of the problem H7ðr, iÞ yield the following equation for the optimal consumptions QCDc,h,r : PQDc,r QCDc,h,r ¼

  X PQDc,r qcdconst c,h,r þ βc,h,r HEXPh,r  PQD qcdconst : cp,r cp,h,r cp

ð2:29Þ The aggregate budget constraint on the Households of Region r obtained from (2.28) has the form X c

PQDc,r QCDc,h,r ¼ HEXPh,r :

ð2:30Þ

Conceptual Description of Government’s Behavior In the course of operation, the Government of Region r (r 6¼ Globe ) annually implements several functions as follows: 1. The Government of Region r collects net tax revenue: • From the imports of Products to domestic Region, import tax (Behavioral rule G1ðr Þ) • From the exports of Products to foreign Regions, export tax (Behavioral rule G2ðr Þ) • From the domestic sales of Products, sales tax and VAT (Behavioral rule G3ðr Þ) • From the Products made by domestic Sectors, output tax (Behavioral rule G4ðr Þ)

88

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

• From the Households’ incomes from their Factors, factor income tax (Behavioral rule G5ðr Þ) • From other Households’ incomes (except for factor income), income tax minus transfers (Behavioral rule G6ðr Þ) • From the use of Factors by domestic Sectors, factor use tax (Behavioral rule G7ðr Þ)

2. 3.

4. 5.

The Government of Region r calculates its income (Behavioral rule G8ðr Þ) based on all these tax revenues. The Government of Region r calculates the quantities of expenditures (Behavioral rule G10ðr Þ) and savings (Behavioral rule G11ðr Þ). Using savings the Government of Region r participates in investments jointly with other Governments and Households of all Regions (see formulas (2.61), (2.72), and (2.22)). The Government of Region r consumes different types of Products in exogenous quantities (Behavioral rule G9ðr Þ). The Government of Region r is represented by a sufficiently large number N G,r of identical Elementary agents implementing the functions of government consumption. Each agent of this type, like the Elementary agents of Sectors and Households of Region r, solves the two nested optimization problems U1ðc, r, iÞ and U2ðc, r, iÞ for calculating the consumption shares of the domestic and imported Product c in Region r, with specification by Regions (see the optimization problems solved by the Elementary agents of Sectors, Households, and Government below).

Note that the structure of the Government’s budget in Model 1 differs from the generally accepted one. More specifically, the government expenditures of Region r (which include government consumption and debt servicing) do not contain transfers and subsidies to Households and Sectors: the corresponding values are deducted from budget revenue in accordance with the concept of net tax revenue. The surplus (or gap) of model’s budget come to aggregate savings of domestic Region, which participate in investments in other Regions. Now, consider formal descriptions of the behavioral rules G1–G11. Behavioral rule G1ðr Þ Calculate net tax revenue (subsidies) on the imports of Products to Region r (r 6¼ Globe). This revenue (MTAX r ) is calculated using the imports of each Product c from foreign Regions w to domestic Region r (QMRw,c,r ) with the corresponding CIF price of imports (PWM w,c,r ) and the corresponding import tariff rates (TM w,c,r ). For the EAEU countries, these tariff rates also cover the redistribution of tax revenue among them in accordance with the Treaty on the EAEU. The next expression of tax revenue also incorporates the exchange rate in Region r (ERr ): MTAX r ¼

X c,wðw6¼rÞ

TM w,c,r PWM w,c,r ERr QMRw,c,r :

ð2:31Þ

2.1 Modeling Procedure

89

Behavioral rule G2ðr Þ Calculate net tax revenue (subsidies) on the exports of Products from Region r (r 6¼ Globe). This revenue (ETAX r ) is calculated using the exports of each Product c from domestic Region r to foreign Regions w (QERc,w,r ) with the world prices of exports of these Products to Region w (PWE c,w,r ) and the corresponding export tariff rates (TE c,w,r ), i.e., ETAX r ¼

X c,wðw6¼rÞ

TE c,w,r PWEc,w,r ERr QERc,w,r :

ð2:32Þ

Behavioral rule G3ðr Þ Calculate net sales tax revenue in Region r (r 6¼ Globe). This revenue (STAX r ) is calculated using the demands for intermediates (QINTDc,r ) and investments (QINVDc,r ) by Product c in domestic Sectors of Region r, the final consumption of Product c by domestic Households of Region r (QCDc,h,r), and by Government (QGDc,r ) with the corresponding supply prices (PQSc,r ) and sales tax rate (TSc,r ), i.e., STAX r ¼

X

  X TS PQS QINTD þ QCD þ QGD þ QINVD c,r c,r c,r c,h,r c,r c,r : c h ð2:33Þ

Remark If Region r uses VAT instead of sales tax, then the term QINTDc,r in formula (2.33) is omitted. Behavioral rule G4ðr Þ Calculate the net input tax revenue in Region r (r 6¼ Globe). This revenue (ITAX r ) is calculated using the outputs of all domestic Sectors a in Region r (QX a,r ) with the corresponding composite prices (PX a,r ) and indirect tax rates (TX a,r ), i.e., ITAX r ¼

X a

TX a,r PX a,r QX a,r :

ð2:34Þ

Behavioral rule G5ðr Þ Calculate the factor income revenue of Households in Region r (r 6¼ Globe). This revenue (FYTAX r ) is calculated using the incomes from Factors f in Region r (YF f ,r ) with the corresponding depreciation coefficients (deprecf ,r ) and direct rates of factor income tax (TYF f ,r ), i.e., FYTAX r ¼

X


 : TYF YF  deprec YF f ,r f ,r f ,r f ,r f

ð2:35Þ

90

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Behavioral rule G6ðr Þ Calculate the net tax revenue from Households’ incomes (except for factor income) in Region r (r 6¼ Globe). This revenue (HTAX r) is calculated using the incomes of Households h in Region r (YH h,r ) with the corresponding direct tax rates (TYH h,r ), i.e., HTAX r ¼

X h

TYH h,r YH h,r :

ð2:36Þ

Behavioral rule G7ðr Þ Calculate the tax revenue from the use of Factors by domestic Sectors in Region r (r 6¼ Globe). This revenue (FTAX r ) is calculated using the demands of domestic Sectors a for Factors f in Region r (FDf ,a,r ) with the corresponding prices (WF f ,r ) and their sectoral proportions (WFDIST f ,a,r ) as well as with the corresponding direct tax rates (TF f ,a,r ), i.e., FTAX r ¼

X f ,a

TF f ,a,r WF f ,r WFDIST f ,a,r FDf ,a,r :

ð2:37Þ

Behavioral rule G8ðr Þ Calculate the income of the Government of Region r (r 6¼ Globe). This income (YGr ) is the sum of all the tax revenues mentioned earlier, i.e., YGr ¼ MTAX r þ ETAX r þ STAX r þ ITAX r þ FYTAX r þ HTAX r þ FTAX r : ð2:38Þ Behavioral rule G9ðc, r Þ Calculate the consumption demand for composite Product c by the Government of Region r (r 6¼ Globe). This consumption (QGDc,r ) is calculated using the Government’s exogenous demand for Product c in Region r (qgdconst c) with the corresponding scaling factor (QGDADJ r ), i.e., QGDc,r ¼ QGDADJ r qgdconst c,r :

ð2:39Þ

The scaling factor allows adjusting different scenarios in Model 1; in basic calculation, QGDADJ r  1. Behavioral rule G10ðr Þ Calculate the Government’s expenditures on the consumption of composite Product in Region r (r 6¼ Globe). These expenditures (EGr ) are the sum of the above consumptions of different types of composite Product (QGDc,r ) multiplied by the corresponding prices (PQDc,r ), i.e.,

2.1 Modeling Procedure

91

EGr ¼

X c

QGDc,r PQDc,r :

ð2:40Þ

Behavioral rule G11ðr Þ Calculate the Government’s current savings in Region r (r 6¼ Globe). The Government’s savings in current year (KAPGOV r ) are calculated as the difference between the Government’s income and expenditures, i.e., KAPGOV r ¼ YGr  EGr :

ð2:41Þ

The government budget of Region r has surplus if KAPGOV r > 0 and gap otherwise. Optimization Problems Solved by Elementary Agents of Sectors, Households, and Government As noted in the preceding paragraphs, the Elementary agents of Sectors, Households, and Government of Region r (r 6¼ Globe) annually solve the two nested optimization problems U1ðc, r, iÞ and U2ðc, r, iÞ. Using these solutions, the agents calculate the demand shares of the foreign and domestic Products of each type c in the aggregate demand for this (composite) Product in domestic Region as well as the shares of imports from each foreign Region w to Region r in the aggregate demand for imported Product c. The problem U2ðc, r, iÞ is solved by an Elementary agent only if the imports of Product с make up a large share of consumption; in the case of small consumption, fixed shares are assigned for the imports of this Product from each foreign Region w. Now, state the corresponding first- and second-level problems as well as establish their P first-order conditions for calculating the desired shares. Denote by N U,r ¼ a N Br,a,r þ N H,r þ N G,r the total number of Elementary agents–consumers in Region r, and let i be the number of such a consumer. First-level problem U1ðc, r, iÞ Calculate the optimal share for the imports of composite Product c in the consumption of Elementary agent i in Region r (r 6¼ Globe). Given the demand for composite (domestic and imported) Product c of Elementary agent i in Region r (QQc,r,i ), the domestic price of competitive imports of Product c in Region r (PM c,r), and the consumer price of domestic supply of Product c in Region r (PDc,r ), the problem is to find the demand for imported Product c (QM c,r,i ) and also the domestic demand for Product c (QDc,r,i ) that minimize the purchase value of this Product for Elementary agent i, min

QM c,r,i , QDc,r,i




 PM c,r QM c,r,i þ PDc,r QDc,r,i ,

ð2:42Þ

subject to the following constraint on the Armington CES consumption function for composite Product c in Region r:

92

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . . 1
c c  c QQc,r,i  acc,r δc,r QM c,r,i ρc,r þ ð1  δc,r ÞQDc,r,i ρc,r ρc,r ¼ 0:

ð2:43Þ

Here δc,r denotes the share parameter (0  δc,r  1), ρcc,r > 0 and acc,r > 0. Solving the problem U1ðc, r, iÞ based on the first-order conditions yields the following optimal relationship (aggregate solution) between the demand for imported Product
c in Region r (QM c,r,i ) and the domestic demand for Product c in Region r QDc,r :  QM c,r ¼ QDc,r

 1 PDc,r δc,r ð1þρcc,r Þ : PM c,r ð1  δc,r Þ

ð2:44Þ

The aggregate CES consumption function for composite Product c in Region r (QQc,r,i ) obtained from (2.43) includes two components—imports (QM c,r ) and domestic (QDc,r )—and has the form 1
c c  c QQc,r  acc,r δc,r QM c,r ρc,r þ ð1  δc,r ÞQDc,r ρc,r ρc,r ¼ 0:

ð2:45Þ

If Region r consumes imported or domestic Product c only (QDc,r ¼ 0 or QM c,r ¼ 0, respectively), then the Elementary agents do not solve the problem U1 and relationships (2.43) and (2.44) are replaced by QQc,r ¼ QDc,r þ QM c,r :

ð2:46Þ

If the imports of Product с to Region r have inconsiderable share in aggregate consumption (do not exceed a given threshold, e.g., 10%), then the imports of such a Product is denoted by QMSc,r : QMSc,r ¼ ioqmsqmc,r QM c,r ,

ð2:47Þ

Otherwise, the imports of Product c (with considerable share in aggregate consumption) is denoted by QMRc,r : QMLc,r ¼ ioqmlqmc,r QM c,r :

ð2:48Þ

In these formulas, ioqmsqmc,r and ioqmlqmc,r are corresponding constant thresholds (in basic calculation, 1). In the case QMSc,r 6¼ 0 , the imports of Product c from Region w to Region r (QMRw,c,r ) are calculated with the exogenous shares ioqmrqmsw,c,r , i.e., QMRw,c,r ¼ ioqmrqmsw,c,r QMSc,r : In this formula, ioqmrqmsw,c,r  0 and

P

w ðw6¼rÞ ioqmrqmsw,c,r

ð2:49Þ ¼ 1.

2.1 Modeling Procedure

93

If the share of imported Product с is not small (QMLc,r 6¼ 0), Elementary agent i solves the following optimization problem. Second-level problem U2ðc, r, iÞ Calculate the optimal consumption structure for the imports of Product c for Elementary agent i of Region r. Given the consumption of imported Product c by Elementary agent i of Region r (QMLc,r,i) and the domestic price of imported Product c from each foreign Region w to Region r (PMRw,c,r), the problem is to find the optimal consumptions of Product c imported from Region w to Region r (QMRw,c,r,i) that minimize the purchase value of this Product for Elementary agent i, min fQMRw,c,r,i g,

X

 PMR QMR w,c,r w,c,r,i , w,w6¼r

w6¼r

ð2:50Þ

subject to the following constraint on the Armington CES import function for Product c in Region r: QMLc,r,i  acr c,r

hX

δr QMRw,c,r,i ρc,r w,w6¼r w,c,r m

i1=ρm

¼ 0:

c,r

ð2:51Þ

P In this formula, δrw,c,r > 0, w,w6¼r δrw,c,r ¼ 1, ρm c,r > 0, and acr c,r > 0. Solving the problem U2ðc, r, iÞ based on the first-order conditions yields the following optimal relationship (aggregate solution) between the imports of Product c from each foreign Region w
to Region  r (QMRw,c,r ) and the consumption of imported Product c by Region r QMLc,r :  QMRw,c,r ¼ QMLc,r

1 m  PMRw,c,r acr c,r ρc,r ðρmc,r þ1Þ : PMLc,r δrw,c,r

ð2:52Þ

In this formula, PMLc,r is the aggregate domestic price of imported Product c in Region r calculated with the corresponding prices of imports from different Regions w (see (2.88) below). The aggregate CES import function for composite Product c in Region r obtained from (2.51) has the form QMLc,r  acr c,r

hX

δr QMRw,c,r ρc,r w,w6¼r w,c,r m

i1=ρm

c,r

¼ 0:

ð2:53Þ

Conceptual Description of Globe’s Behavior Recall that Globe is a special agent (Region) that implements the following function. In accordance with the behavioral rule GL1ðw, c, r Þ, it renders trade and transportation services on the imports of each Product с from each Region w to each other Region r based on corresponding transportation margins.

94

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Behavioral rule GL1ðw, c, r Þ Calculate the quantity of margin services for aggregate imports of Product c from Region w to another Region r (r, w 6¼ Globe). This quantity of margin services imported by Region r from Globe (QT w,c,r) is the product of the corresponding imports (QMRw,c,r ) by the exogenous margin rate margcor w,c,r , (0  margcor w,c,r < 1), i.e., QT w,c,r ¼ QMRw,c,r margcor w,c,r :

ð2:54Þ

Balance and Auxiliary Equations The balance equations presented below guarantee: • Equilibrium in the market of Factors (Labor and Capital) for each Region r 6¼ Globe ðMar1ð f , r ÞÞ • Equilibrium in the markets of each Product c for each Region r 6¼ Globe (Mar2ðc, r Þ) • Equilibrium of savings (the Households and Government of each Region r 6¼ Globe) and investments in Capital (Mar3ðr Þ) • Bilateral current account balances for each pair of Regions (Mar4ðw, r Þ, Mar5ðGlobe, r Þ) • Zero trade balances for each type of Product of Region Globe (Mar6ðc, GlobeÞ) • The export/import equality of Product с for Region Globe (Mar7ðc, GlobeÞ) • Bilateral trade equilibrium for each type of Product in each pair of Regions (Mar8ðw, c, r Þ) Model 1 involves the annual quantities of different Products and Factors at seller’s prices. The auxiliary expressions below are intended to calculate the aggregate quantities and purchase values for each Region r. Auxiliary equation S1ðr Þ The aggregate purchase value of final Products for Region r (r 6¼ Globe) at consumer prices. This value (VFDOMDr ) is composed of the aggregate consumptions of each Product c by Households (QCDc,h,r ), Government (QGDc,r ) of Region r and the investment demands (QINVDc,r ) of Region r with the corresponding consumer prices (PQDc,r ), i.e., VFDOMDr ¼

X

PQDc,r c

X

 QCD þ QGD þ QINVD : c,h,r c,r c,r h

ð2:55Þ

Auxiliary equation S2ðr Þ The investment demand by Product с in Region r (r 6¼ Globe). This demand (QINVDc,r ) is the product of the exogenous shares of investment demand of such Products (qinvdconst c,r ) and the corresponding investment scaling factor (IADJ r ), i.e.,

2.1 Modeling Procedure

95

QINVDc,r ¼ qinvdconst c,r IADJ r :

ð2:56Þ

P In this formula, qinvdconst c,r  0 and c qinvdconst c,r ¼ 1. Note that the investment scaling factor can be adjusted to exogenous investment variations and/or to the quantities of investments available. Auxiliary equation S3ðr Þ The aggregate investment expenditures of Region r (r 6¼ Globe) at consumer prices. These aggregate expenditures (INVEST r ) are composed of the exogenous investment demands by Product c in Region r (QINVDc,r ) with the corresponding consumer prices (PQDc,r ), i.e., INVEST r ¼

X c

PQDc,r QINVDc,r :

ð2:57Þ

Auxiliary equation S4ðr Þ The value share of investments in the aggregate final domestic demand of Region r (r 6¼ Globe). This share ðINVESTSH r Þ is the ratio of the aggregate investment expenditures (INVEST r Þ of Region r to its aggregate purchase value (VFDOMDr ), i.e., INVESTSH r VFDOMDr ¼ INVEST r :

ð2:58Þ

Auxiliary equation S5ðr Þ The value share of Government consumption in the aggregate final domestic demand of Region r (r 6¼ Globe). PThis share ðVGDSH r Þ is the ratio of the final Government consumption ( c PQDc,r QGDc,r Þ to the aggregate purchase value (VFDOMDr ), i.e., VGDSH r VFDOMDr ¼

X c

PQDc,r QGDc,r :

ð2:59Þ

Auxiliary equation S6ðcp, r Þ The aggregate demand for intermediate composite Product cp in Region r (r 6¼ Globe). This aggregate demand (QINTDcp,r ) of all Sectors a in Region r is composed of the aggregate quantities of intermediate Product cp used by Sectors a in Region r (QINT a,r ) with the corresponding exogenous supply/demand shares (ioqint cp,a,r ), i.e., QINTDcp,r ¼ In this formula, ioqint cp,a,r  0 and

X

P

a

ioqint cp,a,r QINT a,r :

cp ioqint cp,a,r

¼ 1.

ð2:60Þ

96

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Auxiliary equation S7ðr Þ The aggregate savings in Region r (r 6¼ Globe). These aggregate savings (TOTSAV r ) are calculated using the savings (SHH h,r ) from the income (YH h,r) of Households after deduction of tax charges with the direct tax rate (TYH h,r ), the income from Factors (YF f ,r ) with the corresponding rate deprecf ,r (for Labor, this rate is deprec1,r ¼ 0 ), the Government savings (KAPGOV r), and the current account balance in domestic currency (KAPWORr ERr ) in Region r, i.e., TOTSAV r ¼ YH h,r ð1  TYH h,r ÞSHH h,r þ

X f

deprecf ,r YF f ,r þ rded r DEDr

þ KAPGOV r þ KAPWORr ERr : ð2:61Þ Auxiliary equation S8ðc, r Þ The aggregate exports of margin services of Region Globe to other Regions. These aggregate exports (QE c,Globe ) are composed of the exports of Product c (margin services) to each foreign Region w (QERc,wGlobe ), i.e., QE c,Globe ¼

X w6¼Globe

QERc,w,Globe :

ð2:62Þ

Auxiliary equation S9ðc, r Þ The imports of margin services by Product c from Region Globe to Region r (r 6¼ Globe). These imports (QERc,r,Globe ) are composed of the margin services on the imports of Product c from each foreign Region w to Region r (QT w,c,r ), i.e., QERc,r,Globe ¼

X w6¼r

QT w,c,r :

ð2:63Þ

Auxiliary equation S10ðc, r Þ The relationship between the prices of margin services on the imports of Product c to Region r (r 6¼ Globe) and the world price of exports of Product c from Region Globe. These prices are equal to each other: PT c,r ¼ PWEc,r,Globe :

ð2:64Þ

Auxiliary equation S11ðr Þ The current account balance of Region r. This balance (KAPWORr ) is the sum of the trade gaps/surpluses of Region r with all other Regions w (KAPREGw,r ) at world prices, i.e.,

2.1 Modeling Procedure

97

X

KAPWORr ¼

w6¼r

KAPREGw,r :

ð2:65Þ

Auxiliary equation S12 The global account balance. This balance is the sum of the corresponding current account balances (KAPWORr ) of all Regions, i.e., KAPWORSYS ¼

X r

KAPWORr :

ð2:66Þ

Note that KAPWORSYS represents an auxiliary variable, possibly with zero value. This equation will be used to test Model 1. Auxiliary equation S13ðr Þ The consumer price index for Region r (r 6¼ Globe). This index (CPI r ) describes the price of final Products consumed in a given Region. It is calculated as the weighted average value of the consumer prices of composite Product c (PQDc,r ), i.e., CPI r ¼

X

comtotshc,r PQDc,r :

c

ð2:67Þ

In this formula, comtotshc,r denotes the exogenous consumption share of Product с in P the aggregate consumption of final Product in Region r, comtotshc,r  0 and c comtotshc,r ¼ 1. Auxiliary equation S14ðr Þ The producer price index for Region r (r 6¼ Globe). This index (PPI r) describes the domestic price of Products made in Region r. It is calculated as the weighted average value of the consumer prices of domestic Product c (PDc,r ), i.e., PPI r ¼

X c

vddtotshc,r PDc,r :

ð2:68Þ

In this formula, vddtotshc,r denotes the exogenous share of Product с in the aggregate P output of Region r, vddtotshc,r  0 and c vddtotshc,r ¼ 1. Auxiliary equation S15ðr Þ The exchange rate index. This index (ERPI) is calculated as the weighted average value of the exchange rates (ERr ) for a chosen collection ref of basic Regions (e.g., the USA and The Rest of World), i.e., ERPI ¼

X r2ref

tradtotshr ERr :

ð2:69Þ

98

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

In this formula, tradtotshr denotes the exogenous share of Region r in the aggregate international trade quantity of basic Regions ref, tradtotshr  0 and P r2ref tradtotshr ¼ 1. Balance equation Mar1ð f , r Þ Equilibrium in the market of Factor f in Region r (r 6¼ Globe). The supply of Factor f in Region r (FSf ,r) is the sum of all demands for this Factor by all Sectors a (FDf ,a,r ), i.e., FSf ,r ¼

X a

FDf ,a,r :

ð2:70Þ

Balance equation Mar2ðc, r Þ Equilibrium in the market of Product c in Region r (r 6¼ Globe). The supply of composite Product c in Region r (QQc,r) is the sum of the aggregate demands for this Product in the following forms: as an intermediate Product for all Sectors (QINTDc,r ), as a final Product consumed by Households (QCDc,h,r ) and Government (QGDc,r ), as the investments in Region r (QINVDc,r ), i.e., QQc,r ¼ QINTDc,r þ QCDc,h,r þ QGDc,r þ QINVDc,r :

ð2:71Þ

Balance equation Mar3ðr Þ Equilibrium of savings and investments in Region r (r 6¼ Globe). The aggregate savings in Region r (TOTSAV r ) are equal to the aggregate investments in Region r (INVEST r ) plus the so-called slack variable for the Walras’ Law (WALRASr ), i.e., TOTSAV r ¼ INVEST r þ WALRASr :

ð2:72Þ

Note that WALRASr is an exogenous variable equal to 0 in basic calculation of Model 1. Balance equation Mar4ðw, r Þ Bilateral current account balance for each pair of different Regions r and w (r 6¼ Globe and w 6¼ Globe). This trade gap/surplus of Region r with Region w, where w 6¼ r, w 6¼ Globe, and r 6¼ Globe (KAPREGw,r ) is calculated as the imports of all types of Product c from Region w to Region r (QMRw,c,r ) in terms of the corresponding FOB prices (PWMFOBw,c,r ) minus the exports of all types of Product c from Region r to Region w (QERc,w,r ) in terms of the corresponding world prices (PWEc,w,r ), i.e.,

2.1 Modeling Procedure

KAPREGw,r ¼

99

X c

PWMFOBw,c,r QMRw,c,r 

X c

PWE c,w,r QERc,w,r :

ð2:73Þ

Negative sign of gap indicates surplus. Balance equation Mar5ðGlobe, r Þ Bilateral current account balance of each Region r with Region Globe. This trade gap/surplus (KAPREGGlobe,r ) is calculated as the aggregate imports of all types of Product c from all foreign Regions wp (QT wp,c,r ) in terms of the corresponding transportation prices to Region r (PT c,r ) minus the exports of all types of transportation services c from Region r to Region Globe (QERc,Globe,r ) in terms of the corresponding world prices (PWE c,Globe,r ), i.e., KAPREGGlobe,r ¼

X

PT c,r QT wp,c,r  c,wp

X c

PWEc,Globe,r QERc,Globe,r :

ð2:74Þ

Balance equation Mar6ðc, GlobeÞ Zero trade balance for Region Globe. For each type of Product c, the aggregate value of its imports QMRw,c,Globe at the price PWM w,c,Globe from all Regions to Region Globe is equal to the aggregate value of its exports QERc,w,Globe at the price PWE c,w,Globe to all Regions, i.e., X w

PWM w,c,Globe QMRw,c,Globe ¼

X w

PWE c,w,Globe QERc,w,Globe

þ GLOBESLACK:

ð2:75Þ

In this formula, GLOBESLACK is an auxiliary variable equal to 0 in basic calculation of Model 1. Balance equation Mar7ðc, GlobeÞ The equality relationship between the exports and imports of Product с for Region Globe and other Regions. These exports (QE c,Globe ) from Region Globe are equal to the imports to Region Globe (QM c,Globe ), i.e., QE c,Globe ¼ QM c,Globe :

ð2:76Þ

Balance equation Mar8ðw, c, r Þ Bilateral trade equilibrium for each type of Product. The imports of Product c from Region w to Region r (QMRw,c,r ) are equal to the corresponding exports from Region w to Region r (QERc,r,w ), i.e., QMRw,c,r ¼ QERc,r,w :

ð2:77Þ

100

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

The next group of equations describes different price balances. The prices in Model 1 are dimensionless positive quantities that determine the comparative values of corresponding indexes for a given year t. The price relationships below guarantee zero profit for all agents of Model 1 with their behavioral rules. In basic calculation, the output price of each type of Products is 1. The price relationships for Region r regulate the prices of compound (aggregate) Product and its components, taxes, transportation margins, and other extra charges as follows: • The equality of the composite price of output by Sector a and the corresponding producer price of the composite domestic output of Product cðaÞ (P1ða, r Þ) • The balance between the output price of Sector a and the prices of value added and intermediate consumption by this Sector (P2ða, r Þ) • The balance between the output price and the prices of domestic consumption and exports of Product c (P3ðc, r Þ) • The balance between the export price of Product c to Region w and the corresponding world price taking into account the export taxes (P4ðc, w, r Þ) • The balance between the export price of Product c and the export price of this Product to each Region w (P5ðc, r Þ) • The balance between the export price of Product c for Region Globe and the export prices of this Product to each other Region w (P6ðcÞ) • The balance between the supply price of composite Product c and the prices of the corresponding domestic and imported Product (P7ðc, r Þ) • The balance between the consumer and seller prices of Product c (P8ðc, r Þ) • The balance between the price of the aggregate intermediate Product for Sector a and the prices of its components (P9ða, r Þ) • The balance between the domestic price of imported Product c from Region w and the corresponding CIF price of this Product (P10ðc, r Þ) • The balance between the domestic price of imported Product c from Region w and the import prices of this Product from each Region w (P11ðc, r Þ) • The balance between the domestic price of imported Product с in Region r and the corresponding prices of imported Product from separate Regions in the case of small-share import (P12ðc, r Þ) • The balance between the prices of imported Product c in Region r and the corresponding prices of imported Product with small and large shares (P13ðc, r Þ) • The balance between the CIF price of imported Product с from Region w and the FOB price of this Product (P14ðw, c, rÞ) • The equality of the bilateral FOB prices for each type of Product (P15ðw, c, r Þ) Balance equation P1ða, r Þ The equality relationship between the composite price of output by Sector a in Region r (r 6¼ Globe) and the corresponding producer price of the composite domestic output of Product cðaÞ.

2.1 Modeling Procedure

101

In accordance with Behavioral rule 1 of Sectors and (2.11), the composite price of output by Sector a in Region r (PX a,r ) is equal to the producer price of this Product (PXC cðaÞ,r ), i.e., PX a,r ¼ PXC cðaÞ,r :

ð2:78Þ

Balance equation P2ða, r Þ The balance between the output price of Sector a and the prices of value added and intermediate consumption by Sector a in Region r (r 6¼ Globe). The value of the domestic output by Sector a (QX a,r ) at the corresponding price (PX a,r) before taxation with the corresponding tax rate ðTX a,r Þ is equal to the value of the quantity of value added (QVAa,r ) at the corresponding price (PVAa,r ) plus the value of the intermediate Product used by Sector a (QINT a,r ) at the corresponding price (QINT a,r ), i.e., PX a,r ð1  TX a,r ÞQX a,r ¼ PVAa,r QVAa,r þ PINT a,r QINT a,r :

ð2:79Þ

Balance equation P3ðc, r Þ The balance between the output price of Product c in Region r (r 6¼ Globe) and the prices of domestic consumption and exports of this Product. The value of the domestic output of Product c (QXC c,r ¼ QX a,r ) at the corresponding price (PXC c,r ¼ PX a,r ) is equal to the value of the domestic consumption of Product c in Region r (QDc,r ) at the corresponding price (PDc,r ) plus the value of the exported Product c (QE c,r ) at the corresponding price (PE c,r ), i.e., PXC c,r QXC c,r ¼ PDc,r QDc,r þ PE c,r QE c,r :

ð2:80Þ

Balance equation P4ðc, w, r Þ The balance between the export price of Product c from Region r (r 6¼ Globe) and the corresponding world price taking into account the export taxes. The domestic price of exports of Product c (PERc,r,w) is equal to the corresponding world price for Region w (PWE c,w,r ) after deduction of the export tax with the corresponding rate ðTE c,w,r Þ, i.e., PERc,r,w ¼ PWEc,w,r ð1  TE c,r,w ÞERr :

ð2:81Þ

In this formula, ERr denotes the exchange rate in Region r. Balance equation P5ðc, r Þ The balance between the export price of Product c in Region r (r 6¼ Globe) and the export prices of this Product to each foreign Region w.

102

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

The value of the domestic output of Product c (QE c,r ) at the corresponding price (PE c,r ) is equal to the aggregate value of exports of this Product to all foreign Regions w (QERc,r,w ) at the corresponding prices (PERc,r,w ), i.e., PE c,r QE c,r ¼

X wðw6¼rÞ

PERc,r,w QERc,r,w :

ð2:82Þ

Balance equation P6ðcÞ The balance between the export price of Product c for Region Globe and the export prices of this Product to each other Region w. The export price of Product с from Region Globe (PE c,Globe) is equal to the export price of this Product to any other Region w (PERc,Globe,w ), i.e., PERc,Globe,w ¼ PE c,Globe :

ð2:83Þ

Balance equation P7ðc, r Þ The balance between the supply price of composite Product c and the prices of the corresponding domestic and imported Product in Region r (r 6¼ Globe). The value of the supply of composite Product c in Region r (QQc,r ) at the corresponding price (PQSc,r ) is composed of the value of the domestic supply of this Product (QDc,r ) at the corresponding price (PDc,r ) and the value of the imported Product c (QM c,r ) at the corresponding price (PM c,r ), i.e., PQSc,r QQc,r ¼ PDc,r QDc,r þ PM c,r QM c,r :

ð2:84Þ

Balance equation P8ðc, r Þ The balance of the consumer and seller prices of Product c in Region r (r 6¼ Globe). The consumer price of Product c in Region r (PQDc,r ) is composed of the corresponding seller price (PQSc,r ) and the sales tax (or VAT) at the corresponding rate (TSc,r ), i.e., PQDc,r ¼ PQSc,r ð1 þ TSc,r Þ:

ð2:85Þ

Balance equation P9ða, r Þ The balance between the price of the aggregate intermediate Product for Sector a and the prices of its components in Region r (r ¼ 6 Globe). This price (PINT a,r ) is the weighted average value of the consumer prices of all types of composite Product cp in Region r (PQDcp,r) with the corresponding weights (ioqint cp,a,r ) that specify the demand/supply shares of all types of intermediate Product cp in the aggregate intermediate consumption of Sector a in this Region, i.e.,

2.1 Modeling Procedure

103

PINT a,r ¼ Note that ioqint cp,a,r  0 and

P

X cp

ioqint cp,a,r PQDcp,r :

cp ioqint cp,a,r

ð2:86Þ

¼ 1.

Balance equation P10ðc, w, r Þ The balance between the domestic price of imported Product c from Region w in Region r (r 6¼ Globe) and the corresponding CIF price of this Product. This price (PMRw,c,r ) is calculated using the corresponding CIF price (PWM w,c,r), the tariff rate on the Product c imported from Region w to Region r (TM w,c,r), and the exchange rate ERr , i.e., PMRw,c,r ¼ PWM w,c,r ð1 þ TM w,c,r ÞERr :

ð2:87Þ

Balance equation P11ðc, r Þ The balance between the domestic price of imported Product c from Region w in Region r (r 6¼ Globe ) and the import prices of this Product from each Region w (the case of large-share imports). The value of imports of Product с to Region r (QMLc,r ) at the corresponding domestic price (PMLc,r ) is composed of the values of imported Product c from all foreign Regions w (QMRw,c,r ) at the corresponding domestic prices (PMRw,c,r ), i.e., PMLc,r QMLc,r ¼

X w ðw6¼rÞ

PMRw,c,r QMRw,c,r :

ð2:88Þ

Balance equation P12ðc, r Þ The balance between the domestic price of imported Product с in Region r (r 6¼ Globe) and the corresponding prices of imported Product from separate Regions in the case of small-share imports). This price (PMSc,r ) is the weighted average value of the domestic prices of imported Product c from other Regions w (PMRw,c,r ) with the exogenous weights (ioqmrqmsw,c,r ) that specify the import shares of Product c from Region w to Region r in the aggregate imports of this Product to Region r, i.e., PMSc,r ¼

X

Note that ioqmrqmsw,c,r  0 and

w ðw6¼rÞ

P

ioqmrqmsw,c,r QMRw,c,r :

w ðw6¼rÞ ioqmrqmsw,c,r

ð2:89Þ

¼ 1.

Balance equation P13ðc, r Þ The balance between the prices of imported Product c in Region r (r 6¼ Globe) and the corresponding prices of imported Product with small and large shares. The value of imported Product с in Region r (QM c,r) at the domestic price (PM c,r) is equal to the aggregate value of its imports with large (QMLc,r ) and small (QMSc,r ) shares at the corresponding prices (PMLc,r and PMSc,r ), i.e.,

104

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

PM c,r QM c,r ¼ PMLc,r QMLc,r þ PMSc,r QMSc,r :

ð2:90Þ

Balance equation P14ðw, c, r Þ The balance between the CIF and FOB prices of imported Product с from Region w ðw 6¼ r Þ to Region r (r 6¼ Globe). This CIF price (PWM w,c,r ) is composed of the corresponding FOB price (PWMFOBw,c,r ) and the transportation unit margin of Product c imported to Region r from Region w (margcor w,c,r ) with the price of imported transportation services of Product c to Region r (PT c,r ¼ PWE c,r,Globe ¼ 1), i.e., PWM w,c,r ¼ PWMFOBw,c,r þ margcor w,c,r PT c,r :

ð2:91Þ

Balance equation P15ðw, c, r Þ The equality of the bilateral FOB prices for each type of Product. The FOB price of Product c imported from Region w to Region r (PWMFOBw,c,r) is equal to the FOB price of its export from Region w to Region r (PWE c,r,w ), i.e., PWMFOBw,c,r ¼ PWE c,r,w :

2.1.2

ð2:92Þ

Building of Model 1 and Solution Algorithm

The conceptual descriptions (see the previous subsection) were employed to build a mathematical model (Model 1) by uniting all relevant equations into a single system, namely, the first-order conditions for all optimization problems solved by different agents and their behavioral rules, including the dynamics of technological coefficients for the CES production function of the QVA (2.8) and the supplies of Factors (Labor (2.21) and Capital (2.22)). Also the balance and auxiliary equations were added in Model 1. Next, all variables of Model 1 were divided into two classes— endogenous and exogenous variables. Note that the exchange rates of all regional currencies were considered as endogenous variables (floating). As indicated by structural analysis, Model 1 can be solved with an appropriate algorithm after representation in form of a system of algebraic and dynamic equations. In addition, the solution of these algebraic equations guarantees general equilibrium in the international markets of each type of Product and two Factors (Labor and Capital) for each year t. Three dynamic equations of Model 1 are adopted to calculate the supplies of Labor and Capital and also the technological coefficients for the production functions for each year t using the values of the corresponding variables for the past and current years, which determines the dynamic structure of Model 1. As a result, Model 1 can be described by a system of relationships in form of two subsystems as follows:

2.1 Modeling Procedure

105

1. A subsystem of difference equations relating the values of the dynamic endogenous variables x1 ðt Þ for 2 consecutive years: x1 ðt þ 1Þ ¼ f 1 ðx1 ðt Þ, x2 ðt Þ, uðt Þ, aðt ÞÞ:

ð2:93Þ

The notations are the following: t ¼ 0, 1, . . . , n  1 as a year number (discrete time); x1 ðt Þ and x2 ðt Þ as the vectors of endogenous variables of the system, where xi ðt Þ 2 X i ðt Þ ⊂ Rmi , i ¼ 1, 2 ; the vector x1 ðt Þ consists of the shift parameters (technological coefficients) of the CES production functions for the quantity of aggregate value added (QVA) of Sectors (2.8), Labor supply (2.21), and Capital supply (2.22) for Sectors; the vector x2 ðt Þ consists of all endogenous variables of Model 1 (demands and supplies for different Products, prices, etc.) except for the ones included in the vector x1 ðt Þ; uðt Þ 2 U ðt Þ ⊂ Rq as the vector function of controlled parameters; the coordinates of this vector correspond to different economic tools of government’s policy, e.g., tax rates, the shares of government expenditures on consumption, etc. in the parametric control problem Pr W (see below); aðt Þ 2 A ⊂ Rs as the vector function of uncontrolled parameters; the coordinates of this vector characterize different (exogenous and endogenous) social and economic parameters such as the coefficients of production functions and aggregation functions, the minimum consumption of Products, etc. in the parametric control problem PrW ; X 1 ðt Þ, X 2 ðt Þ, U ðt Þ, and A as compact sets with non-empty interiors; the sets X 1 ðt Þ and X 2 ðt Þ determine state-space constraints while the sets U ðtS Þ control constraints in the parametric control problems based on Model 1; X i ¼ nt¼1 X i ðt Þ , i ¼ 1, 2 ; S U ¼ n1 t¼0 U ðt Þ; Finally, f 1 : X 1  X 2  U  A ! Rm1 as a differentiable mapping. 2. A subsystem of algebraic equations in the unknown variables x2 ðt Þ that describe the behavior and interaction of economic agents in various markets for a considered year: f 2 ðx1 ðt Þ, x2 ðt Þ, uðt Þ, aðt ÞÞ ¼ 0,

ð2:94Þ

where f 2 : X 1  X 2  U  A ! Rm2 is a differentiable mapping; more specifically, this subsystem includes first-order optimality conditions for the optimization problems of economic agents, the behavioral rules of Government and Globe, as well as balance and auxiliary equations. Given some fixed values of the exogenous functions and parameters uðt Þ and aðt Þ, for each time t the CGE model (2.93), (2.94) defines values of the endogenous variables xðt Þ corresponding to the equilibrium demand and supply prices in the markets of Products and Factors.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

106

In accordance with formulas (2.93) and (2.94), Model 1 is described by a difference-algebraic system. It can be solved using the following algorithm. Step 1. Let t ¼ 0 and specify the initial values of the variables x1 ð0Þ. Step 2. For current time t , calculate the values x2 ðt Þ from system (2.94) with the PATH solver, which is supplied with GAMS [38]. Step 3. Using this equilibrium solution for time t, calculate the values x1 ðt þ 1Þ from the dynamic equations (2.93). Assign t ≔ t þ 1 and get back to Step 2. The number of iterations of Steps 2 and 3 depends on the forecasting and parametric control problems on given time intervals. All equations of Model 1 were implemented in GAMS in form of a basic module. This module calculates the values of all endogenous variables of Model 1 under given values of its exogenous variables.

2.2

Adaptation of Model 1 to the Goals of Research

2.2.1

Choice of Economic Regions and Sectors, Retrospective and Forecasting Periods for Computer Simulations

The composition of Model 1 is completely determined by its purpose and includes only Regions that have close economic relations with the Republic of Kazakhstan. This model describes the dynamic operation and interaction of economies of 9 Regions as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Kazakhstan Russia Belarus Armenia Kyrgyzstan the European Union (considered as a single country) the USA China the Rest of World (considered as a single country)

Note that Regions 1–5 form the Eurasian Economic Union (EAEU). The economy of each Region includes 16 Sectors that are crucial for the economy of Kazakhstan as follows: 1. 2. 3. 4.

Mining (ming) Hydrocarbon production and natural gas extraction (crog) Metalworking and machine building (mepe) Metal industry (mind)

2.2 Adaptation of Model 1 to the Goals of Research

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

107

Education, public health, and public administration (ehas) Production and supply of electricity, gas, and hot water (pegw) Food industry, including beverages and tobacco (fpin) Professional, scientific, and technical activities (psta) Other industries (otis) Other services (oths) Agriculture, forestry, and fishery (agff) Construction (buil) Production of textiles, clothes, leather, and associated goods (mtal) Financial services (fins) Chemical and petrochemical industry (cchpp) Transportation (tser)

The simulation period of Model 1 (years 2004–2022) was determined by the available SAMs from the GTAP Data Base (years 2004, 2007, and 2011) and also by the forecasting horizon of the main macroeconomic indexes provided by the IMF (year 2022).

2.2.2

Initial Database and Calibration of Model 1

The parameter estimation (calibration) procedure of Model 1 involved three collections of initial data as follows: 1. The values of the substitution coefficients for different Factors in the production functions of Sectors and the substitution coefficients for different products in the output functions of Sectors, in the utility functions of Households, and also in the aggregation functions describing the consumption of all regional agents of Model 1 for years 2004–2022. For years 2004, 2007, and 2011, these coefficients were taken directly from the GTAP data base [47]. For the other years (2005–2006, 2008–2010, and 2012–2022), the coefficients were extracted from the GTAP data base for a last year available (2004, 2007, or 2011). 2. The initial values of the dynamic variables of the model Equation (2.93). These values were defined using the regional statistical data for year 2003. 3. Collections of social accounting matrices (SAMs) for 9 Regions of Model 1 for years 2004–2015. Now, consider the auxiliary operations and algorithms that were developed for obtaining the collections of SAMs required for calibration of Model 1. The collections of SAMs for years 2004, 2007, and 2011 were extracted from the GTAP data base [47]. Then the unknown SAMs for years 2005–2006, 2008–2010, and 2012–2015 were estimated in the following way. If statistical data in form of input–output tables were available for Region r and year t, then Algorithm 2.1 and a series of other auxiliary algorithms (IOT2, ITB, RAS; see below) were employed for calculating the corresponding SAMs for Region r and years t and (t  1) and some

108

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

international trade indexes for year t. (Recall that an input–output table shows the use of goods and services at consumer prices.) If not, Algorithm 2.2 was employed for calculating the corresponding SAMs for Region r and years t and (t  1) using the main macroeconomic indexes of Regions (GDP; aggregate investments; aggregate imports; aggregate exports; aggregate government expenditures; aggregate government revenues). Algorithm 2.2 was employed for the basic simulation of Model 1 using the forecasted values of the macroeconomic indexes of Regions provided by the International Monetary Fund. Consider a brief description of SAMs for different Regions of Model 1 and also the algorithms to calculate them. Table 2.1 presents the SAM for one Region of Model 1. All data in this table are expressed in million USD. The notations (1..c), (1..а), (1..f ), and (1..r) indicate that a cell is associated with a collection of rows (or columns) in the amount of 16 (Products), 16 (Sectors), 2 (Factors), and 10 (Regions, including Globe), respectively. These rows (or columns) have subscripts c, а, f, or r meaning specific Product, Sector, Factor, or Region of Model 1. Empty cells are filled with zeros. This SAM is square and satisfies the main property: the sum of elements in each row equals the sum of elements in the corresponding column. The other notations in Table 2.1 are as follows: B1 H1 I1 M1 N1 O1 A2 O2 B3 O3 A4 O4 A5 O5 B6 O6 A7 O7 A8 G8

Matrix of dimensions 16  16: Use of intermediate Product c by Sector a Matrix of dimensions 16  10: Exports of Product c to Region r Column of dimension 16: Consumption of Product c by Households Column of dimension 16: Consumption of Product c by Government Column of dimension 16: Gross saving of Product c Column of dimension 16: Aggregate demand for Product c Diagonal matrix of dimensions 16  16: Output of Product c by Sector a Column of dimension 16: Aggregate output of Product c Matrix of dimensions 2  16: Value of using Factor f by Sector a Column of dimension 2: Aggregate income from Factor f Matrix of dimensions 10  16: Tax revenue from imports of Product c from Region r Column of dimension 10: Aggregate tax revenue from imports from Region r Matrix of dimensions 10  16: Tax revenue from exports of Product c to Region r Column of dimension 10: Aggregate tax revenue from exports to Region r Matrix of dimensions 2  16: Tax revenue from use of Factor f by Sector a Column of dimension 2: Aggregate tax revenue from use of Factor f Matrix of dimensions 10  16: Transportation margin for imports of Product c from Region r Column of dimension 10: Aggregate margins for imports from Region r Matrix of dimensions 10  16: Imports of Product c from Region r Matrix of dimensions 10  16 in which 9 upper rows are zero vectors: Transportation margin for imports from Region r to Globe

A5

Export taxes (1..r)

C15

B15

In total

C12

C9

Factors (1..f )

C14

A15

B11

B6

B3

B1

Sectors (1..a)

Capital

Government

Direct tax

Indirect tax on Sector

Sales tax

A10

A8

Regions (1..r)

Households

A7

Transportation margins (1..r)

Taxes on use of Factors (1..f )

A4

A2

Import taxes (1..r)

Factors (1..f )

Sectors (1..a)

Products (1..c)

Products (1..c)

D15

D13

Import taxes (1..r)

E15

E13

Export taxes (1..r)

Table 2.1 Social accounting matrix (SAM) for one Region

F15

F13

Taxes on use of Factors (1..f )

G15

G8

Transportation margins (1..r)

H15

H14

H1

Regions (1..r)

I15

I14

I12

I1

Households

J15

J13

Sales tax

K15

K13

Indirect tax on Sector

L15

L13

Direct tax

M15

M14

M1

Government

N15

N1

Capital

O14

O13

O12

O11

O10

O9

O8

O7

O6

O5

O4

O3

O2

O1

In total

110

O8 C9 O9 A10 O10 B11 O11 C12 I12 O12 D13 E13 F13 J13 K13 L13 O13 C14 H14 I14 M14 O14 A15 B15 C15 D15 E15 F15 G15 H15 I15 J15 K15 L15 M15 N15

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Column of dimension 10: Aggregate imports from Region r Row of dimension 2: Households’ income from Factor f Element: Aggregate income of Households Row of dimension 16: Sales tax revenue by Product c Element: Aggregate sales tax revenue Row of dimension 16: Indirect tax revenue by Sector a Element: Aggregate indirect tax revenue by all Sectors Row of dimension 2: Income tax revenue from Factor f Element: Income tax revenue from Households Element: Aggregate income tax revenue from Households Row of dimension 10: Import tax revenue in Region r Row of dimension 10: Export tax revenue in Region r Row of dimension 2: Tax revenue from use of Factor f Element: Sales tax revenue Element: Indirect tax revenue by Sectors Element: Income tax revenue from Households Element: Aggregate government income Row of dimension 2: Depreciation of Factor f Row of dimension 10: Savings of Region r in given Region Element: Savings of Households Element: Savings of Government Element: Aggregate savings Row of dimension 16: Aggregate supply of Product c Row of dimension 16: Aggregate expenditures of Sector a Row of dimension 2: Aggregate expenditures on Factor f Row of dimension 16: Aggregate value of import tax from Region r Row of dimension 16: Aggregate value of export tax to Region r Row of dimension 2: Aggregate value of tax on use of Factor f Row of dimension 10: Aggregate value of import margins from Region r Row of dimension 10: Aggregate value of export margins to Region r Element: Aggregate expenditures of Households Element: Aggregate expenditures on sales tax Element: Aggregate value of indirect tax on Sectors Element: Aggregate value of income tax on Households Element: Aggregate government expenditures Element: Aggregate investments

The collection of SAMs for year 2015 was restored using Algorithm 2.1 (Kazakhstan) and Algorithm 2.2 (the other Regions) with the input–output table for Kazakhstan and year 2015, the macroeconomic indexes and international trade data for year 2015, and the collection of SAMs for the previous basic year (2011), as described below. Before the restoration procedure, the input–output table “Use of goods and services at consumer prices” (IOT) for Kazakhstan and year 2015 (see [16]) was

2.2 Adaptation of Model 1 to the Goals of Research

111

properly transformed to match the sectoral structure of Model 1 using an auxiliary algorithm (Algorithm IOT2), which includes the following steps: Step 1. In the IOT, choose 16 collections of columns and 16 collections of rows corresponding to the 16 aggregate Sectors in Model 1. Step 2. In each collection a (а ¼ 1, . . ., 16) of rows, replace the chosen rows with a single row for an aggregate Sector calculated as follows: each element of the new row is the sum of elements of the original IOT that lie at the junction of the corresponding column and rows of the collection а. Step 3. In each collection a (а ¼ 1, . . ., 16) of columns, replace the chosen columns with a single column for an aggregate Sector calculated as follows: each element of the new column is the sum of elements of the matrix obtained at Step 2 that lie at the junction of the corresponding row and columns of the collection а. Step 4. Convert all elements of the resulting matrix from million Kazakhstani Tenge (KZT) into million USD using the average exchange rate for year 2015 (221.73 KZT per 1 USD; see [13]). Algorithm IOT2 was employed to obtain a new input–output table (IOT2) with structure matching the 16 aggregate Sectors in Model 1. For eliminating the disproportions of international trade statistics, the available statistical data for year 2015 were balanced using an auxiliary algorithm (Algorithm ITB), which includes the following steps: Step 1. For each ordered pair (r, w) of Regions, extract from the World Bank database [81] the actual data on the aggregate exports of goods (FOB) from Region r to Region w, on the aggregate imports of goods (CIF) from Region w to Region r, and also on the aggregate exports of services from Region r and the aggregate imports of services to Region r (note that information on the exports and imports of services of Region r by its trade partners is not available). Calculate the data for the Rest of World using the aggregate data on global trade and the calculated data on regional trade. Express all data in million USD. Step 2. Correct the data for Kazakhstan (r ¼ kz) with an appropriate factor so that the aggregate exports of goods from Kazakhstan and the aggregate imports of goods to Kazakhstan are equal to the corresponding values from the input–output table “Use of goods and services at consumer prices” (IOT) for Kazakhstan and year 2015 (see [16]). Take the aggregate exports of services from Kazakhstan and the aggregate imports of services to Kazakhstan from this input–output table. Step 3. For each Region r, from the aggregate exports of services extract the exports of transportation services from Region r to Region Globe using the corresponding share obtained from the SAM for Region r and year 2011. Step 4. For each ordered pair (r, w) of Regions, recalculate the data on the exports of services from Region r to Region w using the aggregate exports of services from Region r (minus exports from Region r to Globe) and also the shares of exports of services from Region r to Region w in the aggregate exports of services (minus exports from Region r to Globe) from Region r obtained from the SAM for Region r and year 2011. For this pair of Regions, recalculate the data on the

112

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

imports of services from Region w to Region r using the aggregate imports of services to Region r and also the shares of imports of services from Region w to Region r in the aggregate imports of services to Region r obtained from the SAM for Region r and year 2011. Step 5. For each ordered pair (r, w) of Regions, check two conditions: (a) the exports of goods from Region r to Region w are smaller than the imports of goods from Region r to Region w; (b) the exports of services from Region r to Region w are equal to the imports of services from Region r to Region w. If these conditions fail, sequentially correct the bilateral trade data for each pair of Regions in accordance with the following descending priority: Kazakhstan, Russia, the USA, EU, Belarus, Armenia, Kyrgyzstan, China, and the Rest of World. In other words, whenever necessary, in this pair of Regions modify the trade data for the Region that is closer to the end of this list. (In particular, the international trade data for Kazakhstan were not modified.) Step 6. Correct the exports of transportation services from the Rest of World to Globe so that difference between the aggregate imports of goods and the aggregate exports of goods for all Regions is equal to the aggregate exports of transportation services of all Regions to Region Globe. For all Regions and year 2015, the balanced bilateral trade data yielded by this algorithm are shown in Tables 2.2 and 2.3. The next algorithms also involve an auxiliary algorithm (RAS) that solves the following problem: given a nonnegative matrix F0 of dimensions m  n without zero rows and columns, find positive factors so that the new matrix F* obtained from F0 with multiplication of its rows and columns by these factors has a desired column sum (an m-dimensional column u) and also a desired row sum (an n-dimensional row v). This algorithm was presented in the Handbook of Input-Output Table Compilation and Analysis [49] to update a square intermediate consumption matrix F0 in response to the appearance of new sums of its rows and columns. Consider a modification of the algorithm for a general (possibly not square) matrix F0. Step 1. Specify F0, u, v, and ε (ε > 0 denotes the accuracy of calculations). Assign k ¼ 0 (k is the counter of iterations). Step 2. Calculate the column uk+1 as the column sum of the matrix Fk. Calculate the col13уп umn rk+1 as the elementwise division of the column u by uk+1. Step 3. Multiply each row of the matrix Fk by the corresponding element rk+1 to calculate the matrix br kþ1 F k, where br kþ1 is a diagonal matrix of dimensions m  m obtained from the column rk+1. Step 4. Calculate the row vk+1 as the row sum of the matrix br kþ1 F k . Calculate the row sk+1 as the elementwise division of the row v by vkþ1 . Step 5. Multiply each column of the matrix br kþ1 F k by the corresponding element sk+1 to calculate the matrix F kþ1 ¼ br kþ1 F kbskþ1 , where bskþ1 is the diagonal matrix of dimensions n  n obtained from s1.

0.0

713.5

6 474.7

417.7

0.4

62.0

9 931.7

45 048.2

17

882.8

80 530.9

Kazakhstan

Kyrgyzstan

Russia

USA

Armenia

Belarus

China

EU

Rest of

World

In total

46220.5

483.5

13

12 904.9

731.1

12

783.7

6.4

2 661.9

3 254.2

394.8

0.0

1 252.6

661.3

58.6

32.8

0.1

0.1

2.5

67.4

0.0

368.3

Kyrgyzstan

Exports

Exports

Imports

Kazakhstan

Partner Region

Primary Region

9597.8

5 025.1

620.0

5 504.8

91.4

0.3

234.6

1 851.9

0.0

749.2

Imports

497833.5

138.8

159

259 052.0

37 414.6

16 539.8

534.8

9 553.5

0.0

1 737.7

3 091.5

Exports

Russia

286648.7

78 840.7

118 487.0

50 853.0

12 316.2

314.2

18 594.4

0.0

70.9

7 172.4

Imports

1619742.8

603.9

1 118

365 476.1

123 675.6

74.1

60.5

0.0

10 752.8

71.7

1 009.0

Exports

USA

2410855.4

481 405.7

1

416 984.7

486 296.2

140.6

101.2

0.0

24 465.1

2.5

1 459.5

Imports

1 490.19

446.1

465.2

171.0

9.0

0.0

87.1

304.6

0.3

6.1

Exports

Armenia

4159.5

1 396.7

1 134.8

414.4

31.0

0.0

112.7

1 069.3

0.1

0.6

Imports

27.2

36080.5

8 591.3

10 667.9

639.0

0.0

40502.3

8 061.2

9 448.6

948.0

0.0

9.5

78.0

868.6

700.4 119.3

21

6.5

82.5

Imports

11

88.8

744.5

Exports

Belarus

Table 2.2 Balanced international trade data (goods) for different Regions and year 2015 (in million USD) China

2342343.0

373 964.2

1

386 355.3

0.0

900.6

122.8

397 104.9

48 310.4

5 242.7

12 712.1

Exports

1958021.3

470 363.6

1

275 496.5

0.0

738.3

179.5

159 841.4

41 619.1

55.3

10 428.2

Imports

EU

6027201.5

333 141.8

1

3 859 411.5

228 676.0

8 976.2

964.6

396 135.5

112 562.7

563.2

11 586.1

Exports

5916866.0

037 565.9

4

4 052 382.1

406 689.7

11 201.3

773.6

383 749.9

285 621.8

1 068.2

47 300.6

Imports

Rest of World

8257405.4

845 300.9

3

1 414 341.1

244 662.9

1

7 658.2

1 326.9

1 407 335.4

74 898.6

923.4

12 745.4

Exports

Imports

9 093 011.3

575.6

4 520

1 661 980.8

662.4

1 442

14 642.7

511.0

1 342 771.1

030.7

204

694.3

30 582.0

0.0

1.5

56.3

295.4

0.9

2.5

62.7

1 216.5

1 031.3

2 656.2

5 323.3

Kazakhstan

Kyrgyzstan

Russia

USA

Armenia

Belarus

China

EU

Rest of World

Globe

In total

8 545.0

0.0

2 571.9

4 219.4

197.5

4.3

1.0

1 443.9

105.7

1.4

0.0

896.6

110.9

268.1

377.3

85.8

2.1

0.6

93.0

30.8

0.0

1.4

Exports

1 231.1

0.0

383.6

358.2

44.1

2.1

0.4

161.1

13.9

0.0

1.5

Imports

Kyrgyzstan

Exports

Imports

Kazakhstan

Partner Region

Primary Region

65 744.5

14 158.3

14 889.3

29 048.7

997.5

30.3

10.4

6 461.2

0.0

13.9

105.7

Exports

Russia

121 022.2

0.0

32 514.8

72 762.3

4 301.9

60.2

20.3

11 215.3

0.0

30.8

56.3

Imports

710 565.0

63 069.1

329 054.6

291 473.2

16 745.6

831.8

263.6

0.0

11 215.3

161.1

1 443.9

Exports

USA

477 428.0

0.0

241 149.3

211 682.8

17 868.0

263.6

178.0

0.0

6 461.2

93.0

295.4

Imports

1 620.8

413.7

398.0

567.2

35.9

0.9

0.0

178.0

20.3

0.4

1.0

Exports

Armenia

1 714.2

0.0

577.3

800.2

47.3

1.3

0.0

263.6

10.4

0.6

0.9

Imports

7 888.6

5 770.7

734.3

1 023.5

46.2

0.0

1.3

263.6

60.2

2.1

4.3

Exports

Belarus

Table 2.3 Balanced international trade data (services) for different Regions and year 2015 (in million USD)

5 736.0

0.0

1 965.6

2 695.4

155.1

0.0

0.9

831.8

30.3

2.1

2.5

Imports

280 477.1

98 437.2

92 819.8

58 640.0

0.0

155.1

47.3

17 868.0

4 301.9

44.1

197.5

Exports

China

452 832.1

0.0

277 973.2

131 428.6

0.0

46.2

35.9

16 745.6

997.5

85.8

62.7

Imports

2 238 394.7

417 058.9

494 903.1

1 041 948.4

131 428.6

2 695.4

800.2

211 682.8

72 762.3

358.2

4 219.4

Exports

EU

1 884 733.8

0.0

480 062.0

1 041 948.4

58 640.0

1 023.5

567.2

291 473.2

29 048.7

377.3

1 216.5

Imports

3 769 296.3

4 195 816.0

1 046 681.6

480 062.0

277 973.2

1 965.6

577.3

241 149.3

32 514.8

383.6

2 571.9

Exports

3 600 948.6

0.0

1 046 681.6

494 903.1

92 819.8

734.3

398.0

329 054.6

14 889.3

268.1

1 031.3

Imports

Rest of World

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Step 6. Check that the absolute values of all elements of the column (ukþ1  u) and row (vkþ1  v ) do not exceed ε. If this condition holds, take F kþ1 as the desired matrix F*. Stop. Step 7. If not, assign k ≔ k þ 1 and get back to Step 2. Using the prepared data for year 2015 as an example (see Tables 2.1, 2.2, and 2.3), consider another auxiliary algorithm (Algorithm 2.1А) to restore the international trade matrices H1, А7, and А8 as blocks of the SAMs for all Regions of Model 1 (see Table 2.1). This algorithm includes the following steps. Step 1. Find the elements of the matrix H1 for Kazakhstan. Substep 1.1. The elements H1c,r of this matrix are the exports of Product c from Kazakhstan to Region r; so fill with zeros the first column of H1, which corresponds to Kazakhstan. Also fill with zeros the last column of H1 (the exports of Kazakhstan to Globe) except for the lower element, which describes the exports of transportation services from Kazakhstan to Globe. Find this element from the corresponding element of Table 2.3 (2 656.2). Substep 1.2. Partition the inner part of the matrix H1 (without the first and last columns) into two submatrices, H1goods of dimensions 11  8 (consisting of the rows related to goods) and H1serv of dimensions 5  8 (consisting of the rows related to services). Substep 1.3. Find the column that is the row sum of the submatrix H1goods (the exports of each type of Product from Kazakhstan) from the corresponding elements of the column “Exports of goods and services” in IOT2. Find the row that is the column sum of the submatrix H1goods (the exports of goods from Kazakhstan to each other Region) from the column “Exports (Kazakhstan)” of Table 2.2. Using these data and the corresponding matrix H1goods 2011 obtained from the SAM for Kazakhstan and year 2011 (which was extracted from the GTAP Data Base), calculate all elements of H1goods by Algorithm RAS. Substep 1.4. Find the column that is the row sum of the submatrix H1serv (the exports of each service from Kazakhstan) from the corresponding elements of the column “Exports of goods and services” in IOT2, after deduction of the exports of transportation services to Globe. Find the row that is the column sum of the submatrix H1serv (the exports of services from Kazakhstan to each other Region) from the column “Exports (Kazakhstan)” of Table 2.3. Using these data and the corresponding matrix H1serv 2011 obtained from the SAM for Kazakhstan and year 2011, calculate all elements of H1serv by Algorithm RAS. This fully determines the desired matrix H1 for Kazakhstan. Step 2. Find the elements of the matrices А7 and А8 for Kazakhstan. Substep 2.1. The elements А7r,c of the matrix А7 are the imports (FOB) of Product c from Region r to Kazakhstan; so fill with zeros the first and last rows of this matrix, which correspond to Kazakhstan and Globe. The elements А8r,c of the matrix А8 are the transportation margins for the imports of Product

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c from Region r to Kazakhstan (i.e., the difference between the CIF and FOB prices); so fill with zeros the first and last rows of this matrix, which correspond to Kazakhstan and Globe. Substep 2.2. Partition the inner parts of the matrices А7 and А8 (without the first and last rows) into two submatrices, A7goods and A8goods of dimensions 8  11 (consisting of the columns related to goods) and A7serv and A8serv of dimensions 8  5 (consisting of the columns related to services). Fill with zeros the submatrix A7serv. Substep 2.3. Find the row that is the column sum of the submatrix A8serv (the imports of each service to Kazakhstan) from the corresponding elements of the row “Imports of goods and services” in IOT2. Find the column that is the row sum of the submatrix A8serv (the imports of services from each other Region to Kazakhstan) from the column “Imports (Kazakhstan)” of Table 2.3. Using these data and the corresponding matrix А8serv 2011 obtained from the SAM for Kazakhstan and year 2011, calculate all elements of A8serv by Algorithm RAS. Substep 2.4. Unite the two submatrices A7goods and A8goods into a single matrix A7A8goods of dimensions 16  11 (with A7goods placed above A8goods). Find the row that is the column sum of the submatrix A7A8goods (the CIF imports of each type of Product to Kazakhstan) from the corresponding elements of the row “Imports of goods and services” in IOT2. Find the column that is the row sum of the submatrix A8goods (the FOB imports of Products from each other Region to Kazakhstan) from the columns “Exports (Kazakhstan)” of Table 2.2. Find the column that is the row sum of the submatrix A7goods (the transportation margins for the imports of Products from each other Region to Kazakhstan) as the difference between the imports and exports of goods from each other Region to Kazakhstan from Table 2.2. Using these data and the corresponding matrix А7А8goods 2011 obtained from the SAM for Kazakhstan and year 2011, calculate all elements of A7A8goods by Algorithm RAS. This fully determines the desired matrices А7 and А8 for Kazakhstan. Step 3. Find the elements of the matrix H1 for each Region w different from Kazakhstan. Substep 3.1. Fill with zeros any columns of this matrix that correspond to Region w (the exports of Product с from Region w to Region w) if w differs from EU and the Rest of World. The elements of the first column of this matrix (the exports of Product с from Kazakhstan to Region w) are equal to the corresponding elements of the row w of the matrix А8 for Kazakhstan. Fill with zeros the last column of H1 (the exports from Region w to Globe) except for the lower element. Determine this element (the exports of transportation services from Region w to Globe) as the corresponding element of Table 2.3. Substep 3.2. Partition the inner parts of the matrix H1 (without the abovementioned columns) into two submatrices, H1goods and H1serv, consisting of the corresponding rows of H1 related to goods or services.

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Substep 3.3. Find the row that is the column sum of the submatrix H1goods (the imports of goods from Region w to each other Region) from the corresponding elements of the row “Exports (w)” in Table 2.2. Find the elements of each column of the submatrix H1goods using the known sum of elements of this column and the ratios of each its element to this sum calculated with the matrix Н1goods 2011 obtained from the SAM for Region w and year 2011. Substep 3.4. Find the row that is the column sum of the submatrix H1serv (the imports of services from Region w to each other Region) from the corresponding elements of the row “Exports (w)” in Table 2.3. Find the elements of each column of the submatrix H1serv using the known sum of elements of this column and the ratios of each its element to this sum calculated with the matrix Н1serv 2011 obtained from the SAM for Region w and year 2011. This fully determines the matrices H1 for all Regions. Step 4. Find the elements of the matrices А7 and А8 for Region w different from Kazakhstan. Substep 4.1. The elements of each row r of the matrix А8 (the FOB imports of Product c from Region r to Region w) are equal to the corresponding elements of column w in the matrix H1 for Region r (the exports of Product c from Region r to Region w); see above. Substep 4.2. The sum of each row r of the matrix А7 (the transportation margins for imports of Product с from Region r to Region w) is defined as the difference between the imports and exports of goods from Region r to Region w from Table 2.2. Find the elements of this row using the known sum of elements of this column and the ratios of each its element to this sum calculated with the matrix A72011 obtained from the SAM for Region w and year 2011. This fully determines the matrices A7 and A8 for all Regions. The matrices H1, A7, and A8 necessary for the SAMs for all Regions and years 2005, 2006, 2008–2010, and 2012–2015 were calculated using Algorithm 2.1A in MS Excel. Now, consider the main Algorithm 2.1 for obtaining SAMs for different Regions based on their input–output tables for past years. As an example, take the SAM for Kazakhstan and year 2015 based on IOT2 and the SAM for Kazakhstan and year 2011. This algorithm includes the following steps. Step 1. Calculate the matrices H1, А7, and А8 using Algorithm 2.1А. Step 2. Fill with the column sum of the matrix A7 the column О7 (the aggregate margins for imports from Region r), the row G15 (the aggregate value of import margins from Region r, the same as O7), and also the last row of the matrix G8 corresponding to Globe (the transportation margins for imports from Region r). Fill with zeros the other elements of the matrix G8.

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Step 3. Fill with the column sums of the matrices A8 and G8 the column О8 of the SAM (the aggregate imports from Region r) and also the row H15 (the aggregate value of export margins from Region r, the same as O8). Step 4. Calculate the row Н14 (the savings of Region r) as the difference between the row Н15 and the row sum of the matrix Н1. Step 5. Fill the matrix В1 (the use of intermediate Product с by Sector а) with the corresponding elements of the left upper (16  16) minor of IOT2. Step 6. The column I1 (the consumption of Product с by Households) consists of the 16 upper elements of the sum of the columns “Expenditures on final consumption of households” and “Expenditures on final consumption of NPISHs” from IOT2. Step 7. The column М1 (the consumption of Product с by Government) coincides with the 16 upper elements of the column “Expenditures on final consumption of public authorities, in total” from IOT2. Step 8. Fill the column N1 (the gross saving of Product c) with the corresponding 16 upper elements of the column “Gross saving, in total” from IOT2. Step 9. Fill the elements N15 (aggregate investments) and О14 (aggregate savings, the same as N15) with the sum of the column N1. Step 10. Fill the column О1 (the aggregate demand for Product с) with the corresponding 16 upper elements of the column “Aggregate demand for goods and services” from IOT2. The elements of the row A15 (the aggregate supply of Product с) are equal to the corresponding elements of the column О1. Step 11. Fill the elements of the diagonal matrix A2 (the output of Product с by Sector а) with the corresponding 16 elements of the row “Output at basic prices” from IOT2. The same 16 elements make up the column О2 (the aggregate output of Product с) and the row В15 (the aggregate expenditures of Sector а). Step 12. Fill the elements of the first row of the matrix B3 (the value of using Labor by Sector а) with the corresponding 16 elements of the row “Payment for Labor” from IOT2. Fill the elements of the second row of the matrix B3 (the value of using Capital by Sector а) with the corresponding elements of the sum of two rows from IOT2, “Profits, mixed income” and “Basic capital consumption.” Step 13. The column О3 of two elements (the aggregate income from Factor f ) and the row C15 (the aggregate expenditures on Factor f, the same as O3) are the row sums of the matrix В3. Step 14. Fill with zero the first element of the row C14 (the depreciation of Labor). Fill the second element of this row (the depreciation of Capital) with the last (total) element of the row “Basic capital consumption” from IOT2. Step 15. The row sums of the matrices А4 (the tax revenue from imports of Product с from Region r) and A5 (the tax revenue from exports of Product с from Region r) plus the row А10 (sales tax revenue by Product с) are the sum of the first 16 elements of the rows “Net taxes on goods” and “Margins” from IOT2. Calculate each element of the matrix А4 as the product of the corresponding element of the matrix (А7 + А8) and the import tax rate obtained using the corresponding elements of the sum of the matrices А4, А7, and А8 from the SAM for Kazakhstan and year 2011.

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119

Calculate each element of the matrix А5 as the product of the corresponding element of the matrix Н1 and the export tax rate obtained using the corresponding elements of the matrices А5 and Н1 from the SAM for Kazakhstan and year 2011. Finally, calculate the row А10 as the sum of the rows “Net taxes on goods” and “Margins” from IOT2, minus the row sums of the matrices А4 and А5. Step 16. Fill the elements of the column O4 (the aggregate revenue from the import taxes from Region r) as well as the corresponding elements of the rows D15 (the aggregate value of import taxes from Region r) and D13 (the revenue from the import taxes from Region r, the same as D15) with the elements of the column sum of the matrix А4. Step 17. Fill the elements of the column O5 (the aggregate tax revenue from exports to Region r) as well as the corresponding elements of the rows E15 (the aggregate value of export tax to Region r) and Е13 (export tax revenue in Region r, the same as E15) with the elements of the column sum of the matrix А5. Step 18. Fill the element О10 (the aggregate sales tax revenue), the corresponding element J15 (the aggregate expenditures on sales tax) and the element J13 (sales tax revenue, the same as J15) with the sum of the row А10. Step 19. The sum of the two rows of the matrix В6 (the tax revenue from the use of Factor f by Sector а) and the row В11 (the indirect tax revenue by Sector а) coincides with the first 16 elements of the row “Other taxes on production minus subsidies” from IOT2. Find the elements of the columns of the matrix B6 and the element a of the row B11 that correspond to Sector a using the known sum of these elements and the ratios of each element of B6 and B11 to their sum calculated with the matrices В6 and В11 from the SAM for Kazakhstan and year 2011. Step 20. Fill the elements of the column O6 (the aggregate tax revenue from the use of Factor f ) as well as the corresponding rows F15 (the aggregate value of tax on the use of Factor f ) and F13 (tax revenue from the use of Factor f, the same as F15) with the row sums of the matrix В6. Step 21. Calculate the element О11 (the aggregate indirect tax revenue by all Sectors) and the corresponding elements K15 (the aggregate value of indirect tax on Sectors) and К13 (the indirect tax revenue by Sectors, the same as K15) as the sum of the row В11. Step 22. The sum of the first elements of the rows С9 (the Households’ income from Labor) and row С12 (the income tax revenue from Labor) is the last (sum total) element of the row “Payment for Labor” from IOT2. The sum of the second elements of the rows С9 (the Households’ income from Capital) and row С12 (the income tax revenue from Capital) is the last (sum total) element of the row “Profits, mixed income” from IOT2. Calculate the first desired elements of the rows С9 and С12 as the product of the last element of the row “Payment for Labor” from IOT2 and the share of the first elements of these rows in their sum obtained from the SAM for Kazakhstan and year 2011.

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Calculate the second desired elements of the rows С9 and С12 as the product of the last element of the row “Profits, mixed income” from IOT2 and the share of the second elements of these rows in their sum obtained from the SAM for Kazakhstan and year 2011. Step 23. The element О9 (the aggregate income of Households) and the corresponding element I15 (the aggregate expenditures of Households) are the sum of the row С9. Step 24. Determine the element I12 (the income tax revenue from Households) using the share α of the value О9 (the aggregate income of Households). Estimate the value α using the corresponding share obtained from the SAM for Kazakhstan and year 2011. Step 25. The element О12 (the aggregate income tax revenue from Households), the corresponding element L15 (the aggregate value of income tax on Households), and also the element L13 (the income tax revenue from Households, the same as L15) are the sum of the rows С12 and I12. Step 26. Calculate the element I14 (the savings of Households) as the element I15 minus the element I12 and the sum of the column I1. Step 27. Calculate the element М14 (the savings of Government) as the element М15 minus the sum of the column М1. This fully determines the SAM for Kazakhstan and year 2015. The SAMs for other Regions and year 2015 without available input–output tables were obtained using Algorithm 2.2; see below. In addition, Algorithms 2.1 and 2.2 were employed to calculate all other SAMs for different Regions and the retrospective period 2004–2015 (except for the matrices extracted from the GTAP Data Base for years 2004, 2007, and 2011). Algorithm 2.2 is intended to calculate SAMs for different Regions based on several macroeconomic indicators from [80] (GDP; Aggregate investments; Imports of goods; Imports of services; Exports of goods; Exports of services; Aggregate government expenditures; Aggregate government expenditures for each Region of Model 1; a collection of SAMs for all Regions of Model 1 and past periods). In the case of the SAM for Kazakhstan, this algorithm includes the following steps. Step 1. Determine the share of the aggregate trade and transportation margins in the aggregate imports of goods from each Region for year t as the corresponding share for year t  1. To this effect, for each Region r (except for Globe) calculate the auxiliary ratio {The sum of the matrix А7 + the aggregate imports of goods from the matrix А8}/{The aggregate imports of goods from the matrix А8} using the SAM for the previous year t  1. Actually, this ratio is the CIF imports of goods divided by the FOB imports of goods for Region r, denoted by CIF ðr, t  1Þ=FOBðr, t  1Þ. All transportation margins for the imports of services are 0. Step 2. Calculate the basic values for the elements of the matrix Н1 (the exports of Product с to Region r, including Globe) for year t from the corresponding matrix for the previous year t  1 in the following way: multiply all rows related to

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121

goods by the coefficient {The exports of goods (t)/The exports of goods (t  1)} and all rows related to services by the coefficient {The exports of services (t)/The exports of services (t  1)}. As a result, the sum of Н1 is equal to the exports of goods and services for year t. Next, in order to satisfy the condition formulated at Step 1 of the algorithm for the EAEU countries, correct the elements of the matrix Н1 for three Regions of Model 1 (the Rest of World, EU, and China) and also correct the exports of transportation services from all 9 Regions to Globe. To this effect, using the SAM for year (t  1) calculate the shares of exports of the three Regions in the imports of all their trade partners (except for Globe), separately for goods and for services. Step 3. Calculate the final elements of the matrices Н1 for the three Regions (the Rest of World, EU, and China) and year t by choosing appropriate correction coefficients for their exports to each Region so that (1) these coefficients have minimum deviations from basic values; (2) the ratio of the CIF imports of goods to the FOB imports of goods for an importing Region remains the same; (3) the deviations of the correction coefficients from their basic values for the EU do not exceed those for China, and also the latter do not exceed the corresponding deviations for the Rest of World; and (4) the deviations of the shares of exports of the three Regions in the imports of all their trade partners (except for Globe) do not exceed 10%. (The basic values of the correction coefficients are 1, and in this case the exports have the initial values calculated at Step 2.) Formally, the correction coefficients for the exports of these three Regions can be calculated by solving the nonlinear programming problem X9

min

C ½7, r, C ½8, r, C ½9, r

i¼7

ðC½i, r   1Þ2

ð2:95Þ

subject to the constraints CIF ðr, t Þ=FOBðr, t Þ ¼ CIF ðr, t  1Þ=FOBðr, t  1Þ;

ð2:96Þ

2

2

ð2:97Þ

2

2

ð2:98Þ

ðC ½7, r   1Þ  ðC ½8, r   1Þ ; ðC ½8, r   1Þ  ðC ½9, r   1Þ ; jEX ði, r, t Þ=IM ðr, tÞ  EX ði, r, t  1Þ=IM ðr, t  1Þj  0, 1;

i ¼ 7, 8, 9:

ð2:99Þ

The notations are the following: C ½i, r  as the correction coefficient for the exports of goods (services) from Region i to Region r; CIF ðr, t Þ as given aggregate imports of goods (services) in CIF prices for Region r and year t; FOBðr, t Þ as the variable imports of goods (services) in FOB prices for Region r and year t; and finally, EX ði, r, t Þ=IM ðr, t Þ as the share of the exports of goods (services) from Region i in the aggregate imports of goods (services) from this Region. Note that the ratio CIF ðr, t  1Þ=FOBðr, t  1Þ is calculated at Step 1 of the algorithm. The first constraint can be relaxed if this problem has no solution.

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The final elements of the matrices Н1 for the three Regions are determined as the products of their basic values and the corresponding coefficients С[i, r]. Step 4. The elements of the matrix А8 (the imports of Product с from Region r) are equal to the corresponding exports of Product с to Kazakhstan; see the matrices Н1 for other Regions of Model 1. Step 5. Calculate the aggregate trade and transportation margins for year t as the difference between the aggregate CIF imports for year t and the aggregate FOB imports for year t obtained at Step 3. Calculate all elements of the matrix А7 (the trade and transportation margins for the imports of Product с from Region r) by multiplying the elements of the matrix А7 (the SAM for year (t  1)) by the ratio of the aggregate trade and transportation margins for year t to the sum of the matrix А7 for year (t  1). Step 6. Calculate the matrix А5 (tax revenue from the exports of Product c to Region r) for year t using the effective tax rates (which are determined by the ratio of the corresponding elements of the matrices А5 and Н1 for year t  1) and the matrix Н1. Step 7. The column О5 (the aggregate tax revenue from the exports to Region r) is the column sum of the matrix А5; its elements in the same arrangement also form the rows Е15 (the aggregate value of export tax to Region r) and Е13 (the export tax revenue in Region r). Step 8. The column О7 (the aggregate margins for the imports from Region r) is the column sum of the matrix А7. Its elements in the same arrangement also form the row G15 (the aggregate value of import margins from Region r) and the corresponding row of the matrix G8 (the transportation margin for the imports from Region r ¼ Globe). Fill with zeroes the other rows of the matrix G8. Step 9. The column О8 (the aggregate imports from Region r) is the column sum of the matrices А8 and G8. The row Н15 (the aggregate value of export margins to Region r) is the transposed column О8. Step 10. Write the calculated imports of transportation services from Globe to each Region into the first column of the SAM for Globe. Write the calculated exports of transportation services from each Region to Globe into the first row of the SAM for Globe. Consider these exports of transportation services from Regions to Globe as the basic ones. Next, find the optimal exports of transportation services from Regions to Globe by minimizing the sum of the squared ratios of the exports to their basic values determined at Step 2 above subject to the following constraint: the aggregate exports of Globe are equal to its aggregate imports. The solution of this optimization problem determines the SAM for Globe, the transportation services of Globe traded to other Regions, and also the exports of transportation services of all other Regions to Globe (the left lower elements of the matrices Н1). Step 11. The row Н14 (the savings of Region r) is the difference between the row Н15 and the row sum of the matrix Н1.

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Step 12. Calculate the matrix А4 (tax revenue from the imports of Product c from Region r) for year t using the effective tax rates (which are determined by the ratio of the corresponding elements of the matrix А4 to the sum (A7 + A8) from the SAM for year t  1) and the sum (A7 + A8) for the current year. Step 13. The column О4 (the aggregate tax revenue from imports from Region r) is the column sum of the matrix А4. Its elements in the same arrangement also form the rows D15 (the aggregate value of import tax from Region r) and D13 (the import tax revenue in Region r). Step 14. Using the forecasted data on GDP, the aggregate government income and the well-known relationship (GDP ¼ The aggregate government income + The aggregate income of Households), calculate the elements О9 (the aggregate income of Households) and О13 (the aggregate government income) as well as the elements I15 (the aggregate expenditures of Households) and М15 (the aggregate government expenditures), the same as the elements O9 and O13, respectively. Step 15. The difference between М15 and the forecasted data on government expenditures determines the element М14 (the savings of Government). Step 16. Calculate the column М1 (the consumption of Product с by Government) as the product of the column М1 for the previous year and the coefficient {The aggregate government expenditures for year t}/{The aggregate government expenditures for year (t  1)}. Step 17. Determine the elements О14 (aggregate savings) and N15 (aggregate investments) as the forecasted values of aggregate investments. Step 18. Calculate the column N1 (the gross saving of Product с) as the product of this column for the previous year and the coefficient {N15 for year t}/{N15 for year (t  1)}. Step 19. Determine the elements of the diagonal matrix А2 (the output of Product с by Sector а) by multiplying the matrix А2 for the previous year by the coefficient {GDP for year t}/{GDP for year (t  1)}. These elements in the same arrangement also form the column О2 (the aggregate output of Product с) and the row В15 (the aggregate expenditures of Sector а). Step 20. Calculate the elements of the row C9 (Households’ income from Factor f ) as the products of the above ratio of the elements О9 for the current and previous years and the corresponding elements of the row С9 for the previous year. Step 21. Calculate the elements I14 (the savings of Households) and I12 (income tax revenue from Households) as well as the corresponding elements of the column I1 (the consumption of Product c by Households) as the product of the element I15 and the coefficients that are equal to the ratios of the corresponding elements from the SAM for the previous year—I14, I12, and I1 to I15. Step 22. The first element of the row C14 (the depreciation of Labor) is 0. The second element of this row (the depreciation of Capital) is the difference between the element О14 and the sum of the elements Н14, I14, and М14. Step 23. Calculate the row С12 (income tax revenue from Factor f ) as the elementwise product of the row С12 for the previous year and the ratios of the corresponding elements of the row С9 for the previous and current years.

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2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Step 24. Calculate the elements of the row C15 (the aggregate expenditures on Factor f ) and the column O3 (the aggregate income from Factor f, the same as C15) as the sum of the elements С9, С12, and С14. Step 25. Determine the matrix B3 (the value of using Factor f by Sector а) by multiplying the rows of the matrix В3 for the previous year by the corresponding ratios of the elements of the column О3 for years t and (t  1). Step 26. Calculate the elements of the matrix B6 (tax revenue from the use of Factor f by Sector а) as the product of the corresponding elements of the matrix В3 and the effective tax rates (the ratios of the corresponding elements of the matrices В6 and В3 for the previous year). Step 27. Determine the column О6 (the aggregate tax revenue from the use of Factor f ) and the rows F13 (tax revenue from the use of Factor f ) and F15 (the aggregate value of tax on the use of Factor f ) using the column sum of the matrix В6. Step 28. Calculate the row В11 (the indirect tax revenue by Sector a) as the product of the row В11 from the SAM for the previous year and the ratio of the GDPs for the current and previous years. Step 29. Determine the element О11 (the aggregate indirect tax revenue by all Sectors) as well as the elements K15 (the aggregate value of indirect tax on Sectors) and K13 (the indirect tax revenue by Sectors), both the same as O11, using the sum of the row В11. Step 30. Calculate the element О12 (the aggregate income tax revenue from Households) as well as the elements L15 (the aggregate value of income tax on Households) and L13 (income tax revenue from Households), both the same as O12, as the sum of the row С12. Step 31. Determine the element J13 (sales tax revenue) as well as the elements J15 (the aggregate expenditures on sales tax) and О10 (aggregate sales tax revenue), both the same as J13, as the difference between the element О13 and the sums of D13, E13, F13, K13, and L13. Step 32. Calculate the elements of the row A10 (sales tax revenue by Product c) as the product of the element О10 and the corresponding coefficients {The сth element of the row А10 from the SAM for the current year}/{The сth element of the row А10 from the SAM for the previous year}. Step 33. Calculate the row А15 (the aggregate supply of Product c) as well as the column О1 (the aggregate demand for Product c, the same as A15) as the row sum of the matrices and rows А2, А4, А5, А7, А8, and А10. Step 34. Determine the matrix В1 (the use of intermediate Product c by Sector a) in the following way. Substep 34.1. The column u (the column sum of the matrix В1) is equal to the difference between the column О1 and the column sum of the matrices Н1, I1, M1, and N1. Substep 34.2. The row v (the row sum of the matrix В1) is equal to the difference between the row B15 and the row sum of the matrices В3, В6, and В11.

2.2 Adaptation of Model 1 to the Goals of Research

125

Substep 34.3. Using Algorithm RAS (see Sect. 2.2), calculate the elements of the matrix В1 from the vectors u and v and also the matrix В1 from the SAM for this Region and the previous year. This fully determines all elements of the SAM for the given Region and current year. For simulation modeling in GAMS, Model 1 was adapted to the chosen Regions, Sectors of economy, simulation periods (years), the collection of SAMs, substitution coefficients, and other parameters mentioned. In basic calculation, the computer simulations with the calibrated Model 1 well reproduced the statistical and forecasting data employed to obtain the collections of SAMs for the initial database of Model 1 and the period 2004–2022. The developed calibration module calculates all exogenous variables of Model 1 from the initial database using special expressions. The calibration procedure is performed during each launch of Model 1. As follows from analysis, Model 1 can be calibrated (in the sense of parameter estimation) using the following algorithm: Step 1. Specify the basic values for: • The output prices of Products (equal to 1) • The substitution coefficients for the CES functions of Model 1 Step 2. Specify the initial values for the dynamic variables (2.93). Step 3. Determine the other exogenous variables of Model 1 from the corresponding equations so that, in basic calculation, the endogenous variables of Model 1 are equal to the corresponding macroeconomic indexes from the collection of SAMs (see Sect. 2.4). As an example, consider calculation of the exogenous functions ρtc,r, γ c,r, and at c,r for some year t (2004  t  2022), which appear in the first-order conditions (2.12), (2.13) of the problem Br3ðc, r Þ. Step 3.1. The elasticities of the CET production function for Product c in Region r, denoted by ρtc,r , are determined at Step 1 of this algorithm. Step 3.2. Express γ c,r (the share parameter of the CET distribution function for Product c in Region r) from Equation (2.13) to obtain

γ c,r

 ρt 1 ! PDc,r QE c,r c,r ¼ 1= 1 þ : PE c,r QDc,r

ð2:100Þ

In accordance with the hypotheses of Model 1, the basic output prices of Product are PDc,r ¼ PE c,r ¼ 1; take the values QDc,r (the domestic demand for Product c in Region r) and QE c,r (the domestic output exported by Product c from Region r) from the SAM for Region r and year t. The parameter ρtc,r has been determined earlier.

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2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Step 3.3. Express at c,r (the shift parameter of the CET production function for Product c in Region r) from Equation (2.12) to obtain

at c,r ¼

 t1
 QXC c,r  t t ρ = γ c,r QE c,r ρc,r þ 1  γ c,r QDc,r ρc,r c,r : PXC c,r

ð2:101Þ

In this formula, the values ρtc,r , γ c,r , QE c,r , and QDc,r have been determined earlier, PXC c,r ¼ 1 , while the value QXC c,r (the domestic output by Product c in P Region r) is calculated using the relationship QXC c,r ¼ a SAMGa,c,r , where SAMGa,c,r denote the outputs of Sectors a by Product c in Region r, which are extracted from the SAM. The calibrated Model 1 was simulated in GAMS [38].

2.3

Applicability Testing of Model 1

The applicability of computer simulations with Model 1 to the real macroeconomic systems was tested using three approaches as follows: 1. Estimating the stability indexes of the mappings defined by Model 1 (see Sect. 1. 3.2). In such computer simulations, all exogenous parameters of Model 1 for year 2004 are considered as the input parameters ( p) while the GDPs of all Regions of Model 1 for a current year (from 2004 to 2022) as the output parameters ( y). The estimated values of the stability indexes βðp, αÞ for the basic value p and α ¼ 0:01 are presented in Table 2.4. All values of the stability indexes in Table 2.4 do not exceed 0.4536, which indicates of a rather high stability of Model 1 for the computer simulations up to year 2022. In particular, as shown by the value βf ðp, 0:01Þ for year 2004, the image of the ball with center at the point p (corresponding to the basic values of all exogenous parameters for year) and radius 0.01 (in relative values) was transformed into a set of diameter 0.4536 (in relative values) for the output variables (the GDPs of all Regions for year 2004). The calculated values of the limiting indexes βf ðp) were sufficiently close to 0 for α ¼ 0.0001, which estimated the mapping as continuous in the domain А.

Table 2.4 Estimated values of stability indexes in basic scenario Year Index β ( p, α) (in %)

2004 0.4536

2005 0.0813

2006 0.0097

2007 0.006

2008–2022 0.0

2.4 Macroeconomic Analysis Based on Model 1

127

2. Estimating the stability of the differential mappings defined by Model 1 (see Sect. 1.3.3). During computer simulations, a mapping f: A!B (dimA ¼ 5 and dimB ¼ 9) was studied in which the VAT rates for the five EAEU countries (Kazakhstan, Russia, Belarus, Armenia, Kyrgyzstan) and year 2015 were considered as the arguments and the GDPs of all 9 Regions of Model 1 for year 2022 as the output variables. The bounds of the five-dimensional parallelepiped A with center at the point p ¼ ( p1, . . ., p5) corresponding to the basic values of the above tax rates have deviations of 0.5pi from the values pi. Note that the computing time of the stability estimation algorithms demonstrated almost exponential growth for higher dimension dimA of the domain A. This fact essentially restricts the use of the second approach; for achieving acceptable computing time, a collection of most significant factors was chosen for solving the specific problems of macroeconomic analysis or parametric control based on Model 1. In accordance with the results of computer simulations, the mapping f had no singularities in the domain А and was estimated as a stable immersion. Therefore, both approaches proved that Model 1 was applicable to the real macroeconomic systems.

2.4

Macroeconomic Analysis Based on Model 1

The following macroeconomic characteristics in USD terms were analyzed for the simulation period 2004–2022 in the baseline scenario (which corresponds to basic calculation) and other scenarios: • A series of SNA and budget indexes for all nine Regions of Model 1 such as GDP, per capita GDP, the consumption of Households, government consumption, exports, imports, government income, and others • The economic indexes of the 16 Sectors for all 9 Regions of Model 1 This period consists of the retrospective period (2004–2015), which corresponds to the observed values of the economic indexes of Model 1, and the forecasting period (2016–2022), which corresponds to their forecasted values provided by the IMF [80].

2.4.1

Comparative Snapshot Analysis Based on Model 1

This subsection gives some examples of comparative snapshot analysis based on Model 1 (in the baseline scenario) at the following levels: • Some macroeconomic indexes for five regions (EAEU, EU, China, the USA, the Rest of World) • Some macroeconomic indexes for the five EAEU countries • Some sectoral indexes for the five EAEU countries

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2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Figures 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, and 2.7 below demonstrate the dynamics of some macroeconomic indexes of the five regions, namely, per capita GDP, the ratio of aggregate investments to GDP, the ratios of Households’ consumption and government consumption to GDP, the ratios of exports and imports, and trade gap of goods and services to GDP. The simulation results presented in Figs. 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, and 2.7 led to the following conclusions: 1. On the whole simulation period 2004–2022, the largest per capita GDPs among the Regions under study were observed for the USA and European Union (by year 2022, 69.7 thousand USD for the USA and 47.9 thousand USD for the EU). 8 7 6 chn

5

usa

4

eu

3

row 2

EAU

1 0

Fig. 2.1 Per capita GDP for the Eurasian Economic Union (EAU), the European Union (eu), China (chn), the USA (usa), and Rest of World (row), in thousand USD

0.6 0.5 0.4 0.3

chn usa eu

0.2

row EAU

0.1 0

Fig. 2.2 The ratio of aggregate investments to GDP for the Eurasian Economic Union (EAU), the European Union (eu), China (chn), the USA (usa), and the Rest of World (row)

0.8 0.7 0.6 chn

0.5

usa

0.4

eu

0.3

row

0.2

EAU

0.1 0

Fig. 2.3 The ratio of Households’ consumption to GDP for the Eurasian Economic Union (EAU), the European Union (eu), China (chn), the USA (usa), and the Rest of World (row)

0.3 0.25 0.2 0.15

chn usa eu

0.1

row

0.05

EAU

0

Fig. 2.4 The ratio of government consumption to GDP for the Eurasian Economic Union (EAU), the European Union (eu), China (chn), the USA (usa), and the Rest of World (row) 0.6 0.5 0.4 0.3

chn usa eu

0.2 0.1

row EAU

0

Fig. 2.5 The ratio of exports to GDP for the Eurasian Economic Union (EAU), the European Union (eu), China (chn), the USA (usa), and the Rest of World (row)

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

130 0.5 0.45 0.4 0.35 0.3 0.25

chn usa

0.2

eu

0.15

row

0.1

EAU

0.05 0

Fig. 2.6 The ratio of imports to GDP for the Eurasian Economic Union (EAU), the European Union (eu), China (chn), the USA (usa), and the Rest of World (row) 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

chn usa eu row EAU

-0.1 -0.12 -0.14

Fig. 2.7 The ratio of trade gap to GDP for the Eurasian Economic Union (EAU), the European Union (eu), China (chn), the USA (usa), and the Rest of World (row). Negative values indicate of trade surplus

By year 2022, this index for China would reach the corresponding value for the EAEU—11.7 thousand USD. 2. On the whole simulation period 2004–2022, the largest ratio of aggregate investments to GDP was observed for China, varying from 0.41 to 0.50. For the other Regions of Model 1, this ratio lay within the range 0.17–0.26. 3. On the whole simulation period 2004–2022, the smallest ratio of Households’ consumption to GDP was observed for China, varying from 0.28 to 0.41. For the other Regions of Model 1, this ratio lay within the range 0.52–0.70. 4. For year 2022, the largest ratios of government consumption to GDP were observed for the USA (0.23) and European Union (0.21) while the smallest ratios

2.4 Macroeconomic Analysis Based on Model 1

131

for the Eurasian Economic Union (0.15), China, and the Rest of World (both 0.17). 5. The most export-oriented economy was the European Union (the ratio of exports to GDP lay within the range 0.32–0.49). The least export-oriented economy was the USA (the ratio of exports to GDP lay within the range 0.08–0.13). 6. The economy least dependent on imports was the USA (the ratio of imports to GDP lay within the range 0.13–0.18). The economy most dependent on imports was the European Union (the ratio of imports to GDP lay within the range 0.32–0.47). 7. On the whole simulation period 2004–2022, the largest trade gap was observed for the USA (the ratio of trade gap to GDP lay within the range 0.03–0.06). Since year 2014, trade surpluses were observed for China, the European Union, and the Eurasian Economic Union. By year 2022, the ratio of trade surplus to GDP for the Eurasian Economic Union would reach 0.027. Next, Figs. 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, and 2.14 show the dynamics of the same macroeconomic indexes for the five EAEU countries. The simulation results presented in Figs. 2.8, 2.9, 2.10, 2.11, 2.12, 2.13, and 2.14 led to the following conclusions: 1. On the whole simulation period 2004–2022, the largest per capita GDPs among the Regions under study were observed for Russia and Kazakhstan (by year 2022, 12.6 thousand USD for Russia and 11.8 thousand USD for Kazakhstan). The smallest values of this index were observed for Kyrgyzstan, from 0.41 thousand USD for year 2004 to 1.19 thousand USD for year 2020. 2. On the forecasting period 2016–2022, the largest ratio of aggregate investments to GDP (see Fig. 2.9) was observed for Kyrgyzstan, varying from 0.33 to 0.36. 18 16 14 12 blr 10

rus

8

kaz

6

kgz arm

4 2 0

Fig. 2.8 Per capita GDP for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm), in thousands USD

0.45 0.4 0.35 0.3

blr

0.25

rus

0.2

kaz

0.15

kgz

0.1

arm

0.05 0

Fig. 2.9 The ratio of aggregate investments to GDP for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

blr rus kaz kgz arm

0

Fig. 2.10 The ratio of Households’ consumption to GDP for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

0.45 0.4 0.35 0.3

blr

0.25

rus

0.2

kaz

0.15

kgz

0.1

arm

0.05 0

Fig. 2.11 The ratio of government consumption to GDP for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

2.4 Macroeconomic Analysis Based on Model 1

133

0.8 0.7 0.6 0.5 0.4 0.3

blr rus kaz kgz

0.2

arm

0.1 0

Fig. 2.12 The ratio of exports to GDP for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

1.8 1.6 1.4 1.2

blr

1

rus

0.8

kaz

0.6

kgz

0.4

arm

0.2 0

Fig. 2.13 The ratio of imports to GDP for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

For Armenia, this ratio took the smallest values on the forecasting period within the range 0.18–0.19. 3. On the forecasting period 2016–2022, the smallest ratios of Households’ consumption to GDP (see Fig. 2.10) were observed for Kyrgyzstan and Armenia, varying from 0.79 to 0.84. For the other EAEU countries, this ratio lay within the range 0.51–0.60. 4. On the forecasting period 2016–2022, the largest ratios of government consumption to GDP (see Fig. 2.11) were observed for Kyrgyzstan (0.36–0.41) while the smallest ratios for Kazakhstan (0.08–0.13). 5. On the forecasting period 2016–2022, the most export-oriented economy was Belarus (the ratio of exports to GDP lay within the range 0.51–0.55; see

134

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

1.4 1.2 1 0.8 0.6

blr rus kaz

0.4 0.2

kgz arm

0 -0.2 -0.4

Fig. 2.14 The ratio of trade gap to GDP for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

Fig. 2.12). The least export-oriented economy was Russia (the ratio of exports to GDP lay within the range 0.24–0.26). 6. The economy most dependent on imports was Kyrgyzstan (on the forecasting period 2016–2022, the ratio of imports to GDP lay within the range 0.89–0.92; see Fig. 2.13). The economies least dependent on imports were Kazakhstan and Russia (on the forecasting period 2016–2022, the ratio of imports to GDP lay within the range 0.21–0.23). 7. On the whole simulation period 2004–2022, the largest trade gap was observed for Kyrgyzstan (the ratio of trade gap to GDP lay within the range 0.23–1.15; see Fig. 2.14). On the whole simulation period, Kazakhstan had trade surplus; by year 2022, the ratio of trade surplus to GDP for Kazakhstan would reach 0.10. Figures 2.15, 2.16, 2.17, 2.18, 2.19, 2.20, 2.21, 2.22, 2.23, 2.24, 2.25, and 2.26 below illustrate the dynamics of some fiscal policy tools for the EAEU countries. The simulation results presented in Figs. 2.15, 2.16, 2.17, 2.18, 2.19, 2.20, 2.21, 2.22, 2.23, 2.24, 2.25, and 2.26 led to the following conclusions: 1. The effective rates of net Labor income tax (see Fig. 2.15) for the five EAEU countries had inconsiderable variations on the retrospective period and were treated as constants on the forecasting period. These rates for Kazakhstan and Russia were significantly larger than for the other EAEU countries. For Armenia, these rates were close to 0. 2. The effective rates of net Capital income tax (see Fig. 2.16) for the five EAEU countries had inconsiderable variations on the retrospective period and were

2.4 Macroeconomic Analysis Based on Model 1

135

0.3500 0.3000 0.2500 0.2000

blr rus

0.1500

kaz kgz

0.1000

arm

0.0500 -0.0500

Fig. 2.15 Effective rates of net Labor income tax for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm) 0.16 0.14 0.12 0.1

blr rus

0.08 0.06 0.04

kaz kgz arm

0.02 0

Fig. 2.16 Effective rates of net Capital income tax for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

treated as constants on the forecasting period. The largest values of these rates were for Belarus while the smallest for Kyrgyzstan. 3. The effective rates of net Households’ income tax (see Fig. 2.17) were positive for Kyrgyzstan and negative for the other EAEU countries (with the smallest negative value for Armenia). Negative values of these rates indicated that the government’s transfers to Households (pensions, social security benefits) were greater than the taxes collected from them.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

136 0.2 0.1 0

blr rus

-0.1

kaz -0.2

kgz arm

-0.3 -0.4 -0.5

Fig. 2.17 Effective rates of net Households’ income tax for Kazakhstan (kaz), Russia (rus), Belarus (blr), Kyrgyzstan (kgz), and Armenia (arm)

0.90 0.80

cming ccrog

0.70 0.60

cmepe cmind cehas

0.50

cpegw cfpin

0.40 0.30

cpsta cos coths

0.20

cagff cbuil

0.10 -

cmtal cfins cchpp

(0.10) (0.20)

Fig. 2.18 Effective rates of net sales tax (VAT) for Kazakhstan, by 16 types of Product

ctser

0.25

cming 0.2

ccrog cmepe cmind

0.15

cehas cpegw cfpin

0.1

cpsta cos coths cagff

0.05

cbuil cmtal cfins

0

cchpp ctser

-0.05

Fig. 2.19 Effective rates of net sales tax (VAT) for Russia, by 16 types of Product 0.3 cming ccrog 0.25

cmepe cmind

0.2

cehas cpegw cfpin

0.15

cpsta cos

0.1

coths cagff cbuil

0.05

cmtal cfins cchpp

0

Fig. 2.20 Effective rates of net sales tax (VAT) for Belarus, by 16 types of Product

ctser

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

138 0.2

cming ccrog 0.15

cmepe cmind cehas cpegw

0.1

cfpin cpsta cos

0.05

coths cagff cbuil cmtal

0

cfins cchpp ctser

-0.05

Fig. 2.21 Effective rates of net sales tax (VAT) for Kyrgyzstan, by 16 types of Product

4. The effective rates of net sales tax (including VAT) for the EAEU countries (see Figs. 2.18, 2.19, 2.20, 2.21, and 2.22) had considerable variations by all 16 types of Products. A significant increase of these rates for Kazakhstan since year 2014 was explained by another source of data: for years 2004, 2007, and 2011, they were calculated using the SAMs extracted from the GTAP Data Base and for the period 2014–2015, using the national input–output tables for Kazakhstan. 5. The effective rates of import tax for 10 types of Products from the USA, the European Union, China, and the Rest of World (see Figs. 2.23, 2.24, 2.25, and 2.26) were considered constant since year 2011 because they were determined using the last available SAM extracted from the GTAP Data Base (for year 2011). Figures 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39, 2.40, 2.41, and 2.42 demonstrate the comparative analysis of the shares of exports in the outputs of different Sectors for the five EAEU countries. The simulation results presented in Figs. 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.39, 2.40, 2.41, and 2.42 led to the following conclusions: 1. On almost the whole simulation period, the largest share of exports in the output of Sector 1 (Mining) was observed for Armenia while the smallest for Belarus; see Fig. 2.27.

2.4 Macroeconomic Analysis Based on Model 1

139

0.9

0.8 cming 0.7

ccrog cmepe

0.6

cmind cehas

0.5

cpegw cfpin

0.4

cpsta cos coths

0.3

cagff cbuil

0.2

cmtal cfins

0.1

cchpp ctser

0

-0.1

Fig. 2.22 Effective rates of net sales tax (VAT) for Armenia, by 16 types of Product

2. On the whole simulation period, the largest share of exports in the output of Sector 2 (Hydrocarbon production and natural gas extraction) was observed for Kazakhstan; for year 2008, this share even exceeded 90%. The smallest values of this index were observed for Armenia and Kyrgyzstan (less than 5%); see Fig. 2.28. 3. Since year 2014, a sharp increase in the shares of exports in the output of Sector 3 (Metalworking and machine building) was observed for Armenia, Kyrgyzstan, and Kazakhstan. On the retrospective period, these shares demonstrated a downtrend for Russia and Belarus; see Fig. 2.29. 4. Since year 2014, an increase in the shares of exports in the output of Sector 4 (Metal industry) was observed for Belarus, Armenia, and Kyrgyzstan. This share demonstrated an uptrend for Kazakhstan; for Russia, however, this share was mostly decreasing on the retrospective period and weakly increasing on the forecasting period; see Fig. 2.30. 5. The largest shares of exports in the output of Sector 5 (Education, public health, and public administration) were observed for Armenia and Kyrgyzstan. For Russia and Kazakhstan, these shares did not exceed 1% on the forecasting period; see Fig. 2.31.

140

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

0.6

0.5

cming 0.4

ccrog cmepe cmind

0.3

cpegw cfpin cos

0.2

cagff cmtal cchpp

0.1

0

Fig. 2.23 Effective rates of import tax from the USA to Kazakhstan

0.16 0.14 cming 0.12

ccrog

0.1

cmepe cmind

0.08 0.06

cpegw cfpin cos

0.04 0.02

cagff cmtal cchpp

0

Fig. 2.24 Effective rates of import tax from China to Kazakhstan

2.4 Macroeconomic Analysis Based on Model 1

141

0.25

0.2

cming ccrog cmepe

0.15

cmind cpegw

0.1

cfpin cos cagff

0.05

cmtal cchpp

0

Fig. 2.25 Effective rates of import tax from EU to Kazakhstan 0.12 0.1

cming ccrog

0.08

cmepe cmind

0.06

cpegw cfpin

0.04

cos cagff

0.02

cmtal cchpp

0

Fig. 2.26 Effective rates of import tax from the Rest of World to Kazakhstan

6. The largest share of exports in the output of Sector 6 (Production and supply of electricity, gas, and hot water) was observed for Kyrgyzstan; this share also demonstrated an uptrend for Armenia. For Russia, Kazakhstan, and Belarus, these shares did not exceed 3% on the forecasting period; see Fig. 2.32. 7. The share of exports in the output of Sector 7 (Food industry, including beverages and tobacco) took the largest values among all other shares, reaching 20% on the forecasting period. For Russia, this share did not exceed 5% on the forecasting period; see Fig. 2.33.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

142

0.8 0.7 0.6 blr

0.5

rus

0.4

kaz 0.3

kgz

0.2

arm

0

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

0.1

Fig. 2.27 The share of exports in the output of Sector 1 (Mining) for five EAEU countries

1 0.9 0.8 0.7

blr

0.6

rus

0.5 0.4

kaz

0.3

kgz

0.2

arm

0

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

0.1

Fig. 2.28 The share of exports in the output of Sector 2 (Hydrocarbon production and natural gas extraction) for five EAEU countries

8. The largest shares of exports in the output of Sector 8 (Professional, scientific, and technical activities) were observed for Russia and Belarus, even exceeding 60% on the forecasting period. This share took the smallest value for Kazakhstan (about 3%); see Fig. 2.34. 9. The share of exports in the output of Sector 9 (Other industries) for Armenia was significantly greater than its counterparts for other countries. For Kazakhstan, this share was minimum value, not exceeding 10%; see Fig. 2.35.

2.4 Macroeconomic Analysis Based on Model 1

143

0.8 0.7 0.6 blr

0.5

rus

0.4

kaz

0.3

kgz

0.2

arm

0

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

0.1

Fig. 2.29 The share of exports in the output of Sector 3 (Metalworking and machine building) for five EAEU countries

1 0.9 0.8 0.7

blr

0.6

rus

0.5

kaz

0.4 0.3

kgz

0.2

arm

0

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

0.1

Fig. 2.30 The share of exports in the output of Sector 4 (Metal industry) for five EAEU countries

10. The share of exports in the output of Sector 10 (Other services) took largest values for Kyrgyzstan and Armenia. For Russia and Kazakhstan, this share was about 2% on the forecasting period; see Fig. 2.36. 11. The share of exports in the output of Sector 11 (Agriculture, forestry, and fishery) took largest values for Russia and Kyrgyzstan on the forecasting period (up to 12%). And the smallest share was observed for Belarus (less 2% on the forecasting period); see Fig. 2.37.

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Fig. 2.31 The share of exports in the output of Sector 5 (Education, public health, and public administration) for five EAEU countries

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Fig. 2.32 The share of exports in the output of Sector 6 (Production and supply of electricity, gas, and hot water) for five EAEU countries

12. The largest share of exports in the output of Sector 12 (Construction) was observed for Belarus (about 17%) while the smallest share for Kyrgyzstan (less 1%); see Fig. 2.38. 13. The share of exports in the output of Sector 13 (Production of textiles, clothes, leather, and associated goods) demonstrated a sharp increase for Kazakhstan and Belarus (reaching almost 90% for the latter) and a sharp decrease for Kyrgyzstan and Armenia after year 2013. For Russia, this share took the minimum value— 3% on the forecasting period; see Fig 2.39.

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.33 The share of exports in the output of Sector 7 (Food industry, including beverages and tobacco) for five EAEU countries

0.9 0.8 0.7 0.6

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Fig. 2.34 The share of exports in the output of Sector 8 (Professional, scientific, and technical activities) for five EAEU countries

14. The largest share of exports in the output of Sector 14 (Financial services) was observed for Armenia (even exceeding 50% for year 2016). For Kazakhstan, this share did not exceed 2%; see Fig. 2.40. 15. The shares of exports in the output of Sector 15 (Chemical and petrochemical industry) for the EAEU countries varied within the range 20–70%; see Fig. 2.41. 16. The largest shares of exports in the output of Sector 16 (Transportation) were observed for Kyrgyzstan and Armenia (even exceeding 90% for separate years). For Kazakhstan and Russia, these shares did not exceed 20%; see Fig. 2.42.

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Fig. 2.35 The share of exports in the output of Sector 9 (Other industries) for five EAEU countries

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Fig. 2.36 The share of exports in the output of Sector 10 (Other services) for five EAEU countries

For the 16 types of Products, Figs. 2.43, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.52, 2.53, 2.54, 2.55, 2.56, 2.57, and 2.58 show the comparative analysis of the corresponding ratios of Households’ consumption to the outputs of Sectors for the five EAEU countries. The simulation results led to the following conclusions: 1. The largest ratios of Households’ consumption to the output of Sector 1 (Mining) were observed for Armenia and Kazakhstan (on the forecasting period); the smallest ratios were observed for Russia (about 1%); see Fig. 2.43.

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.37 The share of exports in the output of Sector 11 (Agriculture, forestry, and fishery) for five EAEU countries

0.25 0.2 blr

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Fig. 2.38 The share of exports in the output of Sector 12 (Construction) for five EAEU countries

2. For Armenia, the ratio of Households’ consumption to the output of Sector 2 (Hydrocarbon production and natural gas extraction) was considerably greater than 1; this fact was due to an extremely small output of Sector 1 in Armenia and the dominating consumption of imported goods by Households. For Belarus and Kyrgyzstan, this index did not exceed 1; for Russia and Kazakhstan, it was measured by fractions of percent; see Fig. 2.44. 3. For Kazakhstan, the ratio of Households’ consumption to the output of Sector 3 (Metalworking and machine building) exceeded 1 since year 2014, which testified to a considerable share of imported goods in consumption. The

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Fig. 2.39 The share of exports in the output of Sector 13 (Production of textiles, clothes, leather, and associated goods) for five EAEU countries

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Fig. 2.40 The share of exports in the output of Sector 14 (Financial services) for five EAEU countries

corresponding ratios for the other EAEU countries did not exceed 1; for Belarus, this index did not exceed 11%; see Fig. 2.45. 4. The largest ratios of Households’ consumption to the output of Sector 4 (Metal industry) were observed for Armenia (up to 32%). For Russia and Belarus, this index was close to 0; see Fig. 2.46. 5. The ratios of Households’ consumption to the output of Sector 5 (Education, public health, and public administration) for the EAEU countries varied within the range 2.4–58%, with the largest values observed for Kyrgyzstan (less than 4%); see Fig. 2.47.

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.41 The share of exports in the output of Sector 15 (Chemical and petrochemical industry) for five EAEU countries

1 0.9 0.8 0.7

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Fig. 2.42 The share of exports in the output of Sector 16 (Transportation) for five EAEU countries

6. On the forecasting period, the ratios of Households’ consumption to the output of Sector 6 (Production and supply of electricity, gas, and hot water) for the EAEU countries varied within the range 31–53%, with the largest values observed for Kazakhstan and Armenia and the smallest for the other EAEU countries; see Fig. 2.48. 7. On the forecasting period, the ratios of Households’ consumption to the output of Sector 7 (Food industry, including beverages and tobacco) for Kazakhstan and Kyrgyzstan exceeded 1 (for Kazakhstan, this index is greater than 1.8 since year 2017). The smallest ratio was observed for Armenia (less than 0.8); see Fig. 2.49.

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Fig. 2.43 The ratio of sectoral Households’ consumption to the output of Sector 1 (Mining) for five EAEU countries

120000 100000 80000

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Fig. 2.44 The ratio of sectoral Households’ consumption to the output of Sector 2 (Hydrocarbon production and natural gas extraction) for five EAEU countries

8. The largest ratios of Households’ consumption to the output of Sector 8 (Professional, scientific, and technical activities) were observed for Russia (up to 30% on the forecasting period) while the smallest for Armenia (not greater than 2%); see Fig. 2.50. 9. The largest ratios of Households’ consumption to the output of Sector 9 (Other industries) were observed for Kazakhstan (above 1 since year 2016) while the smallest for Belarus (less than 18%); see Fig. 2.51.

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.45 The ratio of sectoral Households’ consumption to the output of Sector 3 (Metalworking and machine building) for five EAEU countries

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Fig. 2.46 The ratio of sectoral Households’ consumption to the output of Sector 4 (Metal industry) for five EAEU countries

10. The largest ratios of Households’ consumption to the output of Sector 10 (Other services) were observed for Armenia (above 0.6 since year 2016). For Kazakhstan, this index was minimal among all other EAEU countries, showing a downtrend to 25% by year 2022; see Fig. 2.52. 11. On the forecasting period, the largest ratios of Households’ consumption to the output of Sector 11 (Agriculture, forestry, and fishery) were observed for Kazakhstan (up to 98%) while the smallest for Kyrgyzstan (22–25%); see Fig. 2.53.

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Fig. 2.47 The ratio of sectoral Households’ consumption to the output of Sector 5 (Education, public health, and public administration) for five EAEU countries

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Fig. 2.48 The ratio of sectoral Households’ consumption to the output of Sector 6 (Production and supply of electricity, gas, and hot water) for five EAEU countries

12. The largest ratios of Households’ consumption to the output of Sector 12 (Construction) were observed for Armenia (up to 21%) while the smallest for Kyrgyzstan (not greater than 1.2%); see Fig. 2.54. 13. On the forecasting period, the largest ratios of Households’ consumption to the output of Sector 13 (Production of textiles, clothes, leather, and associated goods) were observed for Kazakhstan (5.5–6%, which testified to the dominating consumption of imported goods) while the smallest for Belarus (about 45%); see Fig. 2.55.

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.49 The ratio of sectoral Households’ consumption to the output of Sector 7 (Food industry, including beverages and tobacco) for five EAEU countries

0.35 0.3 0.25 blr 0.2

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Fig. 2.50 The ratio of sectoral Households’ consumption to the output of Sector 8 (Professional, scientific, and technical activities) for five EAEU countries

14. The largest ratios of Households’ consumption to the output of Sector 14 (Financial services) were observed for Armenia (80–104%) while the smallest primarily for Kyrgyzstan (less than 2.6% on the forecasting period); see Fig. 2.56. 15. The largest ratios of Households’ consumption to the output of Sector 15 (Chemical and petrochemical industry) were observed primarily for Kyrgyzstan (above 2 on the forecasting period) while the smallest for Belarus and Russia (within the range 16–22% on the forecasting period); see Fig. 2.57.

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Fig. 2.51 The ratio of sectoral Households’ consumption to the output of Sector 9 (Other industries) for five EAEU countries

0.8 0.7 0.6 blr

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Fig. 2.52 The ratio of sectoral Households’ consumption to the output of Sector 10 (Other services) for five EAEU countries

16. On the forecasting period, the largest ratios of Households’ consumption to the output of Sector 16 (Transportation) were observed for Kyrgyzstan and Armenia (45–48%) while the smallest for Belarus (10–12%); see Fig. 2.58. Next, Figs. 2.59, 2.60, 2.61, 2.62, 2.63, 2.64, 2.65, 2.66, 2.67, 2.68, 2.69, 2.70, 2.71, 2.72, 2.73, and 2.74 illustrate the comparative analysis of GVA shares in sectoral GDPs for the five EAEU countries. These indexes characterize the contribution of each Sector to GDP.

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.53 The ratio of sectoral Households’ consumption to the output of Sector 11 (Agriculture, forestry, and fishery) for five EAEU countries

0.25 0.2 blr

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Fig. 2.54 The ratio of sectoral Households’ consumption to the output of Sector 12 (Construction) for five EAEU countries

The simulation results led to the following conclusions: 1. The largest GVA shares of Sector 1 (Mining) in GDP were observed for Kazakhstan (within the range 2.9–4%) while the smallest for Kyrgyzstan (0.27–0.69%); see Fig. 2.59. 2. On the forecasting period, the largest GVA shares of Sector 2 (Hydrocarbon production and natural gas extraction) in GDP were observed for Russia (about 14.7%) while the smallest for Armenia (close to 0); see Fig. 2.60.

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Fig. 2.55 The ratio of sectoral Households’ consumption to the output of Sector 13 (Production of textiles, clothes, leather, and associated goods) for five EAEU countries

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Fig. 2.56 The ratio of sectoral Households’ consumption to the output of Sector 14 (Financial services) for five EAEU countries

3. On the forecasting period, the largest GVA shares of Sector 3 (Metalworking and machine building) in GDP were observed for Belarus (above 8%) while the smallest for Kazakhstan (about 0.6%); see Fig. 2.61. 4. The largest GVA shares of Sector 4 (Metal industry) in GDP were observed for Kyrgyzstan (within the range 7.3–7.7% on the forecasting period) while the smallest for Belarus (below 0.4%); see Fig. 2.62. 5. On the forecasting period, the largest GVA shares of Sector 5 (Education, public health, and public administration) in GDP were observed for Russia (slightly

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.57 The ratio of sectoral Households’ consumption to the output of Sector 15 (Chemical and petrochemical industry) for five EAEU countries

0.9 0.8 0.7 0.6

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Fig. 2.58 The ratio of sectoral Households’ consumption to the output of Sector 16 (Transportation) for five EAEU countries

less than 15%) while the smallest for Kazakhstan (slightly greater than 5%); see Fig. 2.63. 6. On the forecasting period, the largest GVA shares of Sector 6 (Production and supply of electricity, gas, and hot water) in GDP were observed for Kyrgyzstan (slightly less than 15%) while the smallest for Kazakhstan (about 1.8%); see Fig. 2.64. 7. On the forecasting period, the largest GVA shares of Sector 7 (Food industry, including beverages and tobacco) in GDP were observed for Armenia (about 13.3%) while the smallest for Kazakhstan (about 2.4%); see Fig. 2.65.

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Fig. 2.59 The GVA share of Sector 1 (Mining) in GDP for five EAEU countries

0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

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2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

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Fig. 2.60 The GVA share of Sector 2 (Hydrocarbon production and natural gas extraction) in GDP for five EAEU countries

8. On the forecasting period, the largest GVA shares of Sector 8 (Professional, scientific, and technical activities) in GDP were observed for Russia (about 7.5%) while the smallest for Armenia (less than 0.5%); see Fig. 2.66. A considerable reduction in this index for Kazakhstan between years 2011 and 2014 is explained by the data deviations between the GTAP Data Base (year 2011) and the national input–output tables for Kazakhstan (year 2014); see Fig. 2.66.

2.4 Macroeconomic Analysis Based on Model 1

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

159

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Fig. 2.61 The GVA share of Sector 3 (Metalworking and machine building) in GDP for five EAEU countries

blr rus kaz kgz arm

2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

Fig. 2.62 The GVA share of Sector 2 (Metal industry) in GDP for five EAEU countries

9. On the forecasting period, the largest GVA shares of Sector 9 (Other industries) in GDP were observed for Belarus (about 4.4%) while the smallest for Kyrgyzstan (about 0.23%); see Fig. 2.67. 10. On the forecasting period, the largest GVA shares of Sector 10 (Other services) in GDP were observed for Kazakhstan (about 38%) while the smallest for Belarus (about 8%); see Fig. 2.68. Actually, this was the greatest sector for Kazakhstan; considerable variations of this share for years 2011–2014 had the same reason as described above.

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Fig. 2.63 The GVA share of Sector 5 (Education, public health, and public administration) in GDP for five EAEU countries

0.4 0.35 0.3 blr

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Fig. 2.64 The GVA share of Sector 6 (Production and supply of electricity, gas, and hot water) in GDP for five EAEU countries

11. On the forecasting period, the largest GVA shares of Sector 11 (Agriculture, forestry, and fishery) in GDP were observed for Belarus (about 18%) while the smallest for Russia (slightly greater than 3%); see Fig. 2.69. 12. The largest GVA shares of Sector 12 (Construction) in GDP were observed for Belarus (slightly greater than 22.2% on the forecasting period) while the smallest for Kyrgyzstan (about 3.3%); see Fig. 2.70. 13. The largest GVA shares of Sector 13 (Production of textiles, clothes, leather, and associated goods) in GDP were observed for Belarus (slightly greater than

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.65 The GVA share of Sector 7 (Food industry, including beverages and tobacco) in GDP for five EAEU countries

0.1 0.09 0.08 0.07

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Fig. 2.66 The GVA share of Sector 8 (Professional, scientific, and technical activities) in GDP for five EAEU countries

1.2% on the forecasting period); for the other EAEU countries, this index was measured by fractions of percent; see Fig. 2.71. 14. On the forecasting period, the largest GVA shares of Sector 14 (Financial services) in GDP were observed for Belarus (slightly greater than 4.3%) while the smallest for Kyrgyzstan (below 0.8%); see Fig. 2.72. 15. On the forecasting period, the largest GVA shares of Sector 15 (Chemical and petrochemical industry) in GDP were observed for Belarus (about 3.7%) while the smallest for Kyrgyzstan (about 0.05%); see Fig. 2.73.

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Fig. 2.67 The GVA share of Sector 9 (Other industries) in GDP for five EAEU countries

0.45 0.4 0.35 0.3

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Fig. 2.68 The GVA share of Sector 10 (Other services) in GDP for five EAEU countries

16. On the forecasting period, the largest GVA shares of Sector 16 (Transportation) in GDP were observed for Belarus (slightly less than 10%) while the smallest for Kyrgyzstan (about 0.05%); see Fig. 2.74.

2.4.2

Scenario Analysis Based on Model 1

Some examples of scenario analysis based on Model 1 below are intended to illustrate how numerical variations of the fiscal, trade, and monetary tools as well

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.69 The GVA share of Sector 11 (Agriculture, forestry, and fishery) in GDP for five EAEU countries

0.35 0.3 0.25 blr 0.2

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Fig. 2.70 The GVA share of Sector 12 (Construction) in GDP for five EAEU countries

as of other exogenous parameters (relative to the baseline scenario) affect different macroeconomic indexes. In particular, the following scenarios were considered: a 10% increase of the effective rates for all import taxes in Kazakhstan since year 2017; a 10% increase of the effective rates for sales tax (VAT and excises) in combination with a 10% increase of sectoral taxes, both in the EU since year 2017; a 10% increase of the effective rates for sales tax (VAT and excises) in combination with a 10% increase of sectoral taxes, both in Kazakhstan since year 2017; the establishment of a free-trade zone between the EU and EAEU since year 2017; the tightening of international economic sanctions against Russia; the

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Fig. 2.71 The GVA share of Sector 13 (Production of textiles, clothes, leather, and associated goods) in GDP for five EAEU countries

0.05 0.045 0.04 0.035

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Fig. 2.72 The GVA share of Sector 14 (Financial services) in GDP for five EAEU countries

establishment of a monetary union within the EAEU since year 2017; and the EAEU/CIS collapse in year 2017. In addition, the 2009 and 2016 economic crises in Kazakhstan were thoroughly examined in terms of USD. As an example, Figs. 2.75, 2.76, and 2.77 below present the simulation results under a 10% increase of the effective rates for all import taxes in Kazakhstan since year 2017 (scenario 1). The simulation results of scenario 1 led to the following conclusions: 1. The GDP of Kazakhstan in year 2017 was reduced by 0.11%; by year 2022, this reduction reached 0.13% (both figures in comparison with the baseline scenario).

2.4 Macroeconomic Analysis Based on Model 1

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Fig. 2.73 The GVA share of Sector 15 (Chemical and petrochemical industry) in GDP for five EAEU countries

0.18 0.16 0.14 0.12

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0

Fig. 2.74 The GVA share of Sector 16 (Transportation) in GDP for five EAEU countries

Among other Regions, this scenario had the largest effect on the GDP of Kyrgyzstan—an increase by 0.0044–0.0034% on the period 2017–2022 (see Fig. 2.75). 2. This scenario had positive effect on the GVAs of Sector 13 (Production of textiles, clothes, leather, and associated goods; an increase to 0.2% for year 2017), Sector 9 (Other industries; an increase by 0.11% for year 2017), and Sector 3 (Metalworking and machine building; an increase by 0.10% for year 2017). The GVAs of other Sectors suffered from negative consequences of scenario 1; see Fig. 2.76.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

166 0.0200 2015

2016

2017

2018

2019

2020

2021

2022

-0.0200

chn usa blr

-0.0400

rus

-0.0600

kaz kgz

-0.0800

arm -0.1000

eu

-0.1200

row

-0.1400

Fig. 2.75 GDP variations (in %) for different Regions in scenario 1 0.3000 aming acrog

0.2000

amepe amind 0.1000

aehas apegw afpin

2015

2016

2017

2018

2019

2020

2021

2022

apsta aos

-0.1000

aoths aagff

-0.2000

abuil amtal afins

-0.3000

achpp atser

-0.4000

Fig. 2.76 GVA variations (in %) for different Sectors of Kazakhstan in scenario 1

3. The largest effect on the prices of final Products was observed for Sector 13 (Production of textiles, clothes, leather, and associated goods; about 0.56% of price variations in comparison with the baseline scenario). For other types of final Products, the price variations lay between 0.12% and +0.12% in comparison with the baseline scenario; see Fig. 2.77.

2.4 Macroeconomic Analysis Based on Model 1

167

0.7000 cming

0.6000

ccrog 0.5000

cmepe cmind

0.4000

cehas 0.3000

cpegw

0.2000

cfpin cpsta

0.1000

cos coths

2015

2016

2017

2018

2019

2020

2021

2022

-0.1000

cagff cbuil

-0.2000

Fig. 2.77 Price variations (in %) for final Products in Kazakhstan in scenario 1 0.5000 aming acrog 2015

2016

2017

2018

2019

2020

2021

2022

amepe amind

-0.5000

aehas apegw

-1.0000

afpin apsta aos

-1.5000

aoths aagff

-2.0000

abuil amtal

-2.5000

afins achpp atser

-3.0000

Fig. 2.78 GVA variations (in %) for different Sectors of EU in scenario 2

The next scenario (no. 2) allowed to study the response of regional economies to a 10% increase of the effective rates for sales tax (VAT and excises) in combination with a 10% increase of sectoral taxes, both in the EU since year 2017. Some simulation results are shown in Figs. 2.78, 2.79, 2.80, and 2.81.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

168 2.0000

cming ccrog

1.5000

cmepe cmind

1.0000

cehas cpegw

0.5000

cfpin -

cpsta 2015

2016

2017

2018

2019

2020

2021

2022

-0.5000

cos coths cagff

-1.0000

cbuil cmtal

-1.5000

Fig. 2.79 Price variations (in %) for final Products in EU in scenario 2 0.0500 2015

2016

2017

2018

2019

2020

2021

2022

-0.0500

chn usa blr

-0.1000

rus kaz

-0.1500

kgz arm

-0.2000

eu row

-0.2500 -0.3000

Fig. 2.80 GDP variations (in %) for different Regions in scenario 2

The simulation results of scenario 2 led to the following conclusions: 1. In this scenario, the GVAs of all EU Sectors were decreasing in comparison with the baseline scenario; the largest reduction, 2.5% for year 2017, was observed for Sector 2 (Hydrocarbon production and natural gas extraction). The aggregate GVA in the EU for year 2017 made up about 1%. After year 2017, these reductions became smaller; see Fig. 2.78.

2.4 Macroeconomic Analysis Based on Model 1

169

0.6000 aming acrog 0.4000

amepe amind aehas

0.2000

apegw afpin apsta

2015

2016

2017

2018

2019

2020

2021

2022

aos aoths

-0.2000

aagff abuil amtal

-0.4000

afins achpp atser

-0.6000

Fig. 2.81 GVA variations (in %) for different Sectors of Kazakhstan in scenario 2

2. The price variations for the final Products of the EU were dual depending on type. The largest price increase was observed for the Products of Sector 15 (Chemical and petrochemical industry; above 1.4%). The largest price reduction was observed for Sector 2 (Hydrocarbon production and natural gas extraction; about 1%); see Fig. 2.79. Such a decrease in the output of this Sector and prices can be explained by a higher drop of demand for these Products in comparison with other Sectors. 3. This scenario had negative effect on the GDPs of all Regions, in the first place, on the GDP of the EU (a 0.27% decrease for year 2017 in comparison with the baseline scenario). Subsequently this decrease became smaller, reaching 0.16% by year 2022. The second Region suffering from scenario 2 was Kazakhstan. The smallest effect was exerted on the GDP of China (reduction by 0.014% by year 2022); see Fig. 2.80. 4. The GVA variations for different Sectors of Kazakhstan in scenario 2 were also dual. The largest GVA decrease was observed for Sector 2 (Hydrocarbon production and natural gas extraction; about 0.5%) while the largest GVA increase for Sector 4 (Metal industry; up to 0.43% by year 2022); see Fig. 2.81. Scenario 3 implied a 10% increase of the effective rates for sales tax (VAT and excises) in combination with a 10% increase of sectoral taxes, both in Kazakhstan since year 2017. Some simulation results of this scenario are illustrated in Figs. 2.82, 2.83, and 2.84.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

170 0.5000

aming acrog -

amepe 2015

2016

2017

2018

2019

2020

2021

2022

amind aehas

-0.5000

apegw afpin apsta

-1.0000

aos aoths aagff

-1.5000

abuil amtal afins

-2.0000

achpp atser -2.5000

Fig. 2.82 GVA variations (in %) for different Sectors of Kazakhstan in scenario 3

0.0500 aming acrog -

amepe 2015

2016

2017

2018

2019

2020

2021

2022

amind aehas

-0.0500

apegw afpin

-0.1000

apsta aos aoths

-0.1500

aagff abuil amtal

-0.2000

afins achpp atser

-0.2500

Fig. 2.83 GVA variations (in %) for different Sectors of Kyrgyzstan in scenario 3

2.4 Macroeconomic Analysis Based on Model 1

171

0.05000 2015

2016

2017

2018

2019

2020

2021

2022

-0.05000

chn usa blr

-0.10000

rus kaz

-0.15000

kgz arm

-0.20000

eu row

-0.25000 -0.30000

Fig. 2.84 GDP variations (in %) for different Regions in scenario 3

The simulation results of scenario 3 led to the following conclusions: 1. In this scenario, the GVAs of all Sectors of Kazakhstan were decreasing in comparison with the baseline scenario. The largest reduction, down to 2.1% for year 2020, was observed for Sector 2 (Hydrocarbon production and natural gas extraction). The smallest reduction was observed for Sector 4 (Metal industry; down to 0.22% for year 2019); see Fig. 2.82. 2. This scenario had negligibly small effect on the GVAs for different Sectors of the other EAEU countries, except for Kyrgyzstan. For this country, the GVA variations in scenario 3 were from 0.036% for Sector 14 (Financial services) for year 2017 and 0.22% for Sector 16 (Transportation) for year 2021; see Fig. 2.83. 3. Among all Regions, the largest negative effect of this scenario (quite expectedly) was on the GDP of Kazakhstan (down to – 0.25% by year 2022). The GDP of Kyrgyzstan was decreasing by 0.038% by year 2022. At the same time, scenario 3 had negligibly small effect on the GDPs of other Regions. These facts indicated that the economy of Kyrgyzstan was more sensitive to the changes in the economy of Kazakhstan in comparison with the other Regions; see Fig. 2.84. The goal of the next analysis was to estimate the effects from establishing a (hypothetic) free-trade zone between the EU and EAEU since year 2017 (scenario 4). In this scenario, all rates for the import and export taxes in the bilateral trade of the EU and EAEU countries were equal to 0 since year 2017. Some simulation results of this scenario are demonstrated in Figs. 2.85, 2.86, 2.87, 2.88, 2.89, 2.90, and 2.91.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

172 2.0000

1.5000

chn usa

1.0000

blr rus

0.5000

kaz kgz arm

2015

2016

2017

2018

2019

2020

2021

2022

eu row

-0.5000

-1.0000

Fig. 2.85 GDP variations (in %) for different Regions in scenario 4

4.0000 aming 2.0000 acrog amepe

2015

2016

2017

2018

2019

2020

2021

-2.0000 -4.0000

2022

amind aehas apegw afpin

-6.0000

apsta

-8.0000

aos aoths

-10.0000 -12.0000

aagff abuil amtal

-14.0000

afins achpp

-16.0000 atser -18.0000

Fig. 2.86 GVA variations (in %) for different Sectors of EU in scenario 4

2.4 Macroeconomic Analysis Based on Model 1

173

7.0000 6.0000 aming acrog

5.0000

amepe 4.0000

amind aehas

3.0000

apegw afpin

2.0000

apsta aos

1.0000

aoths aagff

2015

2016

2017

2018

2019

2020

2021

2022

-1.0000

abuil amtal afins

-2.0000

achpp atser

-3.0000 -4.0000

Fig. 2.87 GVA variations (in %) for different Sectors of Kazakhstan in scenario 4

The simulation results of scenario 4 led to the following conclusions: 1. Due to the establishment of the free-trade zone, the largest GDP growth was observed for Russia (more than 1.5% in comparison with the basic value for year 2017, with a subsequent decrease of the GDP increment to 0.25% by year 2022) and also the EU (the GDP increment up to 0.67% by year 2022). The largest GDP drop was observed for Kazakhstan (down to 0.88% by year 2022). The other three EAEU countries had a GDP reduction in this scenario too. For the other Regions of Model 1, the effect of this scenario was insignificant; see Fig. 2.85. 2. Except for Sector 2 (Hydrocarbon production and natural gas extraction), all EU Sectors were increasing their GVAs; a maximum increase of 1.22% for year 2022 occurred for Sector 15 (Chemical and petrochemical industry). The GVA drop for Sector 2 (Hydrocarbon production and natural gas extraction) made up 15.3% for year 2017, reaching 14.6% by year 2022. This drop was explained by the growing exports from Russia; see Fig. 2.86. 3. The GVAs for different Sectors of Kazakhstan had dual dynamics in this scenario. For instance, the GVA of Sector 4 (Metal industry) was increasing by 5.9%

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

174

20.0000

aming 15.0000

acrog amepe amind aehas

10.0000

apegw afpin apsta 5.0000 aos aoths aagff -

abuil 2015 2016 2017 2018 2019 2020 2021 2022

amtal afins

-5.0000

achpp atser

-10.0000 Fig. 2.88 GVA variations (in %) for different Sectors of Russia in scenario 4

by year 2022 while the GVA of Sector 2 (Hydrocarbon production and natural gas extraction) decreasing since year 2017 by approximately 3.3%; see Fig. 2.87. 4. For Russia, the largest increment due to the free-trade zone with the EU was observed for Sector 2 (Hydrocarbon production and natural gas extraction): on the period 2017–2022, the corresponding GVA had a growth of 16.7–17.5%. Except for this Sector and also Sector 15 (Chemical and petrochemical industry), the GVAs for the other Sectors of Russia were reduced in scenario 4. The largest decrease was observed for Sector 4 (Metal industry; down to 7% by year 2022); see Fig. 2.88. 5. Among all Sectors of Belarus, the largest growth was observed for Sector 2 (Hydrocarbon production and natural gas extraction; up to 6.6% by year 2022) while the largest drop for Sector 5 (Education, public health, and public administration; down to 2.18% by year 2022); see Fig. 2.89.

2.4 Macroeconomic Analysis Based on Model 1

175

8.0000 aming

7.0000

acrog 6.0000

amepe amind

5.0000

aehas apegw

4.0000

afpin 3.0000

apsta

2.0000

aos aoths

1.0000

aagff abuil

2015 2016 2017 2018 2019 2020 2021 2022 -1.0000

amtal afins achpp

-2.0000

atser

-3.0000 Fig. 2.89 GVA variations (in %) for different Sectors of Belarus in scenario 4

6. The GVAs of the two remaining EAEU countries (Kyrgyzstan and Armenia) were less sensitive to the establishment of the free-trade zone. For Kyrgyzstan, the GVA of Sector 3 (Metalworking and machine building) was increasing by 1.24–0.94% while the GVA of Sector 1 (Mining) decreasing by 0.78–1.26% on the period 2017–2022. For Armenia, the GVA of Sector 1 (Mining) was increasing by 0.63–0.65% while the GVA of Sector 16 (Transportation) decreasing by 0.50–0.62% on the period 2017–2022; see Figs. 2.90 and 2.91. The following scenario (no. 5) was intended to analyze the effect of international sanctions against separate countries. Let all import prices for bilateral trade between Russia (on the one side) and the EU, the USA, and Rest of World (on the other) increase by 25 percentage points. Recall that the international sanctions imposed on Russia in year 2015 were taken into consideration during the calibration procedure of Model 1. So scenario 5 was tightening the existing trade restrictions in certain sense. Some simulation results of this scenario are presented in Figs. 2.92, 2.93, 2.94, 2.95, 2.96, and 2.97.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

176

1.5000 aming acrog 1.0000

amepe amind aehas

0.5000

apegw afpin apsta

2015 2016 2017 2018 2019 2020 2021 2022

aos aoths

-0.5000

aagff abuil amtal

-1.0000

afins achpp atser

-1.5000 Fig. 2.90 GVA variations (in %) for different Sectors of Kyrgyzstan in scenario 4

The simulation results of scenario 5 led to the following conclusions: 1. In the long run, the additional sanctions dictated by scenario 5 caused GDP decrease for all Regions in comparison with the baseline scenario (the only exception was a small GDP growth for Belarus on the period 2014–2015). The largest GDP drop was observed for Russia (down to 5.52% by year 2022), Kyrgyzstan (down to 1.08% by year 2022), and Belarus (down to 0.92% by year 2022). At the same time, this scenario had a considerably smaller effect on the GDPs of other Regions; see Fig. 2.92. 2. For Russia, these sanctions had positive effect only on the output of Sector 3 (Metalworking and machine building) on the period 2014–2016. By year 2022, the output drops varied from 2.46% for Sector 3 (Metalworking and machine building) to 7.18% for Sector 12 (Construction); see Fig. 2.93. 3. For the EU, this scenario was beneficial for the GVAs of raw material industries, namely, Sector 2 (Hydrocarbon production and natural gas extraction; growth up to 2.57% for year 2022) and Sector 1 (Mining; growth up to 0.77% for year 2022). The GVAs of the other Sectors of the EU were decreasing by year 2022, between 0.078% for Sector 5 (Education, public health, and public administration) and 0.42% for Sector 3 (Metalworking and machine building); see Fig. 2.94.

0.8000

aming

0.6000

acrog amepe 0.4000

amind aehas apegw

0.2000

afpin apsta

2015 2016 2017 2018 2019 2020 2021 2022

aos aoths

-0.2000

aagff abuil

-0.4000

amtal afins achpp

-0.6000

atser

-0.8000 Fig. 2.91 GVA variations (in %) for different Sectors of Armenia in scenario 4 1.0000

-

chn 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 usa

-1.0000 blr -2.0000

rus kaz

-3.0000

kgz arm

-4.0000 eu -5.0000

-6.0000

Fig. 2.92 GDP variations (in %) for different Regions in scenario 5

row

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

178

3.0000

2.0000 aming 1.0000

acrog amepe

-

amind 20122013201420152016201720182019202020212022

-1.0000

aehas apegw afpin

-2.0000

-3.0000

apsta aos aoths

-4.0000

aagff abuil

-5.0000

amtal afins

-6.0000

achpp atser

-7.0000

-8.0000 Fig. 2.93 GVA variations (in %) for different Sectors of Russia in scenario 5

4. For the USA, scenario 5 had slight positive effect on the GVAs of Sector 15 (Chemical and petrochemical industry; growth up to 0.1% by year 2022), Sector 2 (Hydrocarbon production and natural gas extraction; growth up to 0.09% by year 2022), and Sector 4 (Metal industry; growth up to 0.4% by year 2022). The largest GVA drop was observed for Sector 1 (Mining) and year 2014—minus 1.1%; see Fig. 2.95. 5. For the Rest of World, this scenario was beneficial only for the GVA of Sector 2 (Hydrocarbon production and natural gas extraction; growth up to 0.36% by year 2022). The largest GVA drop was observed for Sector 3 (Metalworking and machine building)—minus 0.21% for years 2014 and 2022; see Fig. 2.96. 6. Scenario 5 had insignificant consequences for the GDP of Kazakhstan, but the effect was dual: the GVAs of different Sectors varied from a 2.34% increase for Sector 1 (Mining) and year 2014 to a 0.86% decrease for Sector 2 (Hydrocarbon production and natural gas extraction) and the same year; see Fig. 2.97.

2.4 Macroeconomic Analysis Based on Model 1

179

3.0000

2.5000

2.0000

1.5000

1.0000

0.5000

20122013201420152016201720182019202020212022

-0.5000

aming acrog amepe amind aehas apegw afpin apsta aos aoths aagff abuil amtal afins achpp atser

-1.0000 Fig. 2.94 GVA variations (in %) for different Sectors of EU in scenario 5

Scenario 6 was to estimate the effects from establishing a monetary union within the EAEU since year 2017. In the main setup, the IMF forecasts on the forecasting period [80] were used as the basic values of the endogenous fluctuating exchange rates (domestic per world unit (USD)); some of these data are given in Table 2.5. Note that the net exports of goods and services in each Region were exogenous variables. This scenario was simulated using a special setup of Model 1 in which since year 2017 the exchange rates of all EAEU countries were treated as exogenous variables while the net exports of goods and services of each EAEU country as endogenous variables. Moreover, for making the number of endogenous variables equal to the number of model equations, the net exports of goods and services of the five EAEU countries were treated as endogenous variables while the aggregate net exports of goods and services of all EAEU countries as an exogenous variable. The exchange rates of the other five Regions were determined in accordance with the baseline scenario of Model 1. In further simulations of scenario 6, since year 2018 the exchange rates of the EAEU countries were equal to their counterparts for year

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

180

0.1500 aming acrog 0.1000

amepe amind

0.0500

aehas apegw afpin

-

apsta aos aoths

-0.0500

aagff abuil

-0.1000

amtal afins achpp

-0.1500 Fig. 2.95 GVA variations (in %) for different Sectors of the USA in scenario 5

2017. Some simulation results of this scenario are illustrated in Table 2.5 and Figs. 2.98, 2.99, 2.100, 2.101, 2.102, 2.103, and 2.104. The simulation results of scenario 6 led to the following conclusions: 1. By year 2022, due to the establishment of the monetary union, the KZT–USD exchange rate was increasing by 10.6% in comparison with the baseline scenario while the AMD–USD exchange rate by 0.3%. At the same time, the BYN–USD exchange rate was decreasing by 14.7%, the KZS–USD exchange rate by 4.1%, the RUB–USD exchange rate by 0.4%; see Table 2.5. (The international currency codes mentioned here are as follows: KZT, Kazakhstani Tenge; AMD, Armenian Dram; BYN, Belorussian Ruble; KZS, Kyrgyzstani Som; and RUB, Russian Ruble). 2. The countries obtaining higher exchange rates to USD in scenario 6 were also increasing their GDPs: Belarus by 7.2% and Kyrgyzstan by 3.3% by year 2022. Quite expectedly, the countries obtaining lower exchange rates to USD in scenario 6 were decreasing their GDPs: Kazakhstan by 1.9% and Armenia by 0.2% by year 2022. This scenario had negligibly small effect on the GDPs of other Regions; see Fig. 2.98. 3. The GVAs of different Sectors demonstrated dual response to scenario 6. For the export-oriented Sectors, higher exchange rates to USD were causing higher

2.4 Macroeconomic Analysis Based on Model 1

181

0.4000 aming 0.3000

acrog amepe amind

0.2000

aehas apegw

0.1000

afpin apsta aos

-

aoths aagff

-0.1000

abuil amtal

-0.2000

afins achpp atser

-0.3000 Fig. 2.96 GVA variations (in %) for different Sectors of the Rest of World in scenario 5

GVAs; for the Sectors with large consumption of intermediate Products, higher exchange rates to USD were causing lower GVAs. For Kazakhstan, by year 2022 the largest GVA increase was observed for Sector 2 (Hydrocarbon production and natural gas extraction; 20.7%) while the largest drop for Sector 12 (Construction; 11%); see Fig. 2.99. 4. For Russia, in scenario 6 the exchange rate to USD had an insignificant decrease, and the GVA variations of different Sectors were by an order of magnitude less than for Kazakhstan. By year 2022, the largest GVA growth was observed for Sector 13 (Production of textiles, clothes, leather, and associated goods; 1.2%) while the largest GVA drop for Sector 1 (Mining; 1.7%); see Fig. 2.100. 5. For Belarus, in scenario 6 the exchange rate to USD demonstrated the largest drop, and the GVA variations of different Sectors were considerable. By year 2022, the largest GVA growth was observed for Sector 5 (Education, public health, and public administration; 18.4%) while the largest GVA drop for Sector 2 (Hydrocarbon production and natural gas extraction; 19.3%); see Fig. 2.101. 6. Similar picture but with other values was observed for Kyrgyzstan. The largest GVA growth by year 2022 was observed for Sector 5 (Education, public health, and public administration; 5.9%) while the largest GVA drop for Sector

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

182

2.5000 aming 2.0000

acrog amepe amind

1.5000

aehas apegw

1.0000

afpin apsta 0.5000

aos aoths

-

aagff 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

abuil amtal

-0.5000

afins achpp

-1.0000

atser -1.5000 Fig. 2.97 GVA variations (in %) for different Sectors of Kazakhstan in scenario 5

Table 2.5 Exchange rates in EAEU countries (domestic per world unit (USD)) Country, scenario Kazakhstan, baseline Kazakhstan, no. 6 Russia, baseline Russia, no. 6 Belarus, baseline Belarus, no. 6 Kyrgyzstan, baseline Kyrgyzstan, no. 6 Armenia, baseline Armenia, no. 6

Year 2017 331.00 331.00 58.98 58.98 1.94 1.94 71.60 71.60 498.35 498.35

2018 331.00 338.78 60.42 60.37 2.06 1.98 73.41 73.28 508.31 510.07

2019 331.00 347.56 62.06 61.93 2.20 2.03 75.60 75.18 518.48 523.28

2020 331.00 354.43 63.34 63.15 2.32 2.07 77.81 76.66 529.37 533.62

2021 331.00 359.33 64.21 64.03 2.42 2.10 80.14 77.72 539.93 541.00

2022 331.00 366.02 65.52 65.22 2.51 2.14 82.55 79.17 549.56 551.07

2.4 Macroeconomic Analysis Based on Model 1

183

8.0000

6.0000

chn usa

4.0000

blr rus

2.0000

kaz kgz arm

2016

2017

2018

2019

2020

2021

2022

eu row

-2.0000

-4.0000

Fig. 2.98 GDP variations (in %) for different Regions in scenario 6

25.0000

aming

20.0000

acrog amepe 15.0000

amind aehas apegw

10.0000

afpin apsta

5.0000

aos aoths

-

aagff 2016

2017

2018

2019

2020

2021

2022

-5.0000

abuil amtal afins achpp

-10.0000

-15.0000

Fig. 2.99 GVA variations (in %) for different Sectors of Kazakhstan in scenario 6

atser

1.5000

1.0000

0.5000

2016

2017

2018

2019

2020

2021

2022

-0.5000

-1.0000

-1.5000

aming acrog amepe amind aehas apegw afpin apsta aos aoths aagff abuil amtal afins achpp atser

-2.0000 Fig. 2.100 GVA variations (in %) for different Sectors of Russia in scenario 6 25.0000

20.0000 aming acrog

15.0000

amepe amind

10.0000

aehas apegw

5.0000

afpin apsta 2016

2017

2018

2019

2020

2021

2022

aos aoths

-5.0000

aagff abuil

-10.0000

amtal afins

-15.0000

achpp atser

-20.0000

-25.0000

Fig. 2.101 GVA variations (in %) for different Sectors of Belarus in scenario 6

2.4 Macroeconomic Analysis Based on Model 1

185

8.0000

aming

6.0000

acrog amepe amind

4.0000

aehas apegw afpin

2.0000

apsta aos -

aoths 2016

2017

2018

2019

2020

2021

2022

aagff abuil

-2.0000

amtal afins achpp

-4.0000

atser

-6.0000

Fig. 2.102 GVA variations (in %) for different Sectors of Kyrgyzstan in scenario 6

2 (Hydrocarbon production and natural gas extraction; 3.9%); see Fig. 2.102. The GVA variations for different Sectors of Armenia did not exceed 1% and are omitted on the graphs. 7. The largest price variations for final Products were observed for the countries with most considerable changes of the exchange rate to USD, namely, for Kazakhstan and Belarus. For Kazakhstan, the largest growth of these prices by year 2022 was observed for Sector 3 (Metalworking and machine building; 3.4%) while the largest drop for Sector 4 (Metal industry; 6.8%). For Belarus, the largest growth of these prices by year 2022 was observed for Sector 5 (Education, public health, and public administration; 6.4%) while the largest drop for Sector 2 (Hydrocarbon production and natural gas extraction; 12.5%); see Figs. 2.102 and 2.104. Finally, scenario 7 was intended for estimating the effects from a hypothetic collapse of the EAEU or CIS in year 2018. In this scenario, since year 2018 the zero tax rates for the bilateral exports and imports of the five EAEU countries were replaced by their counterparts for bilateral trade with the Rest of World. Some simulation results of this scenario are shown in Figs. 2.105, 2.106, 2.107, 2.108, 2.109, and 2.110. The simulation results of scenario 7 led to the following conclusions: 1. In comparison with the baseline scenario, due to the EAEU/CIS collapse the largest GDP increase by year 2022 was observed for Belarus (16.4%). The

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

186 4.00

cming 2.00

ccrog cmepe cmind cehas

2016

2017

2018

2019

2020

2021

2022

cpegw cfpin cpsta

(2.00) cos coths cagff (4.00)

cbuil cmtal cfins

(6.00)

cchpp ctser

(8.00)

Fig. 2.103 Price variations (in %) for final Products in Kazakhstan in scenario 6

positive GDP trends in this scenario were also observed for Armenia (1.3%) and Kyrgyzstan (0.7%). However, as the result of this collapse, the GDPs of Kazakhstan and Russia by year 2022 had drops of 0.7% and 0.2%, respectively; see Fig. 2.105. 2. For Kazakhstan, scenario 7 caused GVA growth for Sector 9 (Other industries; 1.81% for year 2018) and Sector 3 (Metalworking and machine building; 0.89% for year 2018). The GVAs of the other Sectors were decreasing, with the largest GVA drop (minus 1.70%) observed for Sector 1 (Mining) by year 2022; see Fig. 2.106. 3. For Russia, scenario 7 was beneficial in terms of the GVA of Sector 13 (Production of textiles, clothes, leather, and associated goods; a 0.72% increase for year 2018) and also of the GVA of Sector 2 (Hydrocarbon production and natural gas extraction; a 0.66% increase for year 2018). This scenario had negative effect on the other Sectors, causing GVA decrease by year 2022. The largest GVA drop

2.4 Macroeconomic Analysis Based on Model 1

187

10.00

cming ccrog 5.00

cmepe cmind cehas cpegw

2016

2017

2018

2019

2020

2021

2022

cfpin cpsta cos

(5.00)

coths cagff cbuil cmtal

(10.00)

cfins cchpp ctser

(15.00) Fig. 2.104 Price variations (in %) for final Products in Belarus in scenario 6

(0.53%) was observed for Sector 5 (Education, public health, and public administration); see Fig. 2.107. 4. For Belarus, this scenario was most influential in terms of GVAs of different Sectors. The largest GVA drop was observed for Sector 2 (Hydrocarbon production and natural gas extraction; a 52.8% decrease by year 2022). The largest GVA growth was observed for Sector 15 (Chemical and petrochemical industry; 33.2% by year 2022); see Fig. 2.108. 5. For Kyrgyzstan, scenario 7 had dual effect on the GVAs of different Sectors— from a 1.37% increase by year 2022 for Sector 11 (Agriculture, forestry, and fishery) to a drop by the same year for Sector 13 (Production of textiles, clothes, leather, and associated goods); see Fig. 2.109. 6. In scenario 7, the GVAs of all Sectors of Armenia were increasing by year 2022 in comparison with the baseline scenario. The largest growth was observed for Sector 16 (Transportation; 3.38%); see Fig. 2.110. At the end of Sect. 2.4, consider a scenario-based approach to analyze the causes of the 2009 and 2016 economic crises of Kazakhstan (in terms of USD) in

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

188

18.0000 16.0000 14.0000 12.0000 blr 10.0000

rus

8.0000

kaz

6.0000

kgz arm

4.0000 2.0000 2016 2017 2018 2019 2020 2021 2022 -2.0000 Fig. 2.105 GDP variations (in %) for EAEU countries in scenario 7 Fig. 2.106 GVA variations (in %) for different Sectors of Kazakhstan in scenario 7

2.0000

1.5000 aming acrog

1.0000

amepe amind aehas

0.5000

apegw afpin apsta

2016 2017 2018 2019 2020 2021 2022

aos aoths aagff

-0.5000

abuil amtal afins

-1.0000

achpp atser

-1.5000

-2.0000

2.4 Macroeconomic Analysis Based on Model 1

189

0.8000

0.6000

0.4000

0.2000

2016 2017 2018 2019 2020 2021 2022 -0.2000

-0.4000

aming acrog amepe amind aehas apegw afpin apsta aos aoths aagff abuil amtal afins achpp atser

-0.6000 Fig. 2.107 GVA variations (in %) for different Sectors of Russia in scenario 7

comparison with the preceding years (2008 and 2015, respectively). For such analysis, choose the static setup of the model, classifying all exogenous variables into seven nonintersecting groups as follows: 1. KAPWOR—the net exports from Kazakhstan 2. VGD—the government’s share in the aggregate consumption of Products in Kazakhstan 3. INVEST—the ratio of investments to the aggregate consumption of Products in Kazakhstan 4. FSL—Labor supply in Kazakhstan 5. FSС—Capital supply (basic assets) in Kazakhstan 6. EAU—all parameters of the other four EAEU countries 7. Rest—all other parameters of Model 1 The aggregate algorithm below is intended for estimating the effects of these groups of factors on the GDP variations of Kazakhstan, from year t to year (t + 1).

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

190

40.0000 30.0000 aming acrog

20.0000

amepe amind

10.0000

aehas apegw

2016 2017 2018 2019 2020 2021 2022 -10.0000 -20.0000

afpin apsta aotis aoths aagff abuil

-30.0000

amtal afins

-40.0000

achpp atser

-50.0000 -60.0000 Fig. 2.108 GVA variations (in %) for different Sectors of Belarus in scenario 7

Step 1. Using the available data for years t and (t + 1), calibrate Model 1 and determine the values of all exogenous parameters for these two years. Step 2. Calculate the GDP of Kazakhstan—the value V0—for the baseline scenario of Model 1 using the above values of the exogenous parameters for year t. Step 3. Simulate the scenario in which the value of the first parameter from the classification list is replaced by the corresponding value for year (t + 1). Denote by V1 the resulting GDP value. Step 4. Simulate the scenario in which the values of the first and second parameters from the classification list are replaced by the corresponding values for year (t + 1). Denote by V2 the resulting GDP value. ... Step 9. Simulate the scenario in which the values of all parameters from the classification list are replaced by the corresponding values for year (t + 1). Denote by V7 the resulting GDP value.

2.5 Parametric Control Based on Model 1: A Series of Problem Statements. . .

191

2.0000

1.5000

aming acrog amepe

1.0000

amind aehas

0.5000

apegw afpin

2016 2017 2018 2019 2020 2021 2022

apsta aos

-0.5000

aoths aagff

-1.0000

abuil amtal

-1.5000

afins achpp

-2.0000

atser

-2.5000 Fig. 2.109 GVA variations (in %) for different Sectors of Kyrgyzstan in scenario 7

Step 10. Calculate the differences ΔV1 ¼ V1  V0, ΔV2 ¼ V2  V1, . . ., ΔV7 ¼ V7  V6 and their shares (in %) in the GDP variation (ΔV ¼ V7  V0) of Kazakhstan, from year t to year (t + 1). Step 11. Stop. The results of calculations using this algorithm are illustrated in Table 2.6. Clearly, the reduced supplies of both Factors (Labor and Capital) had the largest effect on the GDP drop of Kazakhstan for years 2009 and 2016 (13.45% and 24.9% in USD terms, respectively). The reduced supplies can be explained by the domestic currency devaluations performed in years 2009 and 2016.

2.5

Parametric Control Based on Model 1: A Series of Problem Statements and Their Solutions

A series of parametric control design problems were formulated and solved in the course of estimating the optimal economic policy tools for different Regions of Model 1 for the period 2016–2022; also see Sect. 1.4.1.

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

192

4.0000 3.5000

aming acrog

3.0000

amepe amind

2.5000

aehas apegw

2.0000

afpin apsta

1.5000

aos aoths

1.0000

aagff abuil

0.5000

amtal afins

2016 2017 2018 2019 2020 2021 2022

achpp atser

-0.5000 -1.0000 Fig. 2.110 GVA variations (in %) for different Sectors of Armenia in scenario 7

Table 2.6 Relative effects (in %) of factors on GDP drop of Kazakhstan for years 2009 and 2016 Year (st + 1) 2009 2016

2.5.1

Groups of factors KAPWOR VGD 4.54 4.22 8.65 2.64

INVEST 0.49 1.96

FSL 50.75 44.36

FSС 35.09 31.96

EAU 4.17 5.82

Rest 1.72 4.61

Problems of Economic Growth, Food Security, Reduction of Trade Gap, and Regional Development Disproportions

Consider informal statements of 11 such problems Pri , (i ¼ 1, . . . , 9, EA, W ), for elaborating economic policy toward economic growth, higher output of Sector 11 (Agriculture, forestry, and fishery), and reduction of trade gap and regional development disproportions. In these problems, the values of all uncontrolled

2.5 Parametric Control Based on Model 1: A Series of Problem Statements. . .

193

exogenous variables of Model 1 correspond to the baseline scenario. In the sequel, subscripts i, j ¼ 1, . . . , 9 indicate the number of Region as follows: 1, Kazakhstan; 2, Russia; 3, Belarus; 4, Armenia; 5, Kyrgyzstan; 6, the European Union; 7, the USA; 8, China; and 9, the Rest of World. In addition, the subscripts EA and W indicate the Eurasian Economic Union and global economy, respectively. Statements of Parametric Control Problems Pr i For each problem Pri within the framework of Model 1, find the values of corresponding control parameters (the effective rates of all taxes and the effective tariff rates for imports and exports; the shares of government expenditures on consumption) for years 2016–2022 that maximize the criterion K i (2.103), (2.104), and (2.105) subject to constraints (2.102) on some endogenous variables and also subject to the following constraints on control tools. The admissible variations of the tax and tariff rates (except for the zero rates established within the EAEU) are 5 percentage point deviations from their basic values. The admissible range of the aggregate government consumption is from 0 to 2.5 values of the corresponding basic ones. The admissible range of the government consumption of a specific Product is from 0 to 3.5 values of the corresponding basic ones. For the problems Pr r, (r ¼ 1, . . . , 9), the control parameters are the abovementioned government policy tools of Region r; for the problem Pr EA , the same tools of the five EAEU countries; for the problem PrW , the same tools of all nine Regions of Model 1. The constraints imposed on the endogenous variables of Model 1 in the problems Pr i have the form QVAPr ðt Þ  QVAPr ðt Þ,

r ¼ 1, . . . , 9,

t ¼ 2016, . . . , 2022:

ð2:102Þ

In these inequalities, QVAPr ðt Þ is the per capita GDP of Region r with parametric control; the symbol “” denotes the basic values of a corresponding index (without parametric control). In the problem statements, the criterion K r , (r ¼ 1, . . . , 9), describes the average GDP value (in USD) of Region r for the period 2016–2022, i.e., Kr ¼

X2022 t¼2016

½TQVAr ðt Þ þ αTQXA,r TQXAr ðt Þ  αKAPWOR,r KAPWORr ðt Þ: ð2:103Þ

Here TQVAr ðt Þ gives the annual GDP of Region r for year t; TQXAr ðt Þ is the annual output of Sector 11 (Agriculture, forestry, and fishery) for Region r and year t ; KAPWORr ðt Þ denotes the trade gap of goods and services for Region r and year t; finally, αTQXA,r ¼ 0:5 and αKAPWOR,r ¼ 0:1 are adjustable coefficients. The criteria K EA and K W in the problems Pr EA and PrW characterize the GDPs, outputs of Sector 11 (Agriculture, forestry, and fishery), and trade gaps of the EAEU and global economy (in USD) as well as the relative deviations of the per capita GDPs of different Regions from the per capita GDP of the USA (the Region with the highest per capita GDP) for the period 2016–2022:

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

194

K EA ¼

X2022

½TQVAEA ðt Þ þ αTQXA,EA TQXAEA ðt Þ  αKAPWOR,EA KAPWOREA ðt Þ  X   2022 QVAP7 ðt Þ  QVAPEA ðt Þ αconv,EA X5  P5 εr : r¼1 t¼2016 QVAP7 ðt Þ r¼1 εr t¼2016

ð2:104Þ KW ¼

X2022

½TQVAW ðt Þ þ αTQXA,W TQXAW ðt Þ  X   2022 QVAP7 ðt Þ  QVAPW ðt Þ αconv,W X5  P5 εr : r¼1 t¼2016 QVAP7 ðt Þ r¼1 εr t¼2016

ð2:105Þ

The notations are the following: TQVAEA ðt Þ and TQVAW ðt Þ as the annual GDPs of the EAEU and global economy for year t, respectively; TQXAEA ðt Þ and TQXAW ðt Þ as the annual outputs of Sector 11 (Agriculture, forestry, and fishery) of the EAEU and global economy for year t, respectively; KAPWOREA ðt Þ as the trade gap of goods and services in the EAEU for year t; QVAPEA ðt Þ and QVAPW ðt Þ as the per capita GDP of the EAEU and global economy for year t, respectively; and εr as a weight coefficient (εr ¼ 1 for underdeveloped countries such as Belarus, Armenia, Kyrgyzstan, the Rest of World; εr ¼ 0:1 for developing countries such as Kazakhstan, Russia, China, and the EU; εr ¼ 0 for the developed country, only the USA in Model 1). The adjustable coefficients have the following values: αTQXA,EA ¼ αTQXA,W ¼ 0:5; αKAPWOR,EA ¼ 0:1; and αconv,EA ¼ αconv,W ¼ 0:3. The stated problems Prr were solved numerically using an optimization algorithm provided by GAMS. Their solutions in form of the increments of average GDPs of different Regions and the increments of the average output of Sector 11 (Agriculture, forestry, and fishery) of different Regions for the period 2016–2022 (in % to the basic values) are given in Tables 2.7 and 2.8. In accordance with Tables 2.7 and 2.8, the parametric control approach at the level of all Regions (the problem Pr W ) and also at the level of the five EAEU

Table 2.7 Relative increments of average GDPs of different Regions for period 2016–2022 obtained by solving parametric control problems Pr i (in % to basic values) Problem Pr1 Pr2 Pr3 Pr4 Pr5 Pr6 Pr7 Pr8 Pr9 PrEA PrW

Region r¼1 0.95 0.05 0.01 0.00 0.00 0.48 0.04 0.13 0.17 1.57 2.58

r¼2 0.00 1.80 0.01 0.00 0.00 0.64 0.08 0.04 0.17 2.64 3.73

r¼3 0.01 0.47 1.13 0.00 0.00 0.56 0.06 0.04 0.13 5.10 7.74

r¼4 0.00 0.92 0.01 1.89 0.00 1.09 0.12 0.07 0.33 4.21 4.98

r¼5 0.00 0.09 0.00 0.00 2.91 1.13 0.00 1.11 0.41 3.65 10.14

r¼6 0.00 0.00 0.00 0.00 0.00 2.41 0.08 0.05 0.16 0.00 3.02

r¼7 0.00 0.00 0.00 0.00 0.00 0.16 1.27 0.07 0.11 0.00 1.14

r¼8 0.00 0.00 0.00 0.00 0.00 0.21 0.06 1.45 0.16 0.00 1.71

r¼9 0.00 0.00 0.00 0.00 0.00 0.36 0.18 0.09 1.26 0.00 1.70

2.5 Parametric Control Based on Model 1: A Series of Problem Statements. . .

195

Table 2.8 Relative increments of average outputs of Sector 11 (Agriculture, forestry, and fishery) for different Regions and period 2016–2022 obtained by solving parametric control problems Pri (in % to basic values) Problem Pr1 Pr2 Pr3 Pr4 Pr5 Pr6 Pr7 Pr8 Pr9 PrEA PrW

Region r¼1 6.67 0.06 0.01 0.01 0.01 0.80 0.08 0.12 0.29 5.54 5.74

r¼2 0.00 14.48 0.02 0.00 0.00 0.67 0.12 0.13 0.38 12.35 10.37

r¼3 0.03 0.94 8.86 0.00 0.01 0.98 0.17 0.05 0.23 13.73 18.18

r¼4 0.00 0.99 0.01 5.27 0.00 1.44 0.11 0.05 0.32 5.44 7.59

r¼5 1.42 1.25 1.55 0.01 1.87 2.48 1.33 2.18 1.57 2.07 8.74

r¼6 0.00 0.01 0.00 0.00 0.00 12.64 0.20 0.07 0.21 0.01 18.01

r¼7 0.00 0.01 0.00 0.00 0.00 0.61 11.69 0.43 0.55 0.01 12.55

r¼8 0.00 0.00 0.00 0.00 0.00 0.35 0.12 14.29 0.15 0.00 15.21

r¼9 0.00 0.01 0.00 0.00 0.00 0.72 0.21 0.23 9.56 0.01 9.09

Table 2.9 Relative reduction of average trade gaps of different Regions for period 2016–2022 obtained by solving parametric control problems Pr r (r ¼ 1, . . . , 9Þ (in % to basic values) Problem Prr

Region r¼1 1.41

r¼2 2.52

r¼3 2.65

r¼4 1.28

r¼5 0.93

r¼6 0.05

r¼7 3.36

r¼8 1.80

r¼9 1.78

countries (the problem Pr EA) yielded greater effects for each separate Region in comparison with the same approach at the level of each Region considered separately (the problems Prr , r ¼ 1, . . . , 9). In addition, the economic development of all Regions was leveled as the result of solving the problem Pr11 : the maximal-to-minimal per capita GDP ratio over all Regions was reduced by 5.91% for year 2022 in comparison with the baseline scenario without parametric control and by 5.27% for year 2022 in comparison with year 2015. Moreover, in comparison with year 2015, the per capita GDPs for year 2022 were increased by 31.42% (Kyrgyzstan), 34.75% (Armenia), and 22.91% (the Rest of World). All these countries belong to the group of underdeveloped Regions. For the global economy, the solution of the problem PrW guaranteed a 1.38% growth of per capita GDP and a 7.92% growth of the output of Sector 11 (Agriculture, forestry, and fishery) for year 2022 in comparison with the baseline scenario. The per capita GDP growth for year 2022 constituted 24.83% in comparison with the baseline scenario; the per capita output of Sector 11 (Agriculture, forestry, and fishery) demonstrated a 41.38% growth for year 2022 in comparison with the baseline scenario for year 2015. Also the solution of each problem Pr r , (r ¼ 1, . . . , 9 ), allowed to reduce by 0.05–3.36% the trade gaps of goods and services in the corresponding regions. These relative reductions (in % to the basic values) can be observed in Table 2.9.

196

2.5.2

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Structural Adjustment Problem at a Sectoral Level

As is well-known, owing to structural adjustment of national economy many countries have guaranteed a good pace of economic development, created promising sectors, and achieved outstanding results (the so-called economic miracle). Structural adjustment policy produces growth points, stimulates new investments, and also initiates a gradual transfer of different resources (human, financial, and other) to most challenging sectors. Therefore, a topical problem is to propose macroeconomic structural adjustment tools, with focus on economic growth and accelerated advancement of promising sectors using the effect models of such tools. The available literature on structural adjustment modeling of different regions mostly studies the implementation results of structural adjustment programs (SAPs); see [20, 33, 54, 82]. Such investigations par excellence consider the effects from the structural adjustment programs adopted by the IMF, the World Bank, and other international financial institutions in various developing countries since the 1980s. One example is the paper [20], which developed statistical methods for determining the endogenous relationship between the recipient’s decision to join SAPs and its economic growth. The authors [82] presented a static computable general equilibrium model for reproducing the main results of the antirecessionary economic policy in Greece. Other examples of structural adjustment analysis based on CGE models can be found in the survey [33]. A series of CGE models for assessing the poverty effects of trade liberalization were considered in [54]. Some researchers are concerned with the implementation results of national structural adjustment programs. For instance, a large-scale multisectoral econometric model of Japan’s economy with the Leontief input–output structure was introduced in [72]. The model is applicable to study such growth alternatives as import promotion and increased leisure in context of induced technical progress, output, and employment. Naastepad [63] proposed a multiperiod real-financial CGE model of India, which determines the impact of the budget policy on the loan proposal as a possible factor restricting the effectiveness of structural adjustment programs. As indicated by in-depth analysis, the available literature on the modeling of structural adjustment effects mainly contains various examples of scenario analysis. Unfortunately, no examples on the choice of promising industries for the implementation of SAP policies have been presented so far. Moreover, no optimization problems of macroeconomic structural adjustment policy tools have been stated. This subsection demonstrates the possibilities of an approach to choose a set of promising economic sectors as well as to solve one parametric control problem as follows: find the optimal values of fiscal policy tools toward the economic growth and accelerated advancement of chosen promising sectors of Kazakhstan’s economy in the medium term.

2.5 Parametric Control Based on Model 1: A Series of Problem Statements. . .

197

The following steps were proposed for designing optimal fiscal policy measures toward the economic growth and structural adjustment of Kazakhstan’s economy at the sectoral level: 1. Choose a collection of most promising sectors of Kazakhstan’s economy, for which an accelerated increase of output is desired. 2. Solve a dynamic optimization problem on the economic growth and accelerated advancement of each promising sector in terms of output. These steps were implemented on the basis of Model 1. The marginal cost of public funds for all taxes from Sector i (denoted by MCFi) was employed as an index to characterize the prospects of different industries of Kazakhstan (see also [33]). Below MCFi is defined as the variation of the domestic GDP (in KZT) caused by a 1 tenge increase in tax collections from Sector i. This index measures the significance of a given industry for the domestic GDP in case of taxation increase. The results of computer simulations with Model 1 (the values of MCFi for years 2016–2022) and also the average values of this index are shown in Table 2.10. The following collection of three sectors with the highest average values of MCFi was defined using the last column of Table 2.10: • Sector 5 (Education, public health, and public administration), ehas • Sector 10 (Other services), oths • Sector 16 (Transportation), tser Designate as I ¼ {5; 10; 16} the set of corresponding subscripts i of Sectors. Table 2.10 Values of MCFi for Kazakhstan No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Sector ming crog mepe mind ehas pegw fpin psta otis oths agff buil mtal fins chpp tser

Year 2016 0.05 0.10 0.39 0.38 0.44 0.34 3.17 0.28 1.16 0.38 2.00 0.68 1.79 0.23 1.61 0.07

2017 0.58 0.15 1.03 0.05 0.70 0.21 3.63 0.15 1.57 0.51 2.21 0.32 1.32 0.08 1.42 0.24

2018 0.85 0.12 1.41 0.29 0.55 0.14 3.42 0.02 1.81 0.60 2.10 0.07 1.26 0.04 1.43 0.27

2019 1.08 0.07 1.76 0.49 0.62 0.08 3.46 0.05 2.04 0.67 2.11 0.13 1.30 0.02 1.49 0.34

2020 1.34 0.06 2.06 0.67 0.64 0.01 3.57 0.10 2.25 0.75 2.18 0.31 1.26 0.08 1.51 0.41

2021 1.47 0.09 2.26 0.80 0.70 0.03 3.65 0.14 2.36 0.78 2.22 0.45 1.27 0.13 1.49 0.44

2022 1.67 0.19 2.37 0.90 0.71 0.09 3.84 0.16 2.43 0.84 2.34 0.57 1.09 0.18 1.88 0.47

Average 0.88 0.15 1.45 0.31 0.61 0.12 3.49 0.03 1.83 0.61 2.14 0.03 1.30 0.02 1.49 0.29

198

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

A dynamic optimization problem (the parametric control problem SP) was considered for ensuring the economic growth and accelerated output increase of the chosen sectors of Kazakhstan’s economy for the period 2016–2022 using fiscal policy tools. This problem has the following informal statement. Statement of Problem SP Within the framework of Model 1, find the values of the control parameters u(t) (the same as in the problem Pr1 , i.e., the effective rates of 5 taxes and tariffs differentiated by 16 types of Products; the shares of government expenditures on consumption in Kazakhstan) that maximize the criterion K (2.108) subject to the constraints u(t) 2 U(t) (see the above statement of the problem Pr 1 ) and also constraints (2.106) and (2.107) on some endogenous variables. The constraints of the problem SP imposed on the endogenous variables of Model 1 have the form QVAPðtÞ  QVAPðtÞ,

ð2:106Þ

QXAj ðtÞ  QXAj ðtÞ,

ð2:107Þ

In these inequalities, j ¼ 1, . . ., 16; t ¼ 2016, . . ., 2022; QVAP(t) is the per capita GDP of Kazakhstan; QXAj(t) gives the per capita output of Sector j in Kazakhstan; finally, the symbol “” indicates the basic values of an index. The criterion K of this problem describes the growth of GDP and outputs of the three chosen sectors of Kazakhstan for years 2016–2022 (with the corresponding weight coefficients αi ¼ 0:1): K¼

X2022  t¼2016

TQVAðt Þ þ

X

 α TQX ð t Þ i i i2I

ð2:108Þ

Here TQVAðt Þ denotes the annual per capita GDP of Kazakhstan for year t while TQX i ðt Þ the per capita output of Sector i of Kazakhstan for year t (in USD). In this problem, the values of all uncontrolled exogenous variables of Kazakhstan and all exogenous variables of the other Regions match the baseline scenario. The problem SP was solved numerically using an iterative optimization algorithm provided by GAMS [66]. The output increase by different Sectors of Kazakhstan for years 2016–2022 in comparison with the basic forecast can be seen in Table 2.11. As shown by the simulation results, the stimulated Sectors have the following average growth rates on the parametric control period in comparison with the baseline scenario: Sector 5 (Education, public health, and public administration), 5.81%; Sector 10 (Other services), 4.21%; Sector 16 (Transportation), 4.76%. Moreover, considerable growth of outputs can be observed for the Sectors supplying intermediate Products: for example, the average output of Sector 6 (Production and supply of electricity, gas, and hot water) was increased by 4.66%. In addition, higher incomes of Households led to greater average outputs of Sector 7 (Food industry, including beverages and tobacco; 4.00%) and Sector 13 (Production of textiles, clothes, leather, and associated goods; 4.24%).

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199

Table 2.11 Relative output increase by different Sectors of Kazakhstan obtained by solving problem SP (in % to basic forecast) No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Sector ming crog mepe mind ehas pegw fpin psta otis oths agff buil mtal fins chpp tser

Year 2016 0.50 1.10 1.27 0.29 0.99 3.06 1.09 1.34 0.05 0.75 1.72 0.14 1.25 1.33 0.95 5.34

2017 1.41 1.21 1.38 2.24 2.29 2.46 1.90 0.10 0.03 2.23 0.71 3.32 2.38 0.26 1.12 6.37

2018 3.69 3.43 2.27 2.79 6.63 5.55 4.32 2.50 3.63 5.18 2.33 3.90 4.82 1.77 2.47 2.22

2019 3.73 3.57 3.14 4.34 8.54 4.45 5.95 3.76 4.76 5.55 3.89 3.75 5.40 2.23 3.66 2.07

2020 3.73 3.43 2.18 2.16 6.76 5.31 6.03 3.04 3.60 6.00 2.95 3.15 4.16 2.86 4.09 2.94

2021 2.74 3.45 1.45 1.54 5.56 3.80 3.71 1.91 3.38 4.45 2.86 3.14 4.08 2.88 3.97 5.52

2022 3.31 2.69 1.65 2.33 5.88 4.19 4.53 1.93 3.42 5.21 2.56 3.70 4.73 2.41 3.82 4.46

Average 2.75 2.79 1.83 2.16 5.81 4.66 4.00 2.07 2.96 4.21 1.71 2.92 4.24 2.21 3.12 3.72

On the whole, the solution of the problem SP guaranteed the average growth of the three stimulated Sectors with a rate of 4.58%, which exceeds the average growth of the other Sectors of Kazakhstan’s economy (2.88%). The solution of the problem SP also affected macroeconomic indexes: for year 2022, the per capita GDP of Kazakhstan was increased by 1.1% while the per capita output of all Sectors by 3.61% in average in comparison with the baseline scenario. The forecasted growth of the per capita GDP for year 2022 due to parametric control constituted 38.2% in comparison with year 2015. The solutions of the problem SP presented in this subsection (for their testing, see below) demonstrate high potential of the parametric control approach for making recommendations on optimal economic policy toward economic growth and structural adjustment.

2.5.3

Testing of Solutions Applicability

The optimal values of economic policy tools calculated in all parametric control problems considered were tested for implementability in the following way. For each parametric control problem, different stability indexes for the mappings defined by the scenarios of the calibrated Model 1 with the optimal values of economic policy tools were calculated (see the details in Sects. 1.3.2 and 2.5). All exogenous parameters of Model 1 for year 2016 were treated as the input parameters ( p)

200

2 Macroeconomic Analysis and Parametric Control Based on Global Multi. . .

Table 2.12 Values of stability indexes for solutions of parametric control problems Problem Pr1 Pr2 Pr3 Pr4 Pr5 Pr6 Pr7 Pr8 Pr9 PrEA PrW SP

Year 2016 0.8676 0.8738 0.5896 0.7658 0.8912 0.8929 0.8332 0.7872 0.6444 0.8743 0.8539 0.6140

2017 0.1814 0.1635 0.1326 0.1660 0.1114 0.1121 0.1181 0.1553 0.1533 0.1419 0.1409 0.1378

2018 0.0056 0.0049 0.0046 0.0060 0.0060 0.0055 0.0057 0.0042 0.0045 0.0062 0.0063 0.0053

2019 0.0001 0.0001 0.0002 0.0002 0.0002 0.0002 0.0001 0.0002 0.0002 0.0002 0.0002 0.0002

2020 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

2021 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2022 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

while the GDPs of all Regions of Model 1 for current year on the period 2016–2022 as the output parameters ( y). The values of the stability indexes βðp, αÞ obtained for the values p corresponding to the solutions of each parametric control problem and α ¼ 0:01 are combined in Table 2.12. Clearly, all values in Table 2.12 do not exceed 0.8929, which testifies to a rather high stability of Model 1 (in terms of the stability indexes considered) on the period 2016–2022. For α ¼ 0.0001, different combinations of the calculated values of the limiting indexes βf ðpÞ were very close to 0, which estimates the model mapping as continuous in the domain А. The solutions of the optimization problems obtained and tested in Sects. 2.5.2 and 2.5.3 demonstrate high potential of the parametric control approach for making recommendations on coordinated optimal economic policy at the global level as well as at the levels of the EAEU countries and separate countries (or Regions).

2.5.4

Study of Solutions Dependence on Uncontrolled Parameters

The dependence of the solutions of the parametric control problems on the values of uncontrolled parameters can be studied using an aggregate algorithm that includes the following steps. Step 1. Choose a parametric control problem, an uncontrolled parameter а, and a collection of endogenous variables of Model 1 to be analyzed. Step 2. Choose a value set of the uncontrolled parameter а for which the parametric control problem will be solved.

2.5 Parametric Control Based on Model 1: A Series of Problem Statements. . .

201

Table 2.13 Forecasted GDPs and indexes of different Sectors for year 2022 and different technological coefficients, scenarios with and without parametric control (in billion USD) Scenario Without parametric control

With parametric control

Index QVA Qehas Qoths Qtser QVA Qehas Qoths Qtser

Technological coefficient k 0.98 0.99 1.00 227.06 229.56 232.07 11.65 11.81 11.96 87.17 88.09 89.01 20.18 20.37 20.56 229.11 231.70 234.64 12.26 12.45 12.65 90.41 91.57 92.76 20.85 21.09 21.33

1.01 234.58 12.11 89.94 20.75 237.62 12.85 93.92 21.56

1.02 237.09 12.26 90.86 20.94 240.24 13.04 95.10 21.81

Step 3. Find the solutions of the parametric control problem for each value of the parameter а, and calculate the corresponding values of the endogenous variables, including the criterion of the parametric control problem. As an example, Table 2.13 shows how the optimal values of the criterion (and some other endogenous variables of Kazakhstan) obtained by solving the structural adjustment problem SP based on Model 1 depend on the values of the multiplicative coefficient k embedded in the technological coefficients of all production functions for the QVA of Model 1, starting from year 2016. This coefficient was varied by 1% and 2% with respect to the baseline scenario, taking values 0.98, 0.99, 1, 1.01, and 1.02. The following indexes of Kazakhstan for year 2022 were selected as the endogenous variables for the study: GDP (QVA); the output of Sector 5 (Education, public health, and public administration) (Qehas); the output of Sector 10 (Other services) (Qoths); and the output of Sector 16 (Transportation) (Qtser). The simulation results proved that the parametric control approach was efficient, not only for the baseline scenario (see column 5 of Table 2.13, where k ¼ 1) but also for other scenarios of global economy dynamics with technological changes. For example, the solution of the problem SP with the technological coefficients varied by 2%, 1%, +1%, and +2% on the forecasting period gave the GDP variations of +0.90%, +0.93%, +1.3%, and +1.33%, respectively, in comparison with the scenarios without parametric control. Note that, like parametric control in the baseline scenario, the chosen Sectors were well stimulated. More specifically, with the above changes of the technological coefficients, the rates of increase were considerably higher than the output growth for Sector 5 (Education, public health, and public administration; 5.21%, 5.50%, 6.14%, and 6.37%, respectively), Sector 10 (Other services; 3.72%, 3.95%, 4.43%, and 4.66%, respectively), and Sector 16 (Transportation; 3.32%, 3.53%, 3.91%, and 4.14%, respectively). In accordance with the simulation results, the largest effect from parametric control (in the sense of sectoral and macroeconomic growth) was achieved under greater values of the technological coefficients.

Chapter 3

Macroeconomic Analysis and Parametric Control Based on Global Dynamic Stochastic General Equilibrium Model (Model 2)

One example illustrating the efficiency of parametric control is the results obtained within the framework of the global multi-country dynamic stochastic general equilibrium (DSGE) model developed below. Today, single- [64, 69] and multi-country [21, 68] DSGE models are a widespread tool of macroeconomic analysis [75, 79]. In contrast to other classes of macroeconomic models, these models give a detailed description for the operation of central banks and second-level banks. Chapter 3 presents the following results: • A new nonlinear global multi-country DSGE model (further referred to as Model 3), differing from the well-known ones in a specific set of Regions (countries) of the global economy and accepted hypotheses (see hypotheses 4 and 5 in Sect. 3.1.1) • The estimated parameters of the nonlinear Model 2 using the estimated parameters of the linear Model 2 yielded by the log-linearization of the former • The applicability testing of computer simulations with Model 2 to real macroeconomic systems • Mid-term forecasting and macroeconomic analysis for the evolution of economies (countries) • Statements and solutions for a series of parametric control problems • The implementability testing of parametric control laws • The dependence of optimal criteria values on uncontrolled parameters in parametric control problems

© Springer Nature Switzerland AG 2020 A. A. Ashimov et al., Macroeconomic Analysis and Parametric Control of a Regional Economic Union, https://doi.org/10.1007/978-3-030-32205-2_3

203

204

3.1 3.1.1

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Conceptual Description of the Global Economy Within Model 2 Prerequisites for Global Economy Description

Model 2 was built on the basis of the original models from [30, 61]. Model 2 describes the global economy in accordance with the following hypotheses and notations: 1. The global economy is represented by the interacting agents of the economies from a collection R of chosen Regions (including countries) of the world. 2. The economy of each Region is represented by the following Sectors: Patient Households (P); Impatient Households (S), Q ¼ {P, S}; a collection X ¼ {L, M, N} of Producing Sectors; second-level Banks (B); and finally, State (G). 3. Each Sector of each Region r 2 R is represented by a continuum of properly notated agents. The number of an agent of Region r belongs to a range Tr of length nr, where nr are positive values (weights of Regions); they are chosen proportionally to the average GDPs of corresponding Regions so that ∑r2Rnr ¼ 1. The State of Region r consists of Government and Central Bank. 4. Production is not hierarchical [23, 55] in the sense that each Sector consumes intermediate products as well as supplies intermediate and final products. 5. Households have unfixed aggregate nonfinancial assets [52]; these assets are increased if investments exceed depreciation and decreased otherwise. 6. The corresponding agents perform the same functions in all Regions. 7. All agents operate in stochastic conditions defined by a collection of shocks. 8. The agents of different Regions interact through flows of products and capital (foreign loans and direct foreign investments and dividends). 9. The agents of Households, Firms, and second-level Banks are monopolistic competitors in the markets of labor, products, and bank services, respectively. 10. The agents of Households and Firms are perfect competitors in the markets of Capital (basic assets). 11. Wages are inflexible. Depending on Sector and Region, product prices are either inflexible or determined by exogenous world price (e.g., oil prices). 12. State is a monopolist in the market of public bonds. 13. All agents have perfect rationality. 14. All markets are in equilibrium. Unless indicated otherwise, all product variables of Model 2 expressed in monetary terms below are assumed to be real (i.e., matching the prices of the first quarter of 2000) and measured in millions of domestic currency (Euro in the EU and USD in the Rest of World). Time t corresponds to quarter number, starting from the first quarter of 2000.

3.1 Conceptual Description of the Global Economy Within Model 2

3.1.2

205

Conceptual Description of Household’s Behavior

At each time t, in Region r 2 R each household jq,r 2 Tr of type q 2 Q, Q ¼ {P, S}, consumes and purchases products as nonfinancial assets from all firms of all Producing Sectors from all Regions; provides labor to the firms of all Sectors from domestic Region and also sets wages; leases its land to firms; and pays consumption tax (in domestic Region r) and income tax (in each Region) to State. A patient household jP,r has deposits in all second-level banks of domestic Region (and no credits). In addition, all patient households co-own all firms and secondlevel banks (their equity capital) in all Regions. An impatient household has credits only (and no deposits) in all second-level banks of domestic Region. In all countries except for the USA and the Rest of World, deposits and credits are denominated in two currencies—domestic currency and USD. The households of the USA and the Rest of World have deposits and credits only in domestic currency—USD. For the sake of convenience, introduce the currency set Zr ¼ {r, us}, which consists of two elements for all Regions r except for the USA and the Rest of World and one element us for the USA and the Rest of World. Each household jq,r determines its behavior by solving the optimization problems 1q–7q, 8P, and 9S, see below. The statement of each problem is preceded by equations describing associated constraints. At a current time t, the budget balance of a patient household jP,r in Region r is given by X PrC,t C rP,t ðjP,r Þð1 þ τrC,t Þ þ PrI,H,t I rH,P,t ðjP,r Þð1 þ τrI,t Þ þ Sr Dr ðjP,r Þ þ T rP,t ðjP,r Þ z2Z r z,t P,z,t X ¼ ðW rP,x,t ðjP,r ÞLrP,x,t ðjP,r Þð1  τrW,t Þ þ PrO,t Orx,P,t ðjP,r ÞÞ x2X X þ S r Dr ðjP,r ÞRrP,z,t1 z2Z r z,t P,z,t1 X X þ ð1  τrW,t Þ r 2R Srr ,t ½ x2X DivrP,x,r ,t ðjP,r Þ þ DivrP,B,r ,t ðjP,r Þ:

ð3:1Þ At a current time t, the budget balance of an impatient household jS,r in Region r is given by PrC,t C rS,t ðjS,r Þð1 þ τrC,t Þ þ PrI,H,t I rH,S,t ðjS,r Þð1 þ τrI,t Þ X Sr Br ðjS,r ÞRrS,z,t1 þ T rS,t ðjS,r Þ þ z2Z r z,t S,z,t1 X X ðW rS,x,t ðjS,r ÞLrS,x,t ðjS,r Þð1  τrW,t Þ þ PrO,t Orx,S,t ðjS,r ÞÞ þ Sr Br ðjS,r Þ: ¼ x2X z2Z r z,t S,z,t

ð3:2Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

206

The notations in relationships (3.1) and (3.2) are the following: PrC,t as the consumer prices of households; C rq,t ð jq,r Þ as the household’s consumption; τrC,t , τrI,t , and τrW,t as the effective rates of the VAT on consumer and investment goods and the effective rate of the individual income tax, respectively; IH,q,t( jP,r) as the purchase of nonfinancial assets; PrI,H,t as the prices of nonfinancial assets; W rq,x,t ð jq,r Þ as the household’s wage in Sector x 2 X; Lrq,x,t ð jq,r Þ as the household’s labor supply to Sector x; Srz,t as the exchange rate of domestic currency of Region z to the foreign currency of Region r (Srr,t ¼ 1); Orx,q,t ð jq,r Þ as the land of household jq,r leased to Sector х of Region r for period t (exogenous variable); PrO,t as the price of land lease in Region r for period t; DrP,z,t ð jP,r Þ, z 2 Z r , as the household’s aggregate deposits in currency z; RrD,z,t1 as the deposit yield in currency z (1 + deposit rate);   BrS,z,t jS,r , z 2 Z r , as the aggregate credits of an impatient household in currency z; RrB,z,t1 as the banks’ credit yield in currency z (1 + credit rate); Divrx,r ,t ð jP,r Þ and DivrB,r ,t ð jP,r Þ as the household’s dividends from corresponding Producing Sectors and second-level Banks in Region r; and finally, T rt ð jq,r Þ as the aggregate amount of other net taxes (other taxes minus household’s subsidies). For each period t, the nonfinancial assets H rq,t ð jq,r Þ (houses, cottages in the country, etc.) of household jq,r depreciate at a rate δrH , incur cost in case of saving, and suffer from corresponding shocks. The nonfinancial assets are accumulated in accordance with the formula   H rq,t ð jq,r Þ ¼ 1  δrH H rq,t1 ð jq,r Þ þ

1  Ψ H,q

I rH,q,t ð jq,r Þ εrI,H,t r I H,q,t1 ð jq,r Þ

!! I rH,q,t ð jq,r Þ,

ð3:3Þ

where Ψ H,q denotes a cost function (a convex function that satisfies the condition Ψ H,q ð1Þ ¼ Ψ 0H,q ð1Þ ¼ 0); εI,H,t is the shock of purchasing nonfinancial assets [70] given by εrI,H,q,t ¼ εrI,H,q 1ρI,H,q εrI,H,q,t1 ρI,H,q eηI,H,q,t , r

r

r

ð3:4Þ

where εrI,H,q indicates an equilibrium value of this shock, ρrI,H,q is an autoregression coefficient, and ηrI,H,q,t means a Gaussian white noise of this shock.   The aggregate debt (BrS,z,t jS,r ) of each impatient household jS,r for period t must   be secured by its nonfinancial assets (H rS,t jS,r ). Hence, in case of loan, impatient households have the following credit constraint in second-level Banks: Et

X

Sr Br z2Z r z,tþ1 S,z,t



      jS,r RrS,z,t  mrS,t 1  δrH Et QrH,tþ1 H rS,t jS,r :

ð3:5Þ

3.1 Conceptual Description of the Global Economy Within Model 2

207

Hereinafter, Et denotes mathematical expectation given available information at time t (t ¼ 1, 2, . . .); QrH,tþ1 specifies the price of nonfinancial assets; mrS,t is the loans/ nonfinancial assets restriction coefficient written in the form  mrS,t

¼

mrS

BrG,t BrG

rrm,s !1ρrm,s

mrS,t1 ρm,s eηm,s,t : r

r

ð3:6Þ

In addition, mrS indicates an equilibrium value of this coefficient (0 < mrS  1); ρrm,s is an autoregression coefficient, and ηrm,s,t means a Gaussian white noise of this coefficient; finally, BrG,t specifies the public debt of Region r while BrG its equilibrium value. Each household jq,r supplies specialized labor to the domestic Producing Sectors. Being a monopolist for its type of labor, each household jq,r sets wages W rq,x,t ð jq,r Þ for firms х. For period t the demand of Producing Sectors х of Region r for the labor of a given household jq,r is determined by the expression Lrq,x,t ð jq,r Þ

 r  r W q,x,t ð jq,r Þ λW,q r ¼ Lq,x,t : W rq,x,t

ð3:7Þ

Here Lrq,x,t and W rq,x,t denote the aggregate firms’ demand for labor and the aggregate wages in Producing Sector x; λrW,q is the elasticity of substitution for different labor types. The aggregate labor supply Lrq,t ð jq,r Þ by a household jq,r consisting of the components Lrq,x,t ð jq,r Þ is formed by the CES function [67] r

Lrq,t ð jq,r Þ ¼

X

γr x2X L,q,x

1 μr L,q

Lrq,x,t ð jq,r Þ

μr 1 L,q μr L,q

!μrμL,q1 L,q

:

ð3:8Þ

Here μrL,q denotes the elasticity of substitution for different labor supplies, while P γ rL,q,x, > 0 is the labor preference coefficient for Sector x ( γ rL,q,x ¼ 1). x2X

In this model, the wages W rq,x,t ð jq,r Þ are Calvo inflexible, i.e., a household jq,r sets e rq,x,t ð jq,r Þ for each Sector x 2 X with a probability 1  ξrW,q,x , an optimal wage W where ξrW,q,x 2 ½0, 1; otherwise it indexes the previous wage subject to consumer   prices using some factor γ rW,q,x γ rW,q,x 2 ½0, 1 . The relationships defining the Calvo inflexible wages are written in the form

208

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .



r PrC,t1 γW,q,x r W q,x,t1 ð jq,r Þ PrC,t2   r e q,x,t ð jq,r Þ: þ 1  ϱrq,x,t ð jq,r Þ W

W rq,x,t ð jq,r Þ ¼ ϱrq,x,t ð jq,r ÞεrW,q,t

ð3:9Þ

Here εrW,q,t denotes the indexed wage shock (also known as the markup shock); ξrW,q,x is the probability of wage indexation; ϱrq,x,t ð jq,r Þ acts as a random wage variation signal, i.e., an independent discrete random variable that takes value 1 with the probability ξrW,q,x and value 0 with the probability 1  ξrW,q,x . At each time t, a household jq,r obtains an immediate utility U rq,t ð jq,r Þ from its consumption Crq,t ð jq,r Þ, labor supply Lrq,t ð jq,r Þ, and financial assets H rq,t ð jq,r Þ given by 0 B B U rq,t ð jq,r Þ ¼ εrb,q,t B @

ðCrq,t ð jq,r Þχ rq Crq,t1 Þ 1σ rC,q

1σ r C,q

1þσ r L,q

Lr ð jq,r Þ εrL,q,t q,t 1þσr L,q



r H rq,t ð jq,r Þ1σH,q r þεH,q,t 1  σ rH,q

1 C C C: A

ð3:10Þ

Here χ rq denotes the consumer habit coefficient; Crq,t1 is the aggregate consumption of goods by the households of Region r at the previous time; Lrq,t ð jq,r Þ is calculated using (3.8) and (3.7); σ rC,q > 0, σ rL,q > 0, and σ rH,q > 0 mean the inverse values of the elasticities of substitution for consumption, labor supply, and nonfinancial assets. This utility is subjected to three shocks—the preference shock εrb,q,t, th—labor supply shock εrL,q,t , and the nonfinancial assets demand shock εrH,q,t . The listed shocks are specified in conventional way: εrb,q,t ¼ εrb,q 1ρb,q εrb,q,t1 ρb,q eηb,q,t ,

ð3:11Þ

εrL,q,t ¼ εrL,q 1ρL,q εrL,q,t1 ρL,q eηL,q,t ,

ð3:12Þ

r

r

r

εrH,q,t ¼ εrH,q

1ρrH,q

εrH,q,t1

r

r

r

ρrH,q

ηrH,q,t

e

:

ð3:13Þ

Here εrb,q , εrL,q , and εrH,q indicate the equilibrium values of these shocks; ρrb,q , ρrL,q , and ρrH,q are autoregression coefficients; ηrb,q,t, ηrL,q,t, and ηrH,q,t represent the Gaussian white noises of the shocks. For period t the main goal of a household jq,r is to maximize the total expected P  r i r r q,r discounted utility Et 1 i¼0 βq U q,tþi ð j Þ. Here β q denotes the household’s discount factor. Impatient households put more value on current utility than patient ones, which leads to the natural constraint βrS < βrP . q,r r Problem 1q  Given t, q, r, q,rand j 2 T , find the values of the variables q,r 1 C q,tþi ð j Þ i¼0 , where Cq,t1( j ) are specified nonrandom values and

3.1 Conceptual Description of the Global Economy Within Model 2

209

C P, tþi ðjP, r Þ ¼ fC Pr , tþi ðjP, r Þ, DPr , r, tþi ðjP, r Þ, DPr , us, tþi ðjP, r Þ, H Pr , tþi ðjP, r Þ, I Hr , P, tþi ðjP, r Þ, W Pr , M , tþi ðjP, r Þ, W Pr , N , tþi ðjP, r Þ, W Pr , L, tþi ðjP, r Þ, ~ Pr , N , tþi ðjP, r Þ, W ~ Pr , L, tþi ðjP, r Þg ~ Pr , M , tþi ðjP, r Þ, W W for patient households (q ¼ P) and C S,tþi ðjS,r Þ ¼fCrS,tþi ðjS,r Þ, BrS,r,tþi ðjS,r Þ, BrS,us,tþi ðjS,r Þ, H rS,tþi ðjS,r Þ, I rH,S,tþi ðjS,r Þ, W rS,N,tþi ðjS,r Þ, W rS,M,tþi ðjS,r Þ, W rS,L,tþi ðjS,r Þ, ~ rS,M,tþi ðjS,r Þ, W ~ rS,N,tþi ðjS,r Þ, W ~ rS,L,tþi ðjS,r Þg W for impatient households (q ¼ S), that maximize the household’s utility function  X     i    T q,r 1 r r q,r , Z ð j Þ E β U ð j Þ V rq,t Cq,tþi ð jq,r Þ1 ¼ liminf q,tþi t i¼0 q q,tþi i¼0 i¼0 T!1

ð3:14Þ subject to constraints (3.15), (3.17), and (3.19) derived from their relationships (3.1), (3.3), and (3.9) for patient households and constraints (3.16), (3.17), (3.18), and (3.19) derived from their relationships (3.2), 1 (3.3), (3.5), and (3.9) for impatient households. In formula (3.14), Z q,tþi ð jq,r Þi¼0 is the collection of all uncontrolled variables of this optimization problem that consists of the abovementioned variables 1 not included in the list of Cq,tþi ð jq,r Þi¼0 . Write the constraints of the problem 1q as inequalities. From (3.1) it follows that         gr1,P,tþi jP,r ¼ PrC,tþi C rP,tþi jP,r 1 þ τrC,tþi þ PrI,H,tþi I rH,P,tþi jP,r 1 þ τrI,t X  P,r    þ T rP,tþi jP,r þ Sr Dr j z2Z r z,tþi P,z,tþi X       W rP,x,tþi LrP,x,tþi jP,r 1  τrW,tþi þ PrO,tþi Orx,P,tþi jP,r  x2X X  P,r  r  Sr Dr j RP,z,tþi1 z2Z r z,tþi P,z,tþi1 0P r 1 Sr ,tþi DivrP,x,r ,tþi ð jP,r Þ X   r @ x2X A   P,r  1  τW,tþi  0 r 2R r r Sr ,tþi DivP,B,r ,tþi j ð3:15Þ Here LrP,x,t ð jP,r Þ ¼



 r W rP,x,t ð jP,r Þ λW,P r LP,x,t . r W P,x,t

210

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

From (3.2) it follows that      gr1,S,tþi jS,r ¼ PrC,tþi CrS,tþi jS,r 1 þ τrC,tþi    þ PrI,H,tþi I rH,S,tþi jS,r 1 þ τrI,t X  S,r  r  S,r  r r r R þ þ T j r Sz,tþi BS,z,tþi1 j S,z,tþi1 S,tþi z2Z X        x2X W rS,x,tþi LrS,x,tþi jS,r 1  τrW,tþi þ PrO,tþi Orx,S,tþi jS,r X    z2Z r Srz,tþi BrS,z,tþi jS,r  0: ð3:16Þ Here LrS,x,t ð jP,r Þ ¼



 r W rS,x,t ð jP,r Þ λW,P r LS,x,t . r W S,x,t

From (3.3) it follows that   gr2,q,tþi ð jq,r Þ ¼ H rq,tþi ð jq,r Þ  1  δrH H rq,tþi1 ð jq,r Þ !! I rH,q,tþi ð jq,r Þ r  1  Ψ H εI,H,tþi r I rH,q,tþi ð jq,r Þ  0: I H,q,tþi1 ð jq,r Þ

ð3:17Þ

From (3.5) it follows that   X     gr3,S,tþi jS,r ¼ z2Z r Etþi Srz,tþiþ1 BrS,z,tþi jS,r RrS,z,tþi        mrS,tþi 1  δrH H rS,tþi jS,r Etþi QrH,tþiþ1  0:

ð3:18Þ

From (3.9) it follows that gr4,q,x,tþi ð jq,r Þ

¼

W rq,x,tþi ð jq,r Þ

þ

ϱrq,x,tþi ð jq,r ÞεrW,q,tþi



 r r PC,tþi1 γW,q,x r W q,x,tþi1 ð jq,r Þ PrC,tþi2

 r e q,x,tþi ð jq,r Þ  0, þ 1  ϱrq,x,tþi ð jq,r Þ W

ð3:19Þ where x 2 X. For solving the problem 1q, use the infinite-dimensional Kuhn–Tucker theorem. Theorem 3.1 (see [74]) Suppose (i)

hXT i   1 t V C1 β U ð C , C , Z Þ : t t1 t 1 , Z 1 ¼ lim inf E 0 t¼0 T!1

ð3:20Þ

(ii) U is concave, and each element of g(Ct, Ct1, Zt) is convex in Ct and Ct1 for each Zt and all integers t  0.

3.1 Conceptual Description of the Global Economy Within Model 2

211

(iii) There is a sequence of random variables C1 that each Ct is a function only 0 such  1  1 of information available at t, V C1 , Z 1 is finite with the partial sums definingit on the right-hand side of (3.20) converging to a limit, and, for each  t  0, g Ct , Ct1 , Z t  0. (iv) U and g are both differentiable with respect to Ct and Ct  1 for each Zt and the derivatives have finite expectation. , with each λt in the (v) There is a sequence of nonnegative random vectors fλt g1 0  corresponding information set at t, and satisfying λt g Ct , C t1 , Z t ¼ 0 with probability 1 for all t. (vi)   

 ∂U C t , C t1 , Z t ∂U C tþ1 , C t , Z tþ1 þ βEt ∂C t ∂C t    

∂g C t , C t1 , Z t ∂g C tþ1 , Ct , Z tþ1 þ βEt λtþ1 ¼ λt ∂C t ∂C t

ð3:21Þ

for all t (i.e., the Euler equations hold). b 1, (vii) (Transversality) for every feasible C sequence C 0   

 ∂U C t , C t1 , Z t ∂gðC t , Ct1 , Z t Þ b  λt lim supβ E ðC t  Ct Þ  0: ∂Ct ∂Ct t!1 t

ð3:22Þ

Then C 1 0 maximizes V subject to g(Ct, Ct1, Zt)  0 for all t  0 and to the given nonrandom value of C1. Now, verify the assumptions of this theorem for the problem 1q. Condition (i) This condition determines the type of the utility function V and its arguments using relationships (3.14) and (3.10). Condition (ii) This condition will be verified for the problem 1q using the following theoretical fact. Theorem 3.2 (see [5, 11]) Let D be a convex set in the Euclidean space En with a non-empty interior intD 6¼ ∅, and also let a function f(u) be twice continuously differentiable in U. Then the function f(u) is convex in D if and only if its Hessian matrix is nonnegative definite for all u 2 D. For a matrix to be nonnegative definite, a necessary and sufficient condition is that all its eigenvalues are nonnegative. In accordance with this theorem, • A function f(u) is convex in D if and only if all eigenvalues of its Hessian matrix are nonnegative for all u 2 D.

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

212

• A function f(u) is concave in D if and only if all eigenvalues of its Hessian matrix are nonpositive for all u 2 D. (ii.1) For the problem 1q, the first part of Condition (ii) in Theorem 3.1 (U is concave) means that the function 0 U rq,t ð jq,r Þ

¼

B

ðCrq,t ð jq,r Þχ rq Crq,t1 Þ

1σ r C,q

1σ rC,q

B εrb,q,t B



1þσ r L,q

Lr ð jq,r Þ εrL,q,t q,t 1þσr L,q

r H rq,t ð jq,r Þ1σH,q r þεH,q,t 1  σ rH,q

@

1 C C C, A

r

where Lrq,t ð jq,r Þ ¼

P

r x2X γ L,q,x

1 μr L,q

Lrq,x,t ð jq,r Þ

μr 1 L,q μr L,q

!μrμL,q1 L,q

and Lrq,x,t ð jq,r Þ ¼

W r

λrW,q

q,r q,x,t ð j Þ W rq,x,t

Lrq,x,t

are concave in the variables Cq,t( jP,r) and Cq,t  1( jP,r) for all admissible values of the variables Zt+i( jq,r) and chosen parameter values. (Without loss of generality, let i ¼ 0 for the sake of compact representation.) First, calculate the elements of the Hessian matrix GU of the function U rq,t ð jq,r Þ that are the second partial derivatives with respect to the variables C rq,t ð jq,r Þ, W rq,x,t ð jq,r Þ (x 2 X), and H rq,t ð jq,r Þ:  σrC,q 1 ∂ U rq,t r r r q,r r r < 0;  2 ¼ εb,q,t σ C,q Cq,t ð j Þ  χ q C q,t1 r ∂Cq,t 2

2

∂ U rq,t r r r r q,r σ r 1  2 ¼ σ H,q εb,q,t εH,q,t H q,t ð j Þ H,q < 0; ∂H rq,t 2

∂ U rq,t r r r  2 ¼ εb,q,t εL,q,t λW,q,t r ∂W q,x,t !μr 1 1 μr 1   L,q 1 L,q σ rL,q þμr1 1 X r r 1 μ μ r q,r L,q L,q L,q Lr ð Þ σ rL,q þ r Lrq,t ð jq,r Þ γ j L,q,x q,x,t x2X μL,q  λrW,q 1 1 1 X r μr μr  x2X γ rL,q,x L,q Lrq,x,t ð jq,r Þ L,q Lrq,x,t λrW,q W rq,x,t ð jq,r Þ W rq,x,t λW,q 1 1 X λr1 þ1þλr1 σ rL,q þμr1 μr μr W,q þ Lr ð jq,r Þ L,q  x2X γ rL,q,x L,q Lrq,x,t W,q W rq,x,t Lrq,x,t ð jq,r Þ L,q q,t   1 1 X λr1 þλ r 1 1 1 μr μr  γ rL,q,x L,q Lrq,x,t W,q W rq,x,t r þ 1 þ r Lrq,x,t ð jq,r Þ L,q W,q x2X μL,q λW,q  λrW,q 1 io r W rq,x,t λW,q : Lrq,x,t λrW,q W rq,x,t ð jq,r Þ 

f

3.1 Conceptual Description of the Global Economy Within Model 2 2

Obviously, the inequality

∂ U rq,t

ð∂W rq,x,t Þ

2

 0 holds under positive values of all param-

W,q

Thus, the Hessian matrix GU of the function

2

2

2

∂ U rq,t ∂ U rq,t ∂ U rq,t ¼ ∂Cr ∂H r r ¼ r r ∂W ∂W q,x,t q,t q,x,t ∂H q,t q,t q,t r 1 1 U q,t is diagonal; if 1  μr þ λr W,q L,q

eter models and 1  μ1r þ λr1  0. In addition, ∂Cr L,q

213

2

¼ 0.  0,

2

∂ U rq,t

∂ U rq,t

this matrix has three nonpositive values on the main diagonal ( r 2 , 2 , and ð∂Cq,t Þ ð∂H rq,t Þ 2

∂ U rq,t

2 ) while the other elements are 0. Hence, the Hessian matrix GU of the ð∂W rq,x,t Þ function U rq,t ð jq,r Þ has nonpositive eigenvalues only, which proves its concavity over the domain under the above parameter values.

(ii.2) For the problem 1q, the second part of Condition (ii) in Theorem 3.1 (each element of g(Ct, Ct1, Zt) is convex in Ct and Ct1 for each Zt and all integers   t  0) means the convexity of each of the functions gr1,P,tþi ð jP,r Þ, gr1,S,tþi jS,r ,   gr2,q,tþi ð jq,r Þ, gr3,S,tþi jS,r , and gr4,q,tþi ð jq,r Þ in the corresponding variables. Without loss of generality, let i ¼ 0. 1. Check the convexity of the function gr1,P,t ð jP,r Þ defined by (3.15). Calculate the partial derivatives of the function g1,P,t( jP,r) with respect to the optimized variables for which there exist nonzero partial derivatives of the second order, i.e., with respect to the variables W rP,M,t ð jP,r Þ, W rP,N,t ð jP,r Þ, and W rP,L,t ð jP,r Þ, or, in general case, with respect to the variable W rP,x,t ð jP,r Þ: 2



  r   r  r  P,r λrW,P 1 ∂ gr1,P,t λr r r r  0 for 0 2 ¼ 1  τW,t LP,x,t W P,x,t W,P 1  λW,P λW,P W P,x,t j r ∂W P,x,t  λrW,P  1: Thus, the Hessian matrix Gg1,P,t of the function g1,P,t( jP,r) is diagonal; if 0 

λrW,P  1, this matrix has three positive values on the main diagonal (

2

∂ gr1,P,t

2, where ð∂W rP,x,t Þ x 2 {M, N, L}) while the other elements are 0. Hence, this function is convex over its domain under the above parameter values.

2. The same  considerations as before lead to the conclusion that the function gr1,S,t jS,r defined by (3.16) is convex over its domain under 0  λrW,S  1. 3. Now, check the convexity of the function gr2,q,t ð jq,r Þ defined by (3.17). Calculate the partial derivatives of the function g1,P,t( jP,r) with respect to the optimized variables for which there exist nonzero partial derivatives of the second order, i.e., with respect to the variables I rH,q,t ð jq,r Þ and I rH,q,t1 ð jq,r Þ in the case Ψ H(x) ¼ x2, and find the signs of these derivatives:

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

214

 2 h i ∂ gr2,q,t r r q,r r q,r  2 ¼ 4 εI,H,q,t =I H,q,t1 ð j Þ I H,q,t ð j Þ þ 1 > 0; ∂I rH,q,t 2

2



∂ gr2,q,t ∂I rH,q,t1

 2  3  2  2

r r q,r r q,r r q,r ¼ 2 ε I ð j Þ I ð j Þ 1 þ 2 I ð j Þ > 0; 2 I,H,q,t H,q,t H,q,t1 H,q,t1

2  3  2  ∂ gr2,q,t r r q,r r q,r ¼ 4 ε I ð j Þ I ð j Þ < 0: I,H,q,t H,q,t H,q,t1 ∂I rH,q,t ∂I rH,q,t1 2

As easily observed, the Hessian matrix of the function gr2,q,t ð jq,r Þ consists of nonnegative values and the function gr2,q,t ð jq,r Þ is convex under the inequality I rH,q,t1 ð jq,r Þ

2



εrI,H,q,t

2

I rH,q,t ð jq,r Þ

h i  2

r q,r r q,r I H,q,t ð j Þ þ 1 1 þ 2 I H,q,t1 ð j Þ :

  4. The convexity of the other functions gr3,S,t jS,r and gr4,q,x,t ð jq,r Þ follows from their   e rq,x,t ð jq,r Þ, linearity in the variables BrS,z,t jS,r and W rq,x,t ð jq,r Þ, W rq,x,t1 ð jq,r Þ, W respectively. Condition (iii) In Theorem 3.1, this condition guarantees the existence of at least a  single sequence of vector random variables C1 0 ¼ C 0 , C 1 , . . . of the type C with the following properties: (iii. 1) Each C t is a function only of the available information for period t; in other words, C t does not depend on the variables Zt+1, Zt+2, . . . (although it may depend on their expectations Et). (iii. 2) For Ct ¼ C t , series (3.20) has finite sum. (iii. 3) Ct is an admissible sequence, i.e., it satisfies the optimization constraints g Ct , Ct1 , Z t  0. Note that the sequence C 1 0 becomes optimal under these and other conditions of Theorem 3.1. 1 For the problem 1q, the existence of the vector sequence C q,tþi ð jq,r Þi¼0 is checked, and its elements are obtained by calculating Model 2. Condition iii.1) is guaranteed by the equations of Model 2. Condition iii.2). The convergence of series (3.14) can be estimated numerically as follows. Because the expectations of all variables of Model 2 calculated in Dynare converge to their stationary values as t ! 1, they are bounded in t; particularly, the variables Et U rq,tþi ð jq,r Þ are bounded in i. In this case, the terms of series (3.14) in modulus do not exceed the terms of converging geometric

3.1 Conceptual Description of the Global Economy Within Model 2

215

progression, and hence series (3.14) is absolutely convergent. For calculating the series sums V rq,t , Model 2 can be augmented by the equation V rq,t ¼ U rq,t þ βrq Et V rq,tþ1 .   Condition iii.3) holds since Model 2 contains the equation g C t , Ct1 , Z t ¼ 0. Condition (iv) For the problem 1q, the functions U rq,tþi , gr1,P,tþi , gr1,S,tþi , gr2,q,tþi , gr3,S,tþi, and gr4,q,x,tþi are differentiable. For checking that the partial derivatives under consideration have finite expectations, add new variables corresponding to these derivatives and their formulas. As noted earlier, the calculated expectations of all variables of Model 2 are bounded in time. Condition (v) In Theorem 3.1, this condition implies the following: (v.1) There exists a sequence of vector variables each of which is independent of the variables Zt+1, Zt+2, . . ., (e.g., the Lagrange multipliers {λ0, λ1, . . .} that satisfy (3.21)).   (v.2) The equalities λt g C t , C t1 , Z t ¼ 0 hold for t ¼ 0, 1, . . . . For the problem 1q, the existence of the Lagrange multipliers is checked, and they are obtained by calculating Model 2. As indicated by calculations, these multipliers are nonzero.  Condition v.2) is surely the case because Model 2 contains the equations g C t , Ct1 , Z t ¼ 0. Conditions (vi) Write these first-order conditions of form (3.21) for the problem 1q with i ¼ 0. 1. For q 2 {P, S}, differentiation with respect to C rq,t ð jq,r Þ yields  σ rC,q   εrb,q,t Crq,t ð jq,r Þ  χ rq C rq,t1 ¼ Λrq,t ð jq,r ÞPrC,q,t 1 þ τrC,q,t ,

ð3:23Þ

where Λrq,t ð jq,r Þ is the Lagrange multiplier corresponding to the balance constraint. 2. Differentiation with respect to DrP,r,t ð jP,r Þ yields     ΛrP,t jP,r  βrP RrP,r,t Et ΛrP,tþ1 jP,r ¼ 0:

ð3:24Þ

3. Differentiation with respect to DrP,us,t ð jP,r Þ yields     ΛrP,t jP,r Srus,t  βrP RrP,us,t Et ΛrP,tþ1 jP,r Srus,tþ1 ¼ 0:

ð3:25Þ

216

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

  4. Differentiation with respect to BrS,r,t jS,r yields        ΛrS,t jS,r ¼ RrS,r,t βrS Et ΛrS,tþ1 jS,r þ μrS,t jS,r ,

ð3:26Þ

  where μrS,t jS,r is the Lagrange multiplier corresponding to the loan constraint.   5. Differentiation with respect to BrS,us,t jS,r yields        ΛrS,t jS,r  βrS RrS,us,t Et Srus,tþ1 ΛrS,tþ1 jS,r  μrS,t jS,r RrS,us,t Et Srus,tþ1 ¼ 0: ð3:27Þ 6. Differentiation with respect to H rP,t ð jP,r Þ yields   σ rH,P       εrb,P,t εrH,P,t H rP,t jP,r  QrH,P,t jP,r þ 1  δrH βrP Et QrH,P,tþ1 jP,r ¼ 0,

ð3:28Þ

where QrH,q,t ð jP,r Þ is the Lagrange multiplier corresponding to the nonfinancial assets saving constraint.   7. Differentiation with respect to H rS,t jS,r yields   σrH,S       εrb,S,t εrH,S,t H rS,tþi jS,r  QrH,S,t jS,r þ 1  δrH βrS Et QrH,S,tþ1 jS,r     þμrS,t jS,r mrS,t 1  δrH Et QrH,tþ1 ¼ 0:

ð3:29Þ

8. In the case Ψ H(x) ¼ x2, differentiation with respect to I rH,q,t ð jq,r Þ, where q 2 {P, S}, yields 2



Λrq,t PrI,H,t 1 þ τrI,t



!2 3 r q,r I ð j Þ H,q,t 5 þQrH,q,t ð jq,r Þ41 þ 3 εrI,H,q,t r I H,q,t1 ð jq,r Þ 8 !3 9 q,r < =  2 I r ð j Þ H,q,tþ1 2βrq Et QrH,q,tþ1 ð jq,r Þ εrI,H,q,tþ1 ¼ 0: I rH,q,t ð jq,r Þ : ; ð3:30Þ

3.1 Conceptual Description of the Global Economy Within Model 2

217

9. Differentiation with respect to W rq,x,t ð jq,r Þ, where q 2 {P, S} and x 2 {L, M, N}, yields   Λrq,t ð jq,r ÞLrq,x,t ð jq,r Þ 1  τrW,t  M rq,x,t ð jq,r Þ  r γrW,q,x   P r r þβq εW,q,tþi r C,t Et M rq,x,tþ1 ð jq,r Þϱrq,x,tþ1 ð jq,r Þ ¼ 0: PC,t1

ð3:31Þ

Here M rq,x,t ð jq,r Þ is the Lagrange multiplier corresponding to the Calvo inflexibility condition of the wages. r

e q,x,tþi ð jq,r Þ, where q 2 {P, S} and x 2 {L, M, N}, 10. Differentiation with respect to W yields   M rq,x,t ð jq,r Þ 1  ϱrq,x,t ð jq,r Þ ¼ 0:

ð3:32Þ

Transversality (vii) Along with other conditions of Theorem 3.1, transversality is sufficient for proving that the sequence C1 0 represents the solution of the optimizab 1 of the Kuhn–Tucker theorem tion problem 1q. The set of appropriate sequences C t  1 consider the set of all sequences C q,tþi ð jq,r Þi¼0 satisfying the constraints of the b tþi are bounded in time (like this holds for problem 1q such that the expectations Et C thehnumerical solutions obtained in Dynare). In this i case, the expressions  ∂UðCtþi , C tþi1 , Z tþi Þ ∂gðC tþi , Ctþi1 , Z tþi Þ b Et  λt ðC tþi  C tþi Þ are bounded in time and ∂C tþi ∂Ctþi i h  , C tþi1 , Z tþi Þ ∂gðCtþi , C tþi1 , Z tþi Þ b tþi  C tþi Þ ¼ 0 since 0 <  λ lim ðβrq Þt Et ∂UðCtþi∂C ð C t ∂C tþi tþi i!1

βrq < 1 and the expectations are bounded in i. Hence, transversality holds. Therefore, all conditions of Theorem 3.1 are true for the problem 1q, and the firstorder conditions above (including the problem constraints) define its solution. Solving the next triplet of nested optimization problems, a household jq,r determines the optimal consumptions of each product x 2 X (the problem 2q) made in  each Region r 2 R (the problem 3q) by a firm hx,r (the problem 4q). For consumptive use, during period t, a household jq,r purchases goods of different types C rq,t ð jq,r Þ aggregated in accordance with the CES function

Crq,t ð jq,r Þ ¼

X

1 μr C

γ r Crq,x,t ð jq,r Þ x2X C,x

μr 1 C μr C,q

! r μrC μ

C,q

1

:

ð3:33Þ

218

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Here γ rC,x > 0 is the preference coefficient for the consumption of goods from Sector P x ( x2X γ rC,x ¼ 1); μrC denotes the elasticity of substitution for the consumption of goods from different Sectors. Problem 2q Given t, q 2 Q, r 2 R, jq, r 2 Tr, the consumptions Crq,t ð jq,r Þ, and the prices PrC,x,t , find the consumptions of sectoral goods Crx,q,t ð jq,r Þ by minimizing the purchase cost max fCr

q,r x,q,t ð j Þgx2X

 X   x2X PrC,x,t Crq,x,t ð jq,r Þ

subject to the following constraint derived from (3.33): μr

gðxÞ ¼ C rq,t ð jq,r Þ 

X

γr x2X C,q,x

1 μr C,q

Crq,x,t ð jq,r Þ

μr 1 C,q μr C,q

!μr C,q1 C,q

 0:

ð3:34Þ

Solve the problem 2q using the Kuhn–Tucker theorem as follows. Theorem 3.3 (see [8]) Consider a nonlinear programming problem maxf(x) subject to constraints gi(x)  0, i ¼ 1, . . ., m. Let the objective function f be differentiable and pseudoconcave (concave), while the constraint functions gi be differentiable and quasiconcave (concave). In addition, assume x satisfies the following Kuhn–Tucker conditions: 1. x is an admissible point. 2. There exist multipliers λi  0, i ¼ 1, . . ., m, such that 2.1 λigi(x) ¼ 0, Pi ¼ 1, . . ., n; 2.2 ∇f ðx Þ þ ni¼1 λi ∇gi ðx Þ ¼ 0, where ∇ denotes gradient. Then x is the solution of this nonlinear programming problem. 

Now, verify the conditions of Theorem 3.3 for the problem 2q in which x ¼  Crq,L,t ð jq,r Þ, C rq,M,t ð jq,r Þ, C rq,N,t ð jq,r Þ and the number of constraints is n ¼ 1.

(i) The concavity and differentiability of the objective function f ðxÞ ¼ P  x2X PrC,x,t Crq,x,t ð jq,r Þ follows from its linearity. (ii) Obviously, the function g(x) defined by (3.34) is differentiable. Check its concavity using the following result, see [26]. 1 P Proposition 3.4 Let M r ðxÞ ¼ ð ni¼1 xri Þr , xi > 0 i ¼ 1, . . ., n. The function Mr(x) is concave for r  1 and convex for r  1. Hence, the function g(x) is concave if μrC,q < 1 and μrC,q 6¼ 0.

3.1 Conceptual Description of the Global Economy Within Model 2

219

For the problem 2q, find the Lagrange multiplier Λt from the Euler equation (3), writing the xth coordinate (x 2 X) of this vector equation as

PrC,x,t

þ Λt

X

1

μr q,r C,q C r γr q,x,t ð j Þ x2X C,q,x

μr 1 C,q μr C,q

!μr 1 1 C,q

γ rC,q,x

1 μr C,q

1 μr

Crq,x,t ð jq,r Þ C,q ¼ 0 ð3:35Þ

In view of the economic sense of the price PrC,x,t > 0 , which directly implies Λt > 0, require g(x) ¼ 0 (the Kuhn–Tucker condition 2.1)), i.e., μr

Crq,t ð jq,r Þ ¼

X

γr x2X C,q,x

1 μr C,q

Crq,x,t ð jq,r Þ

μr 1 C,q μr C,q

!μr C,q1 C,q

:

ð3:36Þ

Using this relationship, write (3.35) as   r1 1 1 μ μr μr PrC,x,t ¼ Λt C rq,t ð jq,r Þ C,q γ rC,q,x C,q Crq,x,t ð jq,r Þ C,q :

ð3:37Þ

Expressing Crq,x,t ð jq,r Þ from (3.37) gives  C rq,x,t ð jq,r Þ ¼ γ rC,q,x

PrC,x,t Λt

μrC,q

Crq,t ð jq,r Þ:

ð3:38Þ

Next, raise both sides of (3.37) to the power (1  μrC,q ), multiply them by γ rC,q,x , and sum up over x (x 2 X) to get X

r

r γ r Pr 1μC,q x2X C,q,x C,x,t

¼ Λt

1μrC,q

r

1μ 1  1μr C,q X 1  μrC,q μ μr r q,r r q,r C,q C r C,q : C q,t ð j Þ C,q γ ð j Þ q,x,t x2X C,q,x

Due to (3.36), this leads to the equality Λt ¼

X

r

γ r Pr 1μC,q x2X C,q,x C,x,t



1 1μr C,q

:

ð3:39Þ

Finally, multiply both sides of (3.38) by PrC,x,t , and sum up the resulting expressions over x 2 X using (3.39) to obtain X

Pr Cr ð jq,r Þ x2X C,x,t q,x,t

¼ Λt Crq,t ð jq,r Þ:

ð3:40Þ

220

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Consequently, the variable Λt acts as the consumer product price C rq,t ð jq,r Þ ; so rename PrC,t ¼ Λt . Now, verify the Kuhn–Tucker conditions for this problem.   1. The point x ¼ C rq,L,t ð jq,r Þ, C rq,M,t ð jq,r Þ, Crq,N,t ð jq,r Þ is determined using relationships (3.38) and (3.40). It is admissible (g(x) ¼ 0) on the strength of (3.36). The multiplier Λt defined by (6.37) satisfies the inequality Λt  0. 2.1 The condition λigi(x) ¼ 0 holds due to (3.36). 2.2 The point x and multiplier λ1 ¼ Λt satisfy the condition ∇f ðx Þ þ Pm  i¼1 λi ∇gi ðx Þ ¼ 0 in accordance with the considerations above. Thus, the first-order conditions determining the solution of the problem 2q under μrC,q < 1 and μrC,q 6¼ 0 yield the following optimal consumer demands of a household jq,r for the goods of each Sector x (x 2 X) and prices Prx,t of these goods: Crq,x,t ð jq,r Þ ¼ γ rC,q,x PrC,t ¼

 r μrC,q PC,x,t C rq,t ð jq,r Þ; PrC,t

X

r

γ r Pr 1μC,q x2X C,q,x C,x,t



1 1μr C,q

ð3:41Þ

:

ð3:42Þ

In turn, the goods Crq,x,t ð jq,r Þ of each type x 2 X purchased by a household jq,r are aggregated over the Regions of production as follows: r

C rq,x,t ð jq,r Þ ¼

X

γr  r  2R C,x,r

1 μr C,x

Crq,x,r ,t ð jq,r Þ

μr 1 C,x μr C,x

!μrμC,x1 C,x

:

ð3:43Þ

Here γ rC,x,r > 0 denotes the preference coefficient for the consumption of goods P from Sector x of Region r ( r 2R γ rC,x,r ¼ 1); μrC,x is the elasticity of substitution for the consumption of all goods from Sector x of different Regions. Problem 3q Given t, q 2 Q, r 2 R, x 2 X, jq,r 2 Tr, the consumptions C rq,x,t ð jq,r Þ, and the prices Prx,r ,t , find the consumptions Crq,x,r ,t ð jq,r Þ of the goods of type х made in Regions r 2 R by minimizing the purchase cost 

 X   r 2R Prx,r ,t Crq,x,r ,t ð jq,r Þ

max

Crq,x,r ,t ð jq,r Þ

r 2R

subject to the following constraint derived from (3.43):

3.1 Conceptual Description of the Global Economy Within Model 2

221 μr

X

C rq,x,t ð jq,r Þ 

1

μr q,r C,q,x C r γr  q,x,r  ,t ð j Þ r 2R C,q,x,r

μr 1 C,q,x μr C,q,x

!μr C,q,x1 C,q,x

 0:

ð3:44Þ

The first-order conditions determining the solution of the problem 3q under μrC,q,x < 1 and μrC,q,x 6¼ 0 are obtained as described above. They yield the optimal consumer demands of a household jq,r for all goods from Sector x (x 2 X) made in each Region r:  Crq,x,r ,t ð jq,r Þ

¼

γ rC,q,x,r

Prx,r ,t PrC,x,t

μrC,x,q

Crq,x,t ð jq,r Þ:

ð3:45Þ

The prices Prx,r ,t of these goods satisfy the relationship PrC,x,t ¼

X

r

r 2R

γ rC,q,x,r Prx,r ,t 1μC,q,x



1 1μr C,q,x

:

ð3:46Þ

Each of the above aggregate consumptions C rq,x,r ,t ð jq,r Þ of each type of goods    consists of a continuum of different goods C rq,x,r ,t jq,r , hx,r made by firms hx,r of Region r, which are defined by the integral CES functions 

C rq,x,r ,t ð jq,r Þ ¼



1 nr 

 r1 ð λx

T

r

  Crq,x,r ,t jq,r , hx,r

 λrx 1  λrx

! rλrx dhx,r



λx 1

:

ð3:47Þ



Here λrx denotes the elasticity of substitution for different goods from Sector x of Region r. The next optimization problem gives the consumptions of goods х made in  Region r by a firm hx,r . Problem 4q Given t, q 2 Q, x 2 X, r 2 R, r 2 R, jq,r 2 Tr, the consumptions   C rq,x,r ,t ð jq,r Þ and prices Prx,r ,t hx,r of different types of goods made by firms     hx,r 2 Sr of Region r, find the quantities of goods Crq,x,r ,t jq,r , hx,r from these firms by minimizing the purchase cost ð min 

C rq,x,r ,t ð jq,r , hx,r

Þ, br 2T r

Tr

 x,r  r  q,r x,r  r r P ,h C dh  ,t h  ,t j x,r q,x,r 

subject to the following constraint derived from (3.47):

ð3:48Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

222



C rq,x,r ,t ð jq,r Þ 



1 nr 

 r1 ð λx

T

r

! rλrx

r



Crq,x,r ,t jq,r , h

λx 1 x,r λr x

dhx,r

λx 1



:

ð3:49Þ

  Considering the period T r as the interval 0, nr and representing the integrals in (3.48), (3.49) as the limits of the corresponding Riemann sums ð nr 

     Prx,r ,t hx,r C rq,x,r ,t jq,r , hx,r dhx,r  r     r nr XN r kn r q,r kn ¼ lim P  Cq,x,r ,t j , , k¼1 x,r ,t N N N!1 N

0

ð3:50Þ





1 nr 

 r1 ð



λx

T

r

0

Crq,x,r ,t jq,r , h

! rλrx

r

λx 1 x,r λr x

dhx,r



λx 1







λrx r1 r

 1  kn B 1 λr nr XN ¼ lim @ r x C r  jq,r , k¼1 q,x,r ,t N N n N!1

λx

1 rλrx C A

λx 1

ð3:51Þ

,

formulate the following auxiliary problem 4q. (In fact, its solution converges to that of the problem 4q as N ⟶ 1.) 

Problem 4q Given t, q 2 Q, x 2 X, r 2 R, r 2 R, jq,r 2 Tr, N, Crq,x,r ,t ð jq,r Þ, nr , and  r    r Prx,r ,t knN , where k ¼ 1, . . ., N, find the values C rq,x,r ,t jq,r , knN by maximizing the function r X  r   

r N n kn r r q,r kn  P j , C  q,x,r  ,t k¼1 x,r ,t N N N , k¼1, ..., N

 max  r

Crq,x,r ,t jr , knN

ð3:52Þ

subject to the following constraint derived from (3.49) and (3.51): 0

   1  r B 1 λr nr XN r q,r kn C j , C rq,x,r ,t ð jq,r Þ  @ r x  k¼1 q,x,r ,t N N n

 λrx 1  λrx

1 C A

 λrx  λrx 1

 0:

ð3:53Þ

For obtaining the first-order conditions of this problem, take advantage of Theorem 3.2. Verify the conditions of this theorem for the problem 4q with x ¼        r r  Crq,x,r ,t jq,r , nN , Crq,x,r ,t jq,r , 2nN , . . . , Crq,x,r ,t jq,r , nr and n ¼ 1 constraints.

3.1 Conceptual Description of the Global Economy Within Model 2

(i) The r

 nN

223

f ð xÞ ¼

concavity and differentiability of the function   PN r knr  r r q,r kn follow from its linearity. k¼1 Px,r  ,t N C q,x,r ,t j , N

(ii) Obviously, the function 

0

 λrx 1  λrx

   1  r B 1 λ r  nr X N r q,r kn C j , gðxÞ ¼ C rq,x,r ,t ð jq,r Þ  @ r x  k¼1 q,x,r ,t N N n 

1 rλrx C A

λx 1



is differentiable too. For λrx,t < 1 and λrx,t 6¼ 0, its concavity is checked similar to the problem 2q. For the problem 4q, find the Lagrange multiplier Λt from the Euler equation (3), writing the mth coordinate (m ¼ 1, 2, . . ., N ) of this vector equation as  Prx,r ,t

mnr N



 r1

Λt

1 nr 

λx



 ¼ 

C rq,x,r ,t jq,r ,

mnr N



 r1 λx

0 

B 1 @ r n

 r1 λx





r

nr XN kn C r  jq,r , k¼1 q,x,r ,t N N 

λrx r1 λx

1

1  λrx 1

C A

ð3:54Þ In view of the economic sense of the price Prx,r ,t



mnr N





> 0 , which directly



implies Λt > 0, require g(x ) ¼ 0 (the Kuhn–Tucker condition 2.1), i.e., 

0

 λrx 1  λrx

   1  r B 1 λ r  nr X N r q,r kn C j , C rq,x,r ,t ð jq,r Þ ¼ @ r x  k¼1 q,x,r ,t N N n

1 rλrx C A

λx 1

:

ð3:55Þ

Using this relationship, write (3.54) as  Prx,r ,t

mnr N







1 ¼ Λt r  n

 Expressing Crq,x,r ,t jq,r ,

 r1 λx

mnr N



 Crq,x,r ,t

mnr j , N q,r



 r1 

 from (3.56) gives

λx

Crq,x,r ,t ð jq,r Þ

 r1 λx

:

ð3:56Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

224

 C rq,x,r ,t jq,r ,

r

mn N

0

 ¼

1 @ nr 

Prx,r ,t



mnr N

Λt



1λrx,t A

C rq,x,r ,t ð jq,r Þ:

ð3:57Þ

   Next, raise both sides of (3.56) to the power 1  λrx,t and sum up over m ¼ 1, . . ., N to get    1λrx,t 1 XN  r 1λrx,t mnr P  N m¼1 x,r ,t N 

¼ Λt

 1λrx,t

   1λrx,t   r1 r XN r  r 1 λx n r q,r r q,r mn C q,x,r ,t ð j Þ λx,t C j ,   m¼1 q,x,r ,t N N nr

 1λrx,t  λr x,t

:

ð3:58Þ Due to (3.55), this leads to the equalities    1λrx,t  1 XN  r 1λrx,t mnr 1λrx,t P ¼ Λ ,  ,t t x,r m¼1 N N

ð3:59Þ

and consequently   r 1λx  1 nr XN mn r Px,r ,t m¼1 N nr  N

r

Λt ¼

Finally, multiply both sides of (3.57) by Prx,r ,t



mnr N



!

1  1λrx

:

ð3:60Þ

 , and sum up the resulting

expressions over m ¼ 1, . . ., N using (3.60) to obtain  r     r nr XN mn r r q,r mn C q,x,r ,t j , ¼ Crq,x,r ,t ð jq,r ÞΛt : P  m¼1 x,r ,t N N N

ð3:61Þ

Therefore, the variable Λt acts as the consumer product price Crq,x,r ,t ð jq,r Þ; so rename Prx,r ,t ¼ Λt . Now, verify the Kuhn–Tucker conditions for this problem.      r  1. The point x ¼ C rq,x,r ,t jq,r , nN , . . . , Crq,x,r ,t jq,r , nr is determined using relationships (3.57) and (3.60). It is admissible (g(x) ¼ 0) on the strength of (3.55). The multiplier Λt defined by (3.60) satisfies the inequality Λt  0. 2.1 The condition λigi(x) ¼ 0 holds due to (3.55). 2.2 The point x and multiplier λ1 ¼ Λt satisfy the condition ∇f ðx Þ þ Pm  i¼1 λi ∇gi ðx Þ ¼ 0 in accordance with the considerations above.

3.1 Conceptual Description of the Global Economy Within Model 2

225

    Assuming that the functions Prx,r ,t hx,r and C rq,x,r ,t jq,r , hx,r are continuous

   r for hr 2 0, nr , pass to the limit of relationship (3.57) as N ! 1 and knN ! hr .  This gives the following first-order condition of the problem 4q under λrx < 1 and  λrx 6¼ 0: 

C rq,x,r ,t jq,r , hx,r





1 ¼ r n

  !λrx Prx,r ,t hx,r C rq,x,r ,t ð jq,r Þ: Prx,r ,t

ð3:62Þ

Here Prx,r ,t is calculated using the limiting value of the integral sums (3.60):  Prx,r ,t ¼

1 nr 

ð Tr

 x,r 1λrx x,r r P dh  x,r ,t h 



1  1λrx

:

ð3:63Þ

Solving the next triplet of nested optimization problems, a household jq,r determines the optimal investments in nonfinancial assets of each product x 2 X (the  problem 5q) made in each Region r 2 R (the problem 6q) by a firm hr (the problem 7q). The first-order conditions of these optimization problems will be established below (their derivation is the same as for the problems 2q, 3q, and 4q). For investment in nonfinancial assets, during period t, a household jq,r purchases goods of different types I rH,q,t ð jq,r Þ aggregated in accordance with the CES function r

I rH,q,t ð jq,r Þ ¼

X

γr x2X I,H,x

1 μr I,H

I rH,q,x,t ð jq,r Þ

μr 1 I,H μr I,H

!μrμI,H1 I,H

:

ð3:64Þ

Here γ rI,H,x > 0 is the preference coefficient for the purchase of goods from Sector P x as nonfinancial assets ( x2X γ rI,H,x ¼ 1); μrI,H denotes the elasticity of substitution for the purchase of goods from different Sectors as nonfinancial assets. Problem 5q Given t, q 2 Q, r 2 R, jq,r 2 Tr, the investments I rq,t ð jq,r Þ, and the prices PrI,H,x,t , find the investments in nonfinancial assets of products I rx,q,t ð jq,r Þ by minimizing the purchase cost max

fI rx,q,t ð jq,r Þgx2X

 X   x2X PrI,x,t I rq,x,t ð jq,r Þ

subject to the following constraint derived from (3.64):

226

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . . r

X

1

μr q,r I,H,q I r γr H,q,x,t ð j Þ x2X I,H,q,x

I rH,q,t ð jq,r Þ 

μr 1 I,H,q μr I,H,q

!μrμI,H,q1 I,H,q

 0:

ð3:65Þ

The first-order conditions determining the solution of the problem 5q under μrI,H,q < 1 and μrI,H,q 6¼ 0 yield the following optimal investment demands of a household jq, r for the goods of each Sector x (x 2 X) and prices PrI,H,t of these goods:  I rH,q,x,t ð jq,r Þ PrI,H,t ¼

¼

γ rI,H,q,x

PrI,H,x,t PrI,H,t

X

μrI,H,q

I rH,q,t ð jq,r Þ;

r

1μI,H,q γr Pr x2X I,H,q,x I,H,x,t



1 1μr I,H,q

ð3:66Þ

:

ð3:67Þ

In turn, the goods I rH,q,x,t ð jq,r Þ of each type x 2 X purchased by a household jq,r for investment are aggregated over the Regions of production as follows: r

I rH,q,x,t ð jq,r Þ ¼

X

1

μr q,r I,H,x I r γr  H,q,x,r ,t ð j Þ r 2R I,H,x,r

μr 1 I,H,x μr I,H,x

!μrμI,H,x1 I,H,x

:

ð3:68Þ

Here γ rI,H,x,r > 0 is the preference coefficient for the purchase of goods from Sector P x of Region r as nonfinancial assets ( r 2R γ rI,H,x,r ¼ 1); μrH,x denotes the elasticity of substitution for the purchase of goods from Sector x of different Regions as nonfinancial assets. Problem 6q Given t, q 2 Q, r 2 R, x 2 X, jq, r 2 Tr, the investments I rq,x,t ð jq,r Þ, and the prices Prx,r ,t , find the investments I rq,x,r ,t ð jq,r Þ in nonfinancial assets of goods from Sector x made in Regions r 2 R by minimizing the purchase cost 

 X   r 2R Prx,r ,t I rq,x,r ,t ð jq,r Þ

max

I rq,x,r ,t ð jq,r Þ

r  2R

subject to the following constraint derived from (3.68): r

I rH,q,x,t ð jq,r Þ 

X

1

μr q,r I,H,q,x I r γr  H,q,x,r ,t ð j Þ r  2R I,H,q,x,r

μr 1 I,H,q,x μr I,H,q,x

!μrμI,H,q,x1 I,H,q,x

:

ð3:69Þ

The first-order conditions determining the solution of the problem 6q under μrI,H,q,x < 1 and μrI,H,q,x 6¼ 0 yield the following optimal investment demands of a household jq,r for the goods of each Sector x (x 2 X) made in each Region r:

3.1 Conceptual Description of the Global Economy Within Model 2

 I rq,x,r ,t ð jq,r Þ ¼ γ rI,H,q,x,r

Prx,r ,t PrI,H,x,t

μrI,H,q,x

227

I rq,x,t ð jq,r Þ:

ð3:70Þ

The prices Prx,r ,t of these goods satisfy the relationship PrI,H,x,t ¼

X

r

1μI,H,q,x r γr P  r  2R I,H,q,x,r x,r ,t



1 1μr I,H,q,x

:

ð3:71Þ

Each of the above investments I rH,q,x,r ,t ð jq,r Þ in nonfinancial assets by the household of each type consists of a continuum of different goods    I rH,q,x,r ,t jq,r , hx,r made by firms hx,r of Region r, which are defined by the integral CES functions: 



I rH,q,x,r ,t ð jq,r Þ ¼

1 nr 

 r1 ð λx

T

r

  I rH,q,x,r ,t jq,r , hx,r

 λrx 1  λrx

! rλrx dhx,r

λx 1



:

ð3:72Þ



Like in (3.47), here λrx denotes the elasticity of substitution for different goods from Sector x of Region r. Problem 7q Given t, 2Q, x 2 X, r 2 R, r 2 R, jq, r 2 Tr, the investments I rq,x,r ,t ð jq,r Þ,     and prices Prx,r ,t hx,r of different types of goods made by firms hr 2 T r of Region    r, find the investments I rq,x,r ,t jq,r , hx,r in the nonfinancial assets of these firms by minimizing the purchase cost  ð       max    Prx,r ,t hx,r I rq,x,r ,t jq,r , hx,r dhx,r I rq,x,r ,t ð jq,r , hx,r Þ Tr

ð3:73Þ

subject to the following constraint derived from (3.72): 

I rH,q,x,r ,t ð jq,r Þ 



1 nr 

 r1 ð λx

T

r

  I rH,q,x,r ,t jq,r , hx,r

 λrx 1  λrx



! rλrx dhx,r



λx 1

 0: 

ð3:74Þ

The first-order conditions of this problem under λrx < 1 and λrx 6¼ 0 yield the following optimal investment demands for the goods of corresponding firms (in fact, similar to formula (3.63)):

228

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

I rq,x,r ,t



q,r

j ,h

x,r 



1 ¼ r n

  !λrx Prx,r ,t hx,r I rq,x,r ,t ð jq,r Þ Prx,r ,t

ð3:75Þ

The prices Prx,r ,t are calculated using (3.63). The solution of the next problem gives the optimal deposits of a patient household jP,r in banks br of domestic Region r. For period t the aggregate deposits DrP,z,t ð jP,r Þ of a patient household in the domestic currency of Region z (z 2 Zr) consist of a continuum of the deposits DrP,z,t ð jP,r , br Þ in different second-level banks br 2 Sr: r

λr 1  λr1 ð  P,r r  D,z r 1 D,z r DP,z,t j , b λD,z dbr nr r T

  DrP,z,t jP,r ¼

!λrλD,z1 D,z

,

ð3:76Þ

where λrD,z is the elasticity of substitution for the deposits in two different banks. Problem 8P Given t, r 2 R, jP, r 2 Tr, z 2 Zr, the aggregate deposits DrP,z,t ð jP,r Þ, and the deposit rates RrP,z,t ðbr Þ in different second-level banks, find the household’s deposits in corresponding banks by maximizing the aggregate remuneration ð max DrP,z,t ð jP,r ,br Þ,br 2Sr

Tr



   RrP,z,t ðbr Þ  1 DrP,z,t jP,r , br dbr

ð3:77Þ

subject to the following constraint derived from (3.76): r

  DrP,z,t jP,r þ

λr 1  λr1 ð  P,r r  D,z r 1 D,z r DP,z,t j , b λD,z dbr nr Tr

!λrλD,z1 D,z

 0:

ð3:78Þ

The first-order conditions of this problem under λrD,z > 1 yield the following optimal deposit demands for the corresponding banks and aggregate deposit yield RrP,z,t of the household (in fact, similar to formulas (3.62) and (3.63)):   r   1 RrP,z,t ðbr Þ  1 λD,z r  P,r  DP,z,t j , DrP,z,t jP,r , br ¼ r RrP,z,t  1 n  RrP,z,t ¼

1 nr

ð Tr



RrP,z,t ðbr Þ  1

1λrD,z

 r 1 dbr

λ

D,z

1

þ 1:

ð3:79Þ ð3:80Þ

The solution of the next problem considered gives the optimal loans of an impatient household jS,r in banks br of domestic Region r. For period t the aggregate  S,r  r of an impatient household jS, r in the domestic currency of Region loans BS,z,t j

3.1 Conceptual Description of the Global Economy Within Model 2

229

  z (z 2 Zr) consist of a continuum of the loans BrS,z,t jS,r , br in different second-level banks br 2 Sr: r

λr 1  λr1 ð  S,r r  B,z r 1 B,z r BS,z,t j , b λB,z dbr nr r T

  BrS,z,t jS,r ¼

!λrλB,z1 B,z

:

ð3:81Þ

Here λrB,z is the elasticity of substitution for the loans in two different banks.   Problem 9S Given t, r 2 R, jS, r 2 Tr, z 2 Zr, the aggregate loans BrP,z,t jS,r , and the loan rates RrS,z,t ðbr Þ in different second-level banks, find the household’s loans in corresponding banks by minimizing the aggregate remuneration of the banks  ð   r     RS,z,t ðbr Þ  1 BrS,z,t jS,r , br dbr maх BrS,z,t ð jS,r , br Þ, br 2Sr Tr

ð3:82Þ

subject to the following constraint derived from (3.81): r

BrS,z,t



λr 1  λr1 ð  S,r r  B,z r 1 B,z r BS,z,t j , b λB,z dbr nr Tr

 jS,r 

!λrλB,z1 B,z

 0:

ð3:83Þ

The first-order conditions of this problem under λrB,z < 1 and λrB,z 6¼ 0 yield the following optimal loan demands in corresponding banks and aggregate loan yield RrS,z,t of the banks (in fact, similar to formulas (3.79) and (3.80)): BrS,z,t



S,r

j ,b 

RrS,z,t

3.1.3

¼

r



1 nr

 r λrB,z r   1 RS,z,t ðb Þ  1 ¼ r BrS,z,t jS,r , RrS,z,t  1 n

ð Tr



RrS,z,t ðbr Þ

1

1λrB,z

ð3:84Þ

 db

r

1 1λr B,z

þ 1:

ð3:85Þ

Conceptual Description of Firm’s Behavior

In each Region r, there exists a continuum of firms of each producing sector x 2 X denoted by hx,r 2 Tr. The behavior of a firm hx,r is determined by the first-order conditions of the optimization problems 1x–11x presented below. The statement of each problem is preceded by associated constraints.

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

230

Each agent hx,r supplies corresponding products using its capital K rx,t ðhx,r Þ, hired labor Lrx,t ðhx,r Þ, leased land Orx,t ðhx,r Þ, and purchased intermediate products V rx,t ðhx,r Þ. The Cobb–Douglas production function of an agent hx,r has the form Y rx,t ðhx,r Þ ¼ εrx,t K rx,t ðhx,r Þαx,K Lrx,t ðhx,r Þαx,L Orx,t ðhx,r Þαx,O V rx,t ðhx,r Þαx,V : r

r

r

r

ð3:86Þ

The notations are the following: Y rx,t ðhx,r Þ as product output; αrx,K > 0 as the elasticity of production in the corresponding factor (αrx,K þ αrx,L þ αrx,O þ αrx,V ¼ 1); and εrx,t as the productivity shock defined by εrx,t ¼ εrx 1ρx εrx,t ρx eηx,t , r

r

r

ð3:87Þ

where εrx is an equilibrium value of this shock, ρrx gives an autoregression coefficient, and ηrx,t represents a Gaussian white noise. During period t a firm hx,r possesses the capital K rx,t ðhx,r Þ , accumulating it in accordance with the law K rx,t ðhx,r Þ



¼ 1

δrx





K rx,t1 ðhx,r Þ

 r  εI,x,tþi I rx,t ðhx,r Þ þ 1  Ψx I rx,t ðhx,r Þ: ð3:88Þ I rx,t1 ðhx,r Þ

The notations are the following: I rx,t ðhx,r Þ as the firm’s investments in main capital during period t; δrx as the depreciation coefficient of main capital; Ψ x as the firm’s cost function—a convex and differentiable functio—that satisfies the condition Ψ x ð1Þ ¼ Ψ 0x ð1Þ ¼ 0; and εI,x,t as the investment shock defined by εrI,x,t ¼ εrI,x 1ρI,x εrI,x,t1 ρI,x eηI,x,t , r

r

r

ð3:89Þ

where εrI,x is an equilibrium value of this shock, ρrI,x gives an autoregression coefficient, and ηrI,x,t represents a Gaussian white noise for the purchase of nonfinancial assets by Sector х. For the purchase of capital and financing of other expenditures, a firm hx,r takes out aggregate net loans (loans minus the deposits Brx,z,t ðhx,r Þ, z 2 Zr) in the secondlevel banks of domestic Region with effective loan rates RrS,z,t . Firms have the following constraint on such loans in second-level banks: the aggregate debt of a given firm must be supported by its capital in accordance with the inequality Et

X

     r r r x,r r r r r x,r S B ð h ÞR S,z,t  mx,t 1  δx Et QK,tþ1 K x,t ðh Þ : z2Z r z,tþ1 x,z,t

In this relationship, mrx,t denotes the debt–equity capital ratio

ð3:90Þ

3.1 Conceptual Description of the Global Economy Within Model 2

mrx,t

¼

mrx

231

 r rrm,x !1ρrm,x BG,t r r mrx,t1 ρm,x eηm,x,t , BrG

ð3:91Þ

where mrx is an equilibrium value of this ratio (0 < mrx  1 ); ρrm,x gives an autoregression coefficient; ηrm,x,t represents a Gaussian white noise of this ratio; and finally, QrK,tþ1 is capital price. Depending on Sector and Region, a firm hx,r is assumed to set the product prices r Px,t ðhx,r Þ in accordance with one of the two scenarios below. 1. The prices Prx,t ðhx,r Þ are equal to the world prices of these products taking into account an exchange rate Srus,t and an export duty τrex,x,t for them in Region r, i.e.,   1 þ τrex,x,t Prx,t ðhx,r Þ ¼ Srus,t Pglobal : x,t

ð3:92Þ

(Note that τrex,x,t ¼ τrex,x,r ,t for all r 6¼ r.) The world price Pglobal of product x in USD x,t (in particular, oil prices) is defined by the autoregression Pglobal ¼ Pglobal x,t x

1ρglobal P,x

Pglobal x,t1

ρglobal P,x ηglobal P,x,t

e

,

ð3:93Þ

where Pglobal is an equilibrium value of this price, ρglobal gives an autoregression P,x x global coefficient, and ηP,x,t represents a Gaussian white noise. World prices are quoted in USD. 2. The prices Prx,t ðhx,r Þ are Calvo inflexible, i.e., a firm hx,r of Sector x sets an optimal   erx,t ðhr Þ with a probability 1  ξrx , where ξrx 2 ½0, 1 and x 2 X; in the other price P cases,  r it indexes the  price subject to market price using some factor r γ x γ x 2 ½0, 1, x 2 X . The relationships defining the Calvo inflexible prices are written in the form

Prx,t ðhx,r Þ

¼

ϱrx,t ðhx,r ÞεrP,x,t

 r γrx   r x,r Px,t1 ex,t ðh Þ: ð3:94Þ Prx,t1 ðhx,r Þ þ 1  ϱrx,t ðhx,r Þ P Prx,t2

Here εrP,x,t denotes the indexed price shock (the markup shock); ϱrq,x,t ð jq,r Þ is a random price variation signal, i.e., an independent discrete random variable  that takes value 1 with the probability ξrx and value 0 with the probability 1  ξrx . For period t the profit of a firm hx,r is the difference between its sales income r Y x,t ðhx,r Þ at the price Prx,t ðhx,r Þ and the following expenditures: the wage W rx,t Lrx,t ðhx,r Þ taking into account an efficient rate τrSC,t of social tax; the lease payments for the land Orx,t ðhx,r Þ at the price PrO,t ; the purchase of investment and intermediate products at

232

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

the prices PrI,x,t and PrV,x,t , respectively; and the loan service payments. That is, the profit has the form Pr rx,t ðhx,r Þ ¼ Y rx,t ðhx,r ÞPrx,t ðhx,r Þ    1 þ τrSC,t W rx,t Lrx,t ðhx,r Þ  PrO,t Orx,t ðhx,r Þ  PrI,x,t I rx,t ðhx,r Þ  PrV,x,t V rx,t ðhx,r Þ X    Sr Br ðhx,r Þ Rrx,z,t1  1 þ Qrx,t K rx,t ðhx,r Þ  Qrx,t1 K rx,t1 ðhx,r Þ, z2Z r z,t x,z,t1

ð3:95Þ where  Y rx,t ðhx,r Þ

¼

Prx,t ðhx,r Þ Prx,t

λrx,t Y rx,t

ð3:96Þ

and Y rx,t specifies the aggregate demand for the products of Sector x. The firm’s equity capital NW rb,t ðhx,r Þ is the difference between the capital value at the price Qrx,t and the aggregate loans: NW rx,t ðhx,r Þ ¼ Qrx,t K rx,t ðhx,r Þ 

X

Sr Br ðhx,r Þ: z2Z r z,t x,z,t

ð3:97Þ

The equity capital dynamics of a firm hx,r is determined by its profit, corporate income tax (CIT) with a rate τrKx,t , and dividends (Divrx,t ðhx,r Þ) paid to the owners: NW rx,t ðhx,r Þ ¼ NW rx,t1 ðhx,r Þ þ Pr rx,t ðhx,r Þ  τrK,x,t Pr rb,t ðhx,r Þ  Divrx,t ðhx,r Þ: ð3:98Þ In accordance with formulas (3.95), (3.97), and (3.98), the dividends paid by a firm hx,r for period t are written as      Divrx,t ðhx,r Þ ¼ 1  τrK,x,t Y rx,t ðhx,r ÞPrx,t ðhx,r Þ  1 þ τrSC,t 1  τrK,x,t W rx,t Lrx,t ðhx,r Þ        1  τrK,x,t PrO,t Orx,t ðhx,r Þ  1  τrK,x,t PrI,x,t I rx,t ðhx,r Þ  1  τrK,x,t PrV,x,t V rx,t ðhx,r Þ X X

   S r Br ðhx,r Þ 1  τrK,x,t Rrx,z,t1 þ τrK,x,t : þ z2Z r Srz,t Brx,z,t ðhx,r Þ  z2Z r z,t x,z,t1

ð3:99Þ For each period t, a firm hx,r solves the main optimization problem 1х, which is to maximize the discounted sum of the expected dividends Divrx,tþi ðhx,r Þ, i ¼ 0, 1, . . ., by calculating the optimal values of the following variables: the product prices (in the case of Calvo inflexible prices; see scenario 2 above); the aggregate debt to the second-level banks (loans); the accumulated capital; investments; hired labor and leased land; and finally, the consumption of intermediate products.

3.1 Conceptual Description of the Global Economy Within Model 2

233

Problem 1x Given t, x, r, and hx,r, find the values of the variables C x,tþi ðhx,r Þj1 i¼0 , where C 1,x,tþi ðhx,r Þ ¼ fBrx,r,tþi ðhx,r Þ, Brx,us,tþi ðhx,r Þ, K rx,tþi ðhx,r Þ, I rx,tþi ðhx,r Þ, Lrx,tþi ðhx,r Þ, Orx,tþi ðhx,r Þ, V rx,tþi ðhx,r Þg for Sector x with flexible product prices and ( C 2,x,tþi ðh Þ ¼ x,r

)

erx,tþi ðhx,r Þ, Brx,r,tþi ðhx,r Þ, Brx,us,tþi ðhx,r Þ, Prx,tþi ðhx,r Þ, P K rx,tþi ðhx,r Þ, I rx,tþi ðhx,r Þ, Lrx,tþi ðhx,r Þ, Orx,tþi ðhx,r Þ, V rx,tþi ðhx,r Þ

,

C x,tþi ðhx,r Þ ¼ C1,x,tþi ðhx,r Þ _ C 2,x,tþi ðhx,r Þ for Sector x with inflexible product prices, that maximize the firm’s utility function (the discounted sum of expected dividends)  XT        x,r 1 r i r x,r V rx,t Cx,tþi ðhx,r Þ1 , Z ð h Þ E β Div ð h Þ ¼ liminf x,tþi t i¼0 x,tþi i¼0 i¼0 x T!1

ð3:100Þ subject to the constraints derived from relationships (3.86), (3.88), and (3.90) for the Sectors with flexible product prices and from relationships (3.86), (3.88), (3.90), and (3.94) for the other Sectors. In formula (3.100), Z x,tþi ðhx,r Þj1 i¼0 is the collection of all uncontrolled variables of this optimization problem that consists of the abovementioned variables not included in the list of Cx,tþi ðhx,r Þj1 i¼0 . Write the constraints of the problem 1x as inequalities. From (3.86) it follows that gr1,x,tþi ðhx,r Þ ¼ Y rx,tþi ðhx,r Þ  εrx,tþi K rx,tþi ðhx,r Þαx,K Lrx,tþi ðhx,r Þαx,L Orx,tþi ðhx,r Þαx,O V rx,tþi ðhx,r Þαx,V  0: r

r

r

r

ð3:101Þ Form (3.88) it follows that   gr2,x,tþi ðhx,r Þ ¼ K rx,tþi ðhx,r Þ  1  δrx K rx,tþi1 ðhx,r Þ   r  εI,x,tþi I rx,tþi ðhx,r Þ  1  Ψx I rx,tþi ðhx,r Þ  0: I rx,tþi1 ðhx,r Þ From (3.90) it follows that

ð3:102Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

234

gr3,x,tþi ðhx,r Þ ¼

X

 r  r E S Bx,z,tþi ðhx,r ÞRrx,z,tþi tþi z,tþiþ1 z2Z      mrx,tþi 1  δrx K rx,tþi ðhx,r ÞEtþi QrK,tþiþ1  0: r

ð3:103Þ

In the case of inflexible product prices, from (3.94) it follows that  gr4,x,tþi ðhx,r Þ ¼  Prx,tþi ðhx,r Þ þ ϱrx,tþi ð jq,r ÞεrP,x,tþi

Prx,tþi1 Prx,tþi2

γrx

Prx,tþi1 ðhx,r Þ

  r ex,tþi ðhx,r Þ  0: þ 1  ϱrx,tþi ðhx,r Þ P

ð3:104Þ Like the problem 1q above, the problem 1x will be solved using Theorem 3.1 (the infinite-dimensional Kuhn–Tucker theorem). To this effect, verify the assumptions of this theorem for the problem 1x. Condition (i) This condition determines the type of the utility function V and its arguments using relationships (3.100), (3.99), and (3.96). Condition (ii) For the problem 1x, the first part of this condition (U is concave) means that the function Divrx,t ðhx,r Þ defined by (6.92) and (6.89) is concave in the       variables Prx,t ðhx,r Þ, Brx,r,t ðhx,r Þ, Brx,us,t ðhx,r Þ , I rx,t hL,r , LrP,x,t hL,r , LrS,x,t hL,r ,       Orx,t hL,r , V rx,t hL,r and Prx,t1 ðhx,r Þ , Brx,us,t1 ðhx,r Þ , Brx,r,t1 ðhx,r Þ, I rx,t1 hL,r ,         LrP,x,t1 hL,r , LrS,x,t1 hL,r , Orx,t1 hL,r , V rx,t1 hL,r for all admissible values of the variables Zx, t( jq, r) and chosen parameter values. Find all nonzero second partial derivatives of the function Divrx,t ðhx,r Þ with respect to the listed variables: 2

   r  r λr  r x,r λr 1 ∂ Divrx,t ðhx,r Þ r r r  r x,r 2 ¼ λx 1  λx 1  τK,x,t Y x,t Px,t x Px,t ðh Þ x : ∂Px,t ðh Þ

ð3:105Þ

This derivative is nonpositive, and hence the function Divrx,t ðhx,r Þ is concave under 0  λrx  1. The second part of Condition (ii) (each element of g(Ct, Ct1, Zt) is convex in Ct and Ct1 for each Zt and all integers t  0) means the convexity of each of the functions gr1,x,tþi, gr2,x,tþi, gr3,x,tþi, and gr4,x,tþi in the corresponding variables Cx,t+i(hx, r). 1. Check the convexity of the function gr1,x,t ðhx,r Þ defined by (3.101). Because 2    λr  λr 2 ∂ Y rx,t ðhx,r Þ ¼ λrx 1 þ λrx Y rx,t Prx,t x Prx,t ðhx,r Þ x  0 for λrx 2 ð1, 0 [ ½1, 1Þ, x,r 2 r ∂P ð h Þ ð x,t Þ then the convexity of gr1,x,t ðhx,r Þ on this union follows from the concavity of its subtrahend (the Cobb–Douglas production function) under the assumption that its exponentials of power are positive and sum to 1.

3.1 Conceptual Description of the Global Economy Within Model 2

235

2. The convexity of the function gr2,x,t ðhx,r Þ defined by (3.102) in the variables K rx,t ðhx,r Þ, K rx,t1 ðhx,r Þ, I rx,t ðhx,r Þ, and I rx,t1 ðhx,r Þ is proved in the same way as the convexity of the function gr2,q,t ð jq,r Þ that participates in the constraint of the problem 1q (of course, with appropriate modification of notations). In the case Ψ x(x) ¼ x2, the function gr2,x,t ðhx,r Þ is convex under the condition

h  2 i  2 I rx,t1 ðhx,r Þ < 2 εrI,x,t I rx,t ðhx,r Þ I rx,t ðhx,r Þ þ 1 1 þ 2 I rx,t1 ðhx,r Þ :

ð3:106Þ

3. The convexity of the other functions gr3,x,t ðhx,r Þ (3.103) and gr4,x,t ðhx,r Þ (3.104) follows from their linearity in the variables Brx,r,t ðhx,r Þ, Brx,us,t ðhx,r Þ, K rx,t ðhx,r Þ and erx,t ðhx,r Þ, respectively. Prx,t ðhx,r Þ, Prx,t1 ðhx,r Þ, P Conditions (iii), (iv), and (v) For the problem 1x, these conditions are verified in the same way as for the problem 1q; see above. Conditions (vi) Write these first-order conditions of form (3.21) for the problem 1x with i ¼ 0. 1. Differentiation with respect to Lrx,t ðhx,r Þ yields 

  1  τrK,x,t 1 þ τrSC,t W rx,t Lrx,t ðhx,r Þ ¼ MC rx,t ðhx,r Þαrx,L Y rx,t ðhx,r Þ,

ð3:107Þ

where MC rx,t ðhx,r Þ is the Lagrange multiplier corresponding to constraint (3.101). 2. Differentiation with respect to V rx,t ðhx,r Þ yields 

 1  τrK,x,t PrV,x,t V rx,t ðhx,r Þ ¼ MC rx,t ðhx,r Þαrx,V Y rx,t ðhx,r Þ:

ð3:108Þ

3. Differentiation with respect to Orx,t ðhx,r Þ yields 

   1  τrK,x,t PrO,x,t Orx,t ðhx,r Þ ¼ MC rx,t hL,r αrx,O Y rx,t ðhx,r Þ:

ð3:109Þ

4. Differentiation with respect to K rx,t ðhx,r Þ together with gr1,x,t ðhx,r Þ ¼ 0 yields     MC rx,t ðhx,r Þαrx,K Y rx,t ðhx,r Þ  QrK,x,t ðhx,r Þ þ βrx 1  δrx Et QrK,x,tþ1 ðhx,r Þ K rx,tþi ðhx,r Þ     þμrx,t ðhx,r Þmrx,t 1  δrx Et QrK,tþ1 ¼ 0,

ð3:110Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

236

where QrK,x,t ðhx,r Þ and μrx,t ðhx,r Þ are the Lagrange multipliers corresponding to constraints (3.102) and (3.103), respectively. 5. Differentiation with respect to I rx,t ðhx,r Þ yields 

 1  τrK,t PrI,x,t ¼ QrK,x,t ðhx,r Þ  r r x,r   

εI,x,t I x,t ðh Þ I rx,t ðhx,r Þ 0 εrI,x,t I rx,t ðhx,r Þ r  1  Ψx  εI,x,t r Ψ I rx,t1 ðhx,r Þ I x,t1 ðhx,r Þ x I rx,t1 ðhx,r Þ " r   # I x,tþ1 ðhx,r Þ 2 0 εrI,x,tþ1 I rx,tþ1 ðhx,r Þ r r þ βx E t εI,x,tþ1 Ψx : I rx,t ðhx,r Þ I rx,t ðhx,r Þ ð3:111Þ

6. Differentiation with respect to Brx,r,t ðhx,r Þ yields     Et 1  βrx 1  τrK,x,tþ1 Rrx,z,t  βrx τrK,x,tþ1 ¼ μrx,t ðhx,r ÞRrx,r,t :

ð3:112Þ

7. Differentiation with respect to Brx,us,tþi ðhx,r Þ yields      Srus,tþi  βrx 1  Etþi τrK,x,tþiþ1 Srus,tþi Rrx,us,t  βrx Srus,tþi Etþi τrK,x,tþiþ1   ¼ μrx,tþi ðhx,r ÞRrx,us,tþi Etþi Srus,tþiþ1 :

ð3:113Þ

8. In the case of inflexible prices, differentiation with respect to Prx,t ðhx,r Þ yields 

    λr  λr 1  λrx 1  τrK,t Y rx,t Prx,t x Prx,t ðhx,r Þ x λr    λr  ¼ MC rx,t ðhx,r Þ 1  λrx Y rx,t Prx,t x Prx,t ðhx,r Þ x  r γrx   P  M rx,t ðhx,r Þ þ βrx r x,t Et M rx,tþ1 ðhx,r Þϱrx,tþ1 ðhx,r Þ , Px,t1

ð3:114Þ

where M rx,t ð jq,r Þ is the Lagrange multiplier corresponding to condition (3.104). r

ex,tþi ðhx,r Þ yields 9. In the case of inflexible prices, differentiation with respect to P

3.1 Conceptual Description of the Global Economy Within Model 2

237

  MC rx,t ðhx,r Þ 1  ϱrx,t ðhx,r Þ ¼ 0:

ð3:115Þ

Transversality (Condition vii) holds under the assumption that the expectations b tþi are bounded in i (see the solution of the problem 1q). Et C Therefore, all conditions of Theorem 3.1 are true for the problem 1x, and the firstorder conditions above define its solution. Solving the next triplet of nested optimization problems, a firm hx,r determines the optimal consumptions of each intermediate product x 2 X (the problem 2x) made in   each Region r 2 R (the problem 3x) by a firm hx ,r (the problem 4x). The statements and solutions of these problems are similar to those of the problems 2q, 3q, and 4q considered above. The intermediate consumption of a firm hx,r consists of the aggregate goods  (x 2 X) of different firms, Sectors, and Regions in accordance with the CES functions described below. For period t a firm hx,r purchases goods of each type V rx,x ,t ðhx,r Þ for intermediate consumptive use in form of the aggregate goods V rx,t ðhx,r Þ in accordance with the CES function r

X

V rx,t ðhx,r Þ ¼

γr  x 2X V,x,x

1 μr V,x

μr 1 V ,x μr V ,x

!μrμV,x1 V,x

V rx,x ,t ðhx,r Þ

Here γ rV,x,x are positive preference coefficients ( elasticities of substitution.

P

r x 2X γ V,x,x

:

ð3:116Þ

¼ 1); μrV,x denote the

Problem 2x Given t, x 2 X, r 2 R, hx,r 2 Tr, the intermediate consumptions V rx,t ðhx,r Þ , and the prices PrV,x ,t of the intermediate goods of Sectors x, find the consumptions V rx,x ,t ðhx,r Þ by minimizing the purchase cost max

fV rx,x ,t ðhx,r Þgx 2X

 X   x 2X PrV,x ,t V rx,x ,t ðhx,r Þ

subject to the following constraint derived from (3.116): r

V rx,t ðhx,r Þ 

X

γr  x 2X V,x,x

1 μr V,x

V rx,x ,t ðhx,r Þ

μr 1 V,x μr V,x

!μrμV,x1 V,x

 0:

ð3:117Þ

The first-order conditions determining the solution of the problem 2x under μrV,x < 1 and μrV,x 6¼ 0 yield the following optimal consumer demands of a firm hx,r for the intermediate goods of each Sector x (x 2 X) and prices PrV,x,x ,t of these goods:

238

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

 r μrV,x PV,x ,t V rx,t ðhx,r Þ, PrV,t

V rx,x ,t ðhx,r Þ ¼ γ rV,x,x PrV,x,t ¼

X

r

γ r  Pr  1μV,x x 2X V,x,x V,x,x ,t



1 1μr V,x

ð3:118Þ

:

ð3:119Þ

For period t the intermediate consumption V rx,x ,t ðhx,r Þ of each type of goods is aggregated over all Regions r 2 R with their consumptions V rx,x ,r ,t ðhx,r Þ, i.e., μr

V rx,x ,t ðhx,r Þ ¼

X r 2R

γ rV,x,x ,r

1 μr V ,x,x

μr 1 V ,x,x μr V,x,x

!μr V,x,x1 V ,x,x

V rx,x ,r ,t ðhx,r Þ

where γ rV,x,x ,r denote positive preference coefficients ( μrV,x,x are the elasticities of substitution for these goods.

,

P

r r 2R γ V,x,x ,r

ð3:120Þ ¼ 1 ) and

Problem 3x Given t, x 2 X, x 2 X, r 2 R, hx,r 2 Tr, the consumptions V rx,x ,t ðhx,r Þ of intermediate goods x, and the prices Prx ,r ,t of these goods made in Regions r 2 R, find the consumptions V rx,x ,r ,t ðhx,r Þ of these intermediate goods by minimizing the purchase cost max

fV rx,x ,r ,t ðhx,r Þgr 2R

 X   r 2R Prx ,r ,t V rx,x ,r ,t ðhx,r Þ

subject to the following constraint derived from (3.120): μr

V rx,x ,t ðhx,r Þ 

X

γr   r  2R V,x,x ,r

1 μr V,x,x

V rx,x ,r ,t ðhx,r Þ

μr 1 V,x,x μr V,x,x

!μr V,x,x1 V,x,x

 0:

ð3:121Þ

The first-order conditions determining the solution of the problem 3x under μrV,x,x < 1 and μrV,x,x 6¼ 0 yield the following optimal demands of a firm hx,r for intermediate goods x made in each Region r and prices Prx ,r ,t of these goods: V rx,x ,r ,t ðhx,r Þ ¼ γ rV,x,x ,r

 r  r Px ,r ,t μV ,x,x r V x,x ,t ðhx,r Þ: PrV,x ,t

ð3:122Þ

The prices PrV,x ,r ,t of these goods satisfy the relationship PrV,x ,t ¼

X

r

r 2R

γ rV,x,x ,r Prx ,r ,t 1μV,x,x



1 1μr V,x,x

:

ð3:123Þ

3.1 Conceptual Description of the Global Economy Within Model 2

239 



x ,r The goods for the intermediate consumption of all firms ~h from each Sector x  in each Region r are aggregated in the form



1 nr 

V rx,x ,r ,t ðhx,r Þ ¼

 ð 1  λrx

Tr

   r x,r ~x ,r V , h  ,r  ,t h x,x 

 λrx 1  λrx

r

d~h

x ,r

! rλx

λx 1

,

ð3:124Þ

  x ,r where V rx,x ,r ,t hx,r , ~h denote the quantities of goods x purchased during period x t from a firm ~ h substitution.



,r



of Region r for consumptive use and λrx is the elasticity of

Problem 4x Given t, x 2 X, x 2 X, r 2 R, r 2 R,hx,r 2 Tr, the intermediate x ,r h of different products made consumptions V rx,x ,r ,t ðhx,r Þ, and the prices Prx ,r ,t ~   x ,r x ,r   2 T r in Region r , find the quantities V rx,x ,r ,t hx,r , ~h by firms ~ h of intermediate products from these firms by minimizing the purchase cost  ð   x ,r     x ,r r r x,r ex ,r e e  P h , h h V d h , max       x ,r x ,r ,t x,x ,r ,t  x ,r  Tr  V rx,x ,r ,t hx,r ,e h , e h 2T r ð3:125Þ subject to the following constraint derived from (3.124):

V rx,x ,r ,t ðhx,r Þ 



1 nr 

 r1 ð λx

T

r

r





V rx,x ,r ,t hx,r , ~h

λrx r1 x ,r λx

d ~h

x ,r

! rλx

λx 1

 0:

ð3:126Þ

The first-order conditions determining the solution of the problem 4x under  < 1 and λrx 6¼ 0 yield the following optimal demands of a firm hx,r for interme x ,r  x ,r diate goods x made by firms ~h and prices Pr   ~h of these goods:  λrx

x ,r ,t



V rx,x ,r ,t hx,r , ~h



x ,r





 x ,r 1λrx r ~ P   x ,r ,t h 1 A V r   ðhx,r Þ, ¼ r @ x,x ,r ,t Prx ,r ,t n

 Prx ,r ,t

¼

0

1 nr 

ð T

r

Prx,r ,t 



~hx



 r ,r  1λx



x d~h



,r 

 r1

λx 1

:

ð3:127Þ

ð3:128Þ

Solving the next triplet of nested optimization problems, a firm hx,r determines the optimal consumptions of each investment product x 2 X (the problem 5x) made in

240

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . . 



x ,r each Region r 2 R (the problem 6x) by a firm ~h (the problem 7x). The statements and solutions of these optimization problems are similar to those of the problems 2x, 3x, and 4x (firms) and of the problems 5q, 6q, and 7q (households). The investments of a firm hx,r consist of goods x 2 X of different firms, Sectors, and Regions with sequential aggregation in accordance with the CES functions below. During period t a firm hx,r purchases investment goods I rx,x ,t ðhx,r Þ in form of the aggregate goods I rx,t ðhx,r Þ in accordance with the CES function r

X

I rx,t ðhx,r Þ ¼

γr  x 2X I,x,x

1 μr I,x

I rx,x ,t ðhx,r Þ

where γ rI,x,x ,r denote positive preference coefficients ( the elasticities of substitution.

μr 1 I,x μr I,x

!μrμI,x1 I,x

,

ð3:129Þ

r x 2X γ I,x,x

¼ 1) and μrI,x are

P

Problem 5x Given t, x 2 X, r 2 R, hx,r 2 Tr, the investments I rx,t ðhx,r Þ, and the prices PrI,x ,t of the investment goods of Sectors x, find the investments I rx,x ,t ðhx,r Þ by minimizing the purchase cost max

fI rx,x ,t ðhx,r Þgx 2X

 X   x 2X PrI,x ,t I rx,x ,t ðhx,r Þ

subject to the following constraint derived from (3.129): r

I rx,t ðhx,r Þ 

X

γr  x 2X I,x,x

1 μr I,x

I rx,x ,t ðhx,r Þ

μr 1 I,x μr I,x

!μrμI,x1 I,x

 0:

ð3:130Þ

The first-order conditions determining the solution of the problem 5x under μrI,x < 1 and μrI,x 6¼ 0 yield the following optimal demands of a firm hx,r for the investment goods of each Sector x (x 2 X) and prices PrI,x,x ,t of these goods:  I rx,x ,t ðhx,r Þ PrI,x,t ¼

¼

γ rI,x,x

X

PrI,x ,t PrI,t

μrI,x

I rx,t ðhx,r Þ, r

x 2X

γ rI,x,x PrI,x,x ,t 1μI,x



1 1μr I,x

:

ð3:131Þ ð3:132Þ

During period t the investment goods I rx,x ,t ðhx,r Þ of each type are the aggregation of such goods I rx,x ,r ,t ðhx,r Þ over all Regions r 2 R:

3.1 Conceptual Description of the Global Economy Within Model 2

241 μr

I rx,x ,t ðhx,r Þ ¼

X

1

μr γ r   I,x,x I rx,x ,r ,t ðhx,r Þ r  2R I,x,x ,r

μr  1 I,x,x μr  I,x,x

where γ rI,x,x ,r denote positive preference coefficients ( are the elasticities of substitution for these goods.

P

!μr I,x,x1 I,x,x

,

ð3:133Þ

r r 2R γ I,x,x ,r

¼ 1) and μrI,x,x

Problem 6x Given t, x 2 X, x 2 X, r 2 R, hx, r 2 Tr, the investments I rx,x ,t ðhx,r Þ of intermediate goods x, and the prices Prx ,r ,t of these goods made in Regions r 2 R, find the investments I rx,x ,r ,t ðhx,r Þ of intermediate goods x made in Regions r by minimizing the purchase cost max

fI rx,x ,r ,t ðhx,r Þgr 2R

 X   r 2R Prx ,r ,t I rx,x ,r ,t ðhx,r Þ

subject to the following constraint derived from (3.133): μr

X

I rx,x ,t ðhx,r Þ 

1

γr   r 2R I,x,x ,r

μr  I,x,x

I rx,x ,r ,t ðhx,r Þ

μr  1 I,x,x μr  I,x,x

!μr I,x,x1 I,x,x

 0:

ð3:134Þ

The first-order conditions determining the solution of the problem 6x under μrI,x,x < 1 and μrI,x,x 6¼ 0 yield the following optimal demands of a firm hx,r for the investment goods x made in each Region r and prices Prx ,r ,t of these goods:  I rx,x ,r ,t ðhx,r Þ ¼ γ rI,x,x ,r

Prx ,r ,t PrI,x ,t

μr

I,x,x

I rx,x ,t ðhx,r Þ:

ð3:135Þ

The prices Prx ,r ,t of these goods satisfy the relationship PrI,x ,t ¼

X

r

γ r   Pr   1μI,x,x r 2R I,x,x ,r x ,r ,t

x The investment goods of all firms ~h aggregated in the form



,r



1 1μr  I,x,x

:

ð3:136Þ

from each Sector x in each Region r are r

I rx,x ,r ,t ðhx,r Þ ¼



1 nr 

 r1 ð



λx

T

r

I rx,x ,r ,t hx,r , ~h



λrx x ,r 

1  λrx

d~h

x ,r

! rλx

λx 1

,

ð3:137Þ

  x ,r where I rx,x ,r ,t hx,r , ~h denote the quantities of goods x purchased during period 





t from a firm hx ,r of Region r for investment and λrx is the elasticity of substitution.

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

242

Problem 7x Given t, x 2 X, x 2 X, r 2 R, r 2 R, hx, r 2 Tr, the investments   x ,r I rx,x ,r ,t ðhx,r Þ, and prices Prx ,r ,t ~h of different types of goods made by firms   x ,r x ,r  ~h 2 T r of Region r, find the quantities of investment goods I rx,x ,r ,t hx,r , ~h of the corresponding firms by minimizing the purchase cost 

max  x ,r 

I rx,x ,r ,t hx,r , ~ h

x , ~h

 ,r

2Sr



 ð  Tr

  x ,r       r r x,r ~x ,r ~ ~hx ,r P h , h h I d ð3:138Þ     x ,r ,t x,x ,r ,t 

subject to the following constraint derived from (3.137):

I rx,x ,r ,t ðhx,r Þ 



1 nr 

r



 r1 ð λx

Tr

λr 1    x  x ,r  λrx r x,r ~x ,r I   h ,h d ~h  x,x ,r ,t

! rλx

λx 1

 0:

ð3:139Þ

The first-order conditions determining the solution of the problem 7x under  < 1 and λrx 6¼ 0 yield the following optimal demands of a firm hx,r for the  x ,r  x ,r investment goods x of firms ~h and prices Pr   ~h of these goods:  λrx

x ,r ,t



I rx,x ,r ,t hx,r , ~h



x ,r





 x ,r 1λrx r ~ P   x ,r ,t h 1 A I r   ðhx,r Þ, ¼ r @ x,x ,r ,t Prx ,r ,t n

 Prx ,r ,t

¼

0

1 nr 

ð T

r

Prx,r ,t 



~hx



 r ,r  1λx



x d~h



,r 

 r1

λx 1

ð3:140Þ

:

ð3:141Þ

Solving the next pair of nested optimization problems, a firm hx,r determines the optimal labor hired from each type q of households in domestic Region (the problem 8x) and also the optimal labor hired from each household jq,r (the problem 9x). The statements and solutions of these optimization problems are similar to the ones considered above. For period t the aggregate labor Lrx,t ðhx,r Þ hired by a firm hx, r is written as r

Lrx,t ðhx,r Þ ¼

1 r

γ rx,P,L μL,x Lrx,P,t ðhx,r Þ

μr 1 L,x μr L,x

1 r

þ γ rx,S,L μL,x Lrx,S,t ðhx,r Þ

μr 1 L,x μr L,x

!μrμL,x1 L,x

:

ð3:142Þ

Here γ rx,P,L and γ rx,S,L are positive preference coefficients (γ rx,P,L þ γ rx,S,L ¼ 1 ); μrL,x denotes the elasticity of substitution for the labor of different households; finally, Lrx,P,t ðhx,r Þ and Lrx,S,t ðhx,r Þ specify the labors hired from a corresponding type of households (P and S).

3.1 Conceptual Description of the Global Economy Within Model 2

243

Problem 8x Given t, x 2 X, r 2 R, hx,r 2 Tr, the hired labors Lrx,t ðhx,r Þ, and the wages W rx,P,t and W rx,S,t for each type of households, find the hired labors Lrx,P,t ðhx,r Þ and Lrx,S,t ðhx,r Þ by minimizing the labor cost max

fLrx,q,t ðhx,r Þgq2Q

 X   q2Q W rx,q,t Lrx,q,t ðhx,r Þ

subject to the following constraint derived from (3.142): r

Lrx,t ðhx,r Þ 

γ rx,P,L

1 μr L,x

Lrx,P,t ðhx,r Þ

μr 1 L,x μr L,x

þ γ rx,S,L

1 μr L,x

Lrx,S,t ðhx,r Þ

μr 1 L,x μr L,x

!μrμL,x1 L,x

 0:

ð3:143Þ

The first-order conditions determining the solution of the problem 8q under μrL,x < 1 and μrL,x 6¼ 0 yield the following optimal demands for the labor of each type of households q (q 2 Q) and corresponding wages: Lrx,q,t ðhx,r Þ

γ rx,q,L

¼

 r μrL,x W x,q,t Lrx,t ðhx,r Þ, W rx,t

ð3:144Þ

1  r r  r W rx,t ¼ γ rx,P,L W rx,P,t 1μL,x þ γ rx,S,L W rx,S,t 1μL,x 1μL,x :

ð3:145Þ

For period t the aggregate labor Lrx,q,t ðhx,r Þ hired from each type q of households in Region r consists of the labors Lrq,x,t ðhx,r , jq,r Þ hired from each household jq,r in accordance with the CES function  λr 1 W,q nr 1

Lrx,q,t ðhx,r Þ ¼

λr

ð Tr

Lrq,x,t ðhx,r , jq,r Þ

λr 1 W,q λr W,q

!λr W,q1 djq,r

W,q

,

ð3:146Þ

where λrW,q denotes the elasticity of substitution for the labor of different households. Problem 9x Given t, 2X, r 2 R, hx, r 2 Tr, q 2 Q, the hired labor Lrx,q,t ðhx,r Þ, and the wages W rq,x,t ð jq,r Þ set by households jq,r, find the hired labors Lrq,x,t ðhx,r , jq,r Þ of these households by minimizing the labor cost max x,r q,r

Lrq,x,t ðh , j Þ, jq,r 2T

 ð  r q,r r x,r q,r q,r  W ð j ÞL ð h , j Þdj q,x,t q,x,t r Tr

subject to the following constraint derived from (3.146):

ð3:147Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

244

r

λ 1 W,q  λr1 ð 1 W,q r x,r q,r λrW,q Lq,x,t ðh , j Þ djq,r nr Tr r

Lrx,q,t ðhx,r Þ 

!λrλW,q1 W,q

 0:

ð3:148Þ

The first-order conditions determining the solution of the problem 9x under λrW,q < 1 and λrW,q 6¼ 0 yield the following optimal demands for the labor of each type of households q (q 2 Q) and corresponding wages W rq,x,t : r  r q,r λW,q 1 W q,x,t ð j Þ Lrq,x,t ðhx,r Þ, nr W rq,x,t

Lrq,x,t ðhx,r , jq,r Þ ¼  W rq,x,t ¼

1 nr

ð

r

Sr

ð3:149Þ

 r

1 λ 1 W,q

W rq,x,t ð jq,r Þ1λW,q djq,r

:

ð3:150Þ

Solving the next problem 10х, a firm hx,r determines the optimal land leased from two types of households and State of domestic Region. The statement and solution of this optimization problem are similar to those of the problem 8х. For period t a firm hx,r leases land from the patient (OrP,x,t ðhx,r Þ) and impatient r (OS,x,t ðhx,r Þ) households and from State (OrG,x,t ðhx,r ÞÞ, which make up its aggregate land lease Orx,t ðhx,r Þ, i.e., r

Orx,t ðhx,r Þ ¼

γ rx,P,O

1 μr O,x

Orx,P,t ðhx,r Þ

μr 1 O,x μr O,x

þ γ rx,S,O

1 μr O,x

Orx,S,t ðhx,r Þ

μr 1 O,x μr O,x

þ γ rx,G,L

1 μr O,x

Orx,G,t ðhx,r Þ

μr 1 O,x μr O,x

!μrμO,x1 O,x

ð3:151Þ Here μrO,x denotes the corresponding elasticity of substitution; γ rx,P,O , γ rx,S,O , and γ rx,G,L are positive preference coefficients; γ rx,P,O þ γ rx,S,O þ γ rx,G,L ¼ 1. Problem 10x Given t, x 2 X, r 2 R, hx, r 2 Tr, the leased land Orx,t ðhx,r Þ, and the lease prices (rates) PrO,P,t , PrO,S,t , and PrO,G,t set by two types of households and State, respectively, find the land leases Orx,P,t ðhx,r Þ, Orx,S,t ðhx,r Þ, and Orx,G,t ðhx,r Þ by minimizing the lease cost f

max

 g

Orx,P,t ðhx,r Þ, Orx,S,t ðhx,r Þ, Orx,G,t ðhx,r Þ

PrO,P,t Orx,P,t ðhx,r Þ  PrO,S,t Orx,S,t ðhx,r Þ  PrO,G,t Orx,G,t ðhx,r Þ

subject to the following constraint derived from (3.151):



3.1 Conceptual Description of the Global Economy Within Model 2 r

0 Orx,t ðhx,r Þ

245

Bγ rx,P,O B B @

1 μr O,x

Orx,P,t ðhx,r Þ

μr 1 O,x μr O,x

þγ rx,G,L

1 μr O,x

þ γ rx,S,O

1 μr O,x

Orx,G,t ðhx,r Þ

Orx,S,t ðhx,r Þ

μr 1 O,x μr O,x

μr 1 O,x μr O,x

1μrμO,x1 O,x

C C C A

 0: ð3:152Þ

The first-order conditions determining the solution of the problem 10x under μrO,x < 1 and μrO,x 6¼ 0 yield the following optimal demands for leased land from each type of owners and corresponding lease rates:  r μrO,x PO,P,t ¼ Orx,t ðhx,r Þ, PrO,t  r μrO,x P Orx,S,t ðhx,r Þ ¼ γ rx,S,O O,S,t Orx,t ðhx,r Þ, PrO,t  r μrO,x P r x,r r Ox,G,t ðh Þ ¼ γ x,G,O O,G,t Orx,t ðhx,r Þ, PrO,t Orx,P,t ðhx,r Þ

γ rx,P,O

ð3:153Þ ð3:154Þ ð3:155Þ

1  r r r  r PrO,t ¼ γ rx,P,O PrO,P,t 1μO,x þ γ rx,S,O PrO,S,t 1μO,x þ γ rx,G,O PrO,G,t 1μO,x 1μO,x :

ð3:156Þ

These conditions were simulated under the same lease rates for all land owners: PrO,t ¼ PrO,P,t ¼ PrO,S,t ¼ PrO,G,t . Finally, solving the problem 11х, a firm hx,r determines the optimal loans in each second-level bank of domestic Region. The statement and solution of this optimization problem are similar to those of the problem 9S. The aggregate loan Brx,z,t ðhx,r Þ of a firm hx,r in currency z consists of a continuum of loans Brx,z,t ðhx,r , br Þ in different second-level banks br of domestic Region: r

Brx,z,t ðhx,r Þ ¼

λr 1  λr1 ð B,z 1 B,z r x,r r λrB,z Bx,z,t ðh , b Þ dbr r n Tr

!λrλB,z1 B,z

,

ð3:157Þ

where λrB,z is the elasticity of substitution for the loans in different banks. Problem 11x Given t, 2X, r 2 R, hx, r 2 Tr, z 2 Zr, the loans Brx,z,t ðhx,r Þ, and the loan rates Rrx,z,t ðbr Þ of different second-level banks in Region r, find the loans of a firm hx,r in the corresponding banks by minimizing their aggregate remuneration

246

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

max

Brx,z,t ðhx,r , br Þ,

br 2S

 ð   r  r r x,r r r  Rx,z,t ðb Þ  1 Bx,z,t ðh , b Þdb r Tr

subject to the following constraint derived from (3.157): r

Brx,z,t ðhx,r Þ 

λr 1  λr1 ð B,z 1 B,z r x,r r λrB,z B ð h , b Þ dbr x,z,t nr Tr

!λrλB,z1 B,z

 0:

ð3:158Þ

The first-order conditions determining the solution of the problem 11x under λrB,z < 1 and λrB,z 6¼ 0 yield the following optimal demands for loans in corresponding banks and aggregate loan rate Rrx,z,t (in fact, they are similar to formulas (3.79) and (3.80)): Brx,z,t ðhx,r , br Þ ¼  Rrx,z,t ¼

3.1.4

1 nr

ð Tr

 r λrB,z r 1 Rx,z,t ðb Þ  1 Brx,z,t ðhx,r Þ: Rrx,z,t  1 nr



Rrx,z,t ðbr Þ  1

1λrB,z

ð3:159Þ

 dbr

1 1λr B,z

þ 1:

ð3:160Þ

Conceptual Description of Second-Level Bank’s Behavior

The Banking Sector of Region r is represented by a continuum of second-level banks br 2 Tr. During period t, each bank br accepts aggregate deposits DrP,z,t ðbr Þ from the patient households of domestic Region in domestic currency and USD with deposit rates RrP,z,t ðbr Þ, z 2 Z r , as well as provides aggregate loans BrS,z,t ðbr Þ in domestic currency and USD to impatient households with loan rates RrS,z,t ðbr Þ. Next, a bank br takes up net loans (loans minus deposits) IDrz,t ðbr Þ, z 2 Z, at aggregate rates RrID,z,t denominated in USD (and also in domestic currency in case of borrowing in Region r) from other second-level banks. Each bank br also takes up net loans DrCB,z,t ðbr Þ at a rate RrCB,z,t from the Central Bank of domestic Region. If IDrz,t ðbr Þ, DrCB,z,t ðbr Þ < 0, then these values are considered as deposits. In addition, a bank br has public bonds of domestic Region denominated in domestic currency and public bonds of other Regions denominated in USD, with aggregate exogenous volumes BrG,b,z,t ðbr Þ and rates RrG,b,z,t . They satisfy the following relationships:

3.1 Conceptual Description of the Global Economy Within Model 2

X r BrG,b,z,t ðbr Þ ¼ Br  ðb Þ; r 2R G,B,z,r t X r r Rr RrG,b,z,t BrG,b,z,t ðbr Þ ¼  B  ðb Þ; r 2R G,b,z,r ,t G,b,z,r ,t BrG,b,r,r ,t ðbr Þ ¼ 0 ðпри r 6¼ r  и r 6¼ us, rowÞ: 

247

ð3:161Þ ð3:162Þ ð3:163Þ



Here BrG,b,z,r ,t ðbr Þ ¼ BrG,B,r,z,t and RrG,b,z,r ,t ¼ RrG,z,t are the volumes and rates of the public bonds of Region r that belong to a bank br. For accepting deposits from the households of domestic Region as well as taking up loans from Central Bank and other banks, a bank br must secure them by the volume of provided loans: Et

X

X

Sr z2Z r z,tþ1

Sr z2Z r z,tþ1



 r  DP,z,t ðbr ÞRrP,z,t ðbr Þ þ θrD,G RrCB,z,t DrCB,z,t ðbr Þ þ θrID RrID,z,t IDrz,t ðbr Þ  mrB,t Et

BrS,z,t ðbr ÞRrS,z,t ðbr Þ þ

X

Br ðbr ÞRrx,z,t ðbr Þ x2X x,z,t

 þ BrG,b,z,t ðbr ÞRrG,b,z,t :

ð3:164Þ Here θrID and θrD,G are special risk parameters; Srz,tþ1 denotes the exchange rate of currencies r to z; mrB,t is the bank’s borrowing-loan ratio defined by mrB,t

¼

mrB

 r rrm,x !1ρrm,B BG,t r r mrB,t1 ρm,B eηm,B,t , BrG

ð3:165Þ

where mrB gives an equilibrium value of this ratio (0 < mrB < 1 ); ρrm,B is an autoregression coefficient; BrG,t specifies the public debt of Region r; BrG is an equilibrium value of the public debt of Region r; r rm,x > 0 denotes the public debt influence parameter; and finally, ηrm,B,t represents a Gaussian white noise. For period t the profit of a bank br is the difference between the payments on loans and deposits: Pr rb,t ðbr Þ ¼

X

Sr z2Z r z,t



 X

 BrS,z,t1 ðbr Þ RrS,z,t1 ðbr Þ  1 þ Br ðbr Þ Rrx,z,t1 ðbr Þ  1 x2X x,z,t1





 þBrG,b,z,t1 ðbr Þ RrG,b,z,t1  1  DrP,z,t1 ðbr Þ RrP,z,t1 ðbr Þ  1  DrCB,z,t1 ðbr Þ RrCB,z,t1  1

 IDrz,t1 ðbr Þ RrID,z,t1  1 Þ

ð3:166Þ In this formula,

248

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

 r  r RP,z,t ðbr Þ  1 λD,P,z r DP,z,t ; RrP,z,t  1

ð3:167Þ

 r  r RS,z,t ðbr Þ  1 λB,S,z r ¼ BS,z,t ; RrS,z,t  1

ð3:168Þ

DrP,z,t ðbr Þ ¼ BrS,z,t ðbr Þ

 Brx,z,t ðbr Þ

¼

Rrx,z,t ðbr Þ  1 Rrx,z,t  1

λrB,x,z Brx,z,t ,

ð3:169Þ

where DrP,z,t , RrP,z,t , BrS,z,t , RrS,z,t , Brx,z,t , and Rrx,z,t are the corresponding average values of these indexes over the Banking Sector of Region r. The bank’s equity capital NW rb,t ðbr Þ is the difference between the provided loans and borrowed finances: NW rb,t ðbr Þ ¼

X

Sr z2Z r z,t

! P BrS,z,t ðbr Þ þ x2X Brx,z,t ðbr Þ þ BrG,b,z,t ðbr Þ : DrP,z,t ðbr Þ  DrCB,z,t ðbr Þ  IDrz,t ðbr Þ

ð3:170Þ

The equity capital dynamics of a firm br is determined by its profit, corporate income tax (CIT) with a rate τrK,b,t , and dividends (Divrb,t ðbr Þ) paid to the owners: NW rb,t ðbr Þ ¼ NW rb,t1 ðbr Þ þ Prrb,t ðbr Þ  τrK,b,t Pr rb,t ðbr Þ  Divrb,t ðbr Þ:

ð3:171Þ

In accordance with formulas (3.166), (3.170), and (3.171), the dividends paid by a bank br for period t can be written as

   1 DrP,z,t ðbr Þ  DrP,z,t1 ðbr Þ RrP,z,t1 ðbr Þ 1  τrK,b,t þ τrK,b,t C B

r    C B þDr r r r r r C B CB,z,t ðb Þ  DCB,z,t1 ðb Þ RCB,z,t1 1  τ K,b,t þ τK,b,t C B C B

   r r r r r r r C B X C B þIDz,t ðb Þ  IDz,t1 ðb Þ RID,z,t1 1  τK,b,t þ þτK,b,t r r r B C: Divb,t ðb Þ ¼ S

   C z2Z r z,t B C B BrS,z,t ðbr Þ þ BrS,z,t1 ðbr Þ RrS,z,t1 ðbr Þ 1  τrK,b,t þ τrK,b,t C B B P 

r    C C B r r r r r r r B  x2X Bx,z,t ðb Þ  Bx,z,t1 ðb Þ Rx,z,t1 ðb Þ 1  τK,b,t þ τK,b,t C A @

r    r r r r r r BG,b,z,t ðb Þ þ BG,b,z,t1 ðb Þ RG,b,z,t1 1  τK,b,t þ τK,b,t 0

ð3:172Þ For further use in the bank’s utility function, introduce the concept of “reduced” dividends with variables relating to one period t only and a discounting factor βrb , 0 < βrb < 1:

3.1 Conceptual Description of the Global Economy Within Model 2

249

9 8 r  

 DP,z,t ðbr Þ 1  τrK,b,tþ1 βrb  βrb 1  τrK,b,tþ1 RrP,z,t ðbr Þ > > > > > > > > > >

   > > > > r r r r r r r > > þDCB,z,t ðb Þ 1  τK,b,tþ1 βb  βb 1  τK,b,tþ1 RCB,z,t > > > > > > > > > >

   > > r r r r r r r > > = < þID R ð b Þ 1  τ β  β 1  τ X b K,b,tþ1 b K,b,tþ1 z,t ID,z,t r r f b,t ðbr Þ ¼ Et : S Div r z,t  

 z2Z > > r r r r r r r r > > R B ð b Þ 1  τ β  β 1  τ ð b Þ > > b K,b,tþ1 b K,b,tþ1 S,z,t S,z,t > > > > > > >   r

 > P  r > > r r r r r r > > >  x2X Bx,z,t ðb Þ 1  τK,b,tþ1 βb  βb 1  τK,b,tþ1 Rx,z,t ðb Þ > > > > > > > > >

   > > ; : r r r r r r r BG,b,z,t ðb Þ 1  τK,b,tþ1 βb  βb 1  τK,b,tþ1 RG,b,z,t

ð3:173Þ For period t a bank br determines: • • • • •

Deposit rates for the patient households of domestic Region (RrP,z,t ð br Þ) Loan rates for the impatient households of domestic Region (RrS,z,t ð br Þ) Loan rates for the firms of different Sectors of domestic Region (Rrx,z,t ðbr Þ) Net debts to the banks of all Regions (IDrz,t ðbr Þ Net debts to the Central Bank of domestic Region (DrCB,z,t ðbr Þ)

in the following way. Solving theoptimization problem 1.1b, a bank br determines the optimal values r r r r r r r r r of the tools Rr ð b Þ, R ð b Þ, R ð b Þ, D ð b Þ, ID ð b Þ . If there exist P,z,t S,z,t x,z,t CB,z,t z,t r recommended upper bounds RP,z,t on the deposit rates of patient households such r r r that Rr P,z,t ð b Þ  RP,z,t for z 2 Z, a bank b uses the optimal collection of its tools. The r upper bounds RP,z,t recommended by Central Bank or a deposit security fund for higher stability of the banking system and economy’s de-dollarization (if necessary) have the form   r RP,r,t ¼ K rb,t RrG,r,t  1 þ 1:

ð3:174Þ

Here RrG,r,t is the Central Bank’s rate on the attracted loans of second-level banks in domestic currency; K rb,t denotes the margin coefficient, 0 < K rb,t  1. In addition,     r RP,us,t ¼ f r Dolrt K rb,t RrG,us,t  1 þ 1,

ð3:175Þ

where RrG,us,t is the Central Bank’s rate on the attracted loans of second-level banks in   USD; Dolrt ¼ Srus,t DrP,us,t = Srus,t DrP,us,t þ DrP,r,t denotes the dollarization parameter in Region r, i.e., the share of USD deposits in the aggregate deposits of all households in this Region; f Dolrt is a decreasing de-dollarization function such that f(x) ¼ 1 if x  Dolr and f(1) ¼ 0; and finally, Dolr specifies an equilibrium (minimal admissible) value of dollarization.

250

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . . r

r

r r r r If Rr P,z,t ð b Þ > RP,z,t , a bank b adjusts RP,z,t ð b Þ ¼ RP,z,t , calculating the other r r r r r r r r tools {DP,z,t ðb Þ, RS,z,t ð b Þ, Rx,z,t ðb Þ, DCB,z,t ðb Þ, IDrz,t ðbr Þ } from the optimization problem 1.2b. Consider the statements of first-order conditions of these problems.

Problem 1.1b Given t, r, and br 2 Tr, find the values of the variables Cb,tþi ðbr Þj1 i¼0, where  Cb,tþi ð br Þ ¼ RrP,z,tþi ð br Þ, RrS,z,tþi ð br Þ, Rrx,z,tþi ðbr Þ, DrCB,z,tþi ðbr Þ, IDrz,tþi ðbr Þ x2X,z2Z , that maximize the discounted sum of dividends  XT        r 1 r ifr r β ð b Þ ð3:176Þ V rb,t C b,tþi ðbr Þ1 Div b,tþi i¼0 , Z b,tþi ðb Þ i¼0 ¼ liminf Et b i¼0 T!1

subject to the following constraint derived from (3.164): grb,tþi ðbr Þ

X

DrP,z,tþi ðbr ÞRrP,z,tþi ðbr Þ ¼ r r þθD,G DCB,z,tþi ðbr ÞRrCB,z,tþi þ θrID IDrz,tþi ðbr ÞRrID,z,tþi 0 1 r r r r BS,z,tþi ðb ÞRS,z,tþi ðb Þ X C  r B X B C  0:  mrB,tþi  r Etþi Sz,tþiþ1 @ z2Z þ x2X Brx,z,tþi ðbr ÞRrx,z,tþi ðbr ÞA þBrG,b,z,tþi ðbr ÞRrG,b,z,tþi 

E Sr z2Z r tþi z,tþiþ1



!

ð3:177Þ In formula (3.177), Z b,tþi ðbr Þj1 i¼0 is the collection of all uncontrolled variables of this optimization problem that consists of the above-mentioned variables not included in the list of C b,tþi ð br Þj1 i¼0 . Relationships (3.176) and (3.177) involve equalities (3.167), (3.168) and (3.169). Like the problems 1q and 1x above, the problem 1.1b will be solved using Theorem 3.1. First, verify all conditions of this theorem for the problem 1.1b. Condition (i) This condition determines the type of the utility function V and its arguments using relationships (3.164), (3.161), and (3.155), (3.156) and (3.157). Condition (ii) For the problem 1.1b, the first part of this condition (U is concave) f rb,t ðbr Þ is concave in the variables RrP,z,t ð br Þ, RrS,z,t ð br Þ, means that the function Div Rrx,z,t ðbr Þ, DrG,z,t ðbr Þ, IDrz,t ðbr Þ, RrP,z,t1 ðbr Þ, RrS,z,t1 ðbr Þ, Rrx,z,t1 ðbr Þ, DrG,z,t1 ðbr Þ, and IDrz,t1 ðbr Þ for all admissible values of the variables Zb,S,t+i(br) and chosen parameter f rb,t ðbr Þ with values. Find all nonzero second partial derivatives of the function Div respect to the listed variables:

3.1 Conceptual Description of the Global Economy Within Model 2 r



 λr  λr 2 ¼ Srz,t DrP,z,t RrP,z,t  1 D,P,z RrP,z,t ðbr Þ  1 D,P,z λrD,P,z λrD,P,z 1  Et τrK,b,tþ1

e b,t ðbr Þ ∂ Div 2

1.

ð

251

Þ     βrb  βrb 1  Et τrK,b,tþ1 RrP,z,t ðbr Þ þ 1  Et τrK,b,tþ1 1  2βrb þ RrP,z,t ðbr Þβrb g. 2 ∂RrP,z,t ðbr Þ

r 2 e q,r    ∂ Div b,q,t ðb Þ 0 Obviously, 1  Et τrK,b,tþ1 1  2βrb þ RrP,z,t ðbr Þβrb > 0 . Hence, q,r 2 r ð∂RP,z,t ðb ÞÞ

ð1Et τr Þ½12βrb þRrP,z,t ðbr Þβrb  for λrD,P,z 2  1E τr K,b,tþ1 ,0 if 1  Et τrK,b,tþ1 βrb  r ½ t K,b,tþ1 βb βrb ð1Et τrK,b,tþ1 ÞRrP,z,t ðbr Þ   or for λrD,P,z 2 ð1, 0  βrb 1  Et τrK,b,tþ1 RrP,z,t ðbr Þ > 0  S ð1Et τr Þ½12βrb þRrP,z,t ðbr Þβrb  , þ1 if 1  Et τrK,b,tþ1 βrb   1E τr K,b,tþ1 r ½ t K,b,tþ1 βb βrb ð1Et τrK,b,tþ1 ÞRrP,z,t ðbr Þ   . In the case 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 RrP,z,t ðbr Þ < 0 r 2 e q,r   ∂ Div b,q,t ðb Þ βrb 1  Et τrK,b,tþ1 RrP,z,t ðbr Þ ¼ 0, the inequality  0 holds for λrD,P,z  0. q,r 2 r ð∂RP,z,t ðb ÞÞ r



 λ r  λr 2 ¼ Srz,t BrS,z,t RrS,z,t  1 B,S,z RrS,z,t ðbr Þ  1 B,S,z λrB,S,z λrB,S,z 1  Et τrK,b,tþ1

e b,t ðbr Þ ∂ Div 2

2.

ð∂RrS,z,t ðbr ÞÞ

2

    βrb  βrb 1  Et τrK,b,tþ1 RrS,z,t ðbr Þ þ 1  Et τrK,b,tþ1 1  2βrb þRrS,z,t ðbr Þβrb g.    Like in the previous case, 1  Et τrK,b,tþ1  1  2βrb þ RrP,z,t ðbr Þβrb > 0 . So the

r

e b,t ðbr Þ ∂ Div 2

inequality

ð∂RrS,z,t ðbr ÞÞ

2

 0 holds for λrB,S,z 2

1, 

ð1Et τrK,b,tþ1 Þ½12βrb þRrS,z,t ðbr Þβrb  ½1Et τrK,b,tþ1 βrb βrb ð1Et τrK,b,tþ1 ÞRrS,z,t ðbr Þ

  [½0, þ1Þ if 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 RrS,z,t ðbr Þ > 0 , or for λrB,S,z 2

  ð1Et τrK,b,tþ1 Þ½12βrb þRrS,z,t ðbr Þβrb  0,  1E τr if 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 ½ t K,b,tþ1 βrb βrb ð1Et τrK,b,tþ1 ÞRrS,z,t ðbr Þ   RrS,z,t ðbr Þ < 0 . At the same time, if 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 r 2 e r ∂ Div b,t ðb Þ RrS,z,t ðbr Þ ¼ 0, then  0 for λrB,S,z  0. r 2 r ð∂RS,z,t ðb ÞÞ r



  λr   ¼ Srz,t Brx,z,t Rrx,z,t  1 B,S,z Rrx,z,t ðbr Þ  1 λrB,x,z λrB,x,z 1  Et τrK,b,tþ1 βrb ð  Þ     βrb 1  Et τrK,b,tþ1 Rrx,z,t ðbr Þ þ 1  Et τrK,b,tþ1 1  2βrb þ Rrx,z,t ðbr Þβrb g.    Again, like in the previous case, 1  Et τrK,b,tþ1 1  2βrb þ Rrx,z,t ðbr Þβrb > 0. As a r 2 e r ∂ Div b,t ðb Þ is true for λrB,x,z 2 result, the inequality 2  0 ∂Rrx,z,t ðbr ÞÞ ð 

ð1Et τr Þ½12βrb þRrx,z,t ðbr Þβrb  1,  1E τr K,b,tþ1 [ ½0, þ1Þ if 1  Et τrK,b,tþ1 βrb  ½ t K,b,tþ1 βrb βrb ð1Et τrK,b,tþ1 ÞRrx,z,t ðbr Þ

  r ð1Et τrK,b,tþ1 Þ½12βrb þRrx,z,t ðbr Þβrb  r r r r βb 1  Et τK,b,tþ1 Rx,z,t ðb Þ > 0 or for λB,x,z 2 0,  ½1E τr βr βr ð1E τr ÞRr ðbr Þ if e b,t ðbr Þ ∂ Div 2

3.

2 ∂Rrx,z,t ðbr Þ

t K,b,tþ1 b

b

t K,b,tþ1

x,z,t

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

252

  1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 Rrx,z,t ðbr Þ < 0 . If 1  Et τrK,b,tþ1 βrb  r 2 e r   ∂ Divb,t ðb Þ r βrb 1  Et τrK,b,tþ1 Rrx,z,t ðbr Þ ¼ 0, then 2  0 for λB,x,z  0. ð∂Rrx,z,t ðbr ÞÞ The dividends function of a bank br has zero second partial derivatives with respect to the other variables DrG,z,t ðbr Þ, IDrz,t ðbr Þ and r r r r RrP,z,t1 ð b Þ, RrS,z,t1 ðb Þ, Rrx,z,t1 ðb Þ, DrG,z,t1 ðb Þ, and IDrz,t1 ðbr Þ. Thus, the concavf rb,q,t ðbq,r Þ are satisfied for the values λrD,P,z , λrB,S,z , ity requirements of the function Div and λrB,x,z from the corresponding intervals, yielding nonpositive values of the corresponding second partial derivatives. The second part of Condition (ii) (each element of g(Ct, Ct  1, Zt) is convex in Ct and Ct  1 for each Zt and all integers t  0) means the convexity of the function grb,t ðbr Þ defined by (3.177) in the above-mentioned variables. Taking into account (3.167), (3.168) and (3.169), check the convexity of this function in the variables             RrP,z,t bS,r , RrS,z,t bS,r , Rrx,z,t bS,r , DrG,z,t bS,r , IDrz,t bS,r , RrP,z,t1 bS,r ,         RrS,z,t1 bS,r , Rrx,z,t1 bS,r , DrG,z,t1 bS,r , and IDrz,t1 bS,r for all admissible values of the variables Zb, S, t + i(br) and chosen parameter values. Find all nonzero second partial derivatives of the function grb,t ðbr Þ with respect to the listed variables. 2    λ r n   λr 2 ∂ grb,t ðbr Þ ¼ Et Srz,tþ1 DrP,z,t RrP,z,t  1 D,P,z λrD,P,z λrD,P,z þ 1 RrP,z,t ðbr Þ  1 D,P,z 1. r 2 r ð∂RP,z,t ðb ÞÞ  λr 1  λr 1 RrP,z,t ðbr Þ  λrD,P,z RrP,z,t ðbr Þ  1 D,P,z  λrD,P,z RrP,z,t ðbr Þ  1 D,P,z g ¼ λrD,P,z  r λrD,P,z 2 r  r  r  r  r r r r RP,z,t ðb Þ  1 λD,P,z λD,P,z RP,z,t ðb Þ þ2  Et Sz,tþ1 DP,z,t RP,z,t  1 RrP,z,t ðbr Þg. 2

∂ gr ðbr Þ

b,t Here RrP,z,t ðbr Þ > 0 and 2  RrP,z,t ðbr Þ > 0. Hence, the inequality 2  0 ∂RrP,z,t ðbr ÞÞ ð  r i r 2R ðb Þ holds for λrD,P,z 2 1,  Rr P,z,tðbr Þ [ ½0, þ1Þ. P,z,t

2.

2 ∂ grb,t ðbr Þ 2 ∂RrS,z,t ðbr Þ





 λr 2 r RrS,z,t  1ÞλB,S,z RrS,z,t ðbr Þ  1 B,S,z λrB,S,z 

¼ mrB,t Et Srz,tþ1 BrS,z,t ð ð Þ λrB,S,z RrS,z,t ðbr Þ þ 2  RrS,z,t ðbr Þ . 2

Consequently, the inequality 3.

2 ∂ grb,t ðbr Þ 2 ∂Rrx,z,t ðbr Þ

ð

∂ grb,t ðbr Þ

ð∂RrS,z,t ðbr ÞÞ

  ¼ mrB,t Et Srz,tþ1

Þ λrB,x,z Rrx,z,t ðbr Þ þ 2  Rrx,z,t ðbr Þ ∂ grb,t ðbr Þ

ð

h 2Rr ðbr Þ i  0 is true for λrB,S,z 2  Rr S,z,tðbr Þ , 0 . S,z,t

  λr  λr 2 Brx,z,t Rrx,z,t  1 B,x,z Rrx,z,t ðbr Þ  1 B,x,z λrB,x,z 

2

As a result, the inequality

2

Þ

2 ∂Rrx,z,t ðbr Þ

h i 2Rr ðbr Þ  0 is the case for λrB,x,z 2  Rr x,z,tðbr Þ , 0 . x,z,t

3.1 Conceptual Description of the Global Economy Within Model 2

253

The constraint function grb,t ðbr Þ of a borrowing bank br has zero second partial derivatives with respect to the other variables DrG,z,t ðbr Þ, IDrz,t ðbr Þ and RrP,z,t1 ð br Þ, RrS,z,t1 ð br Þ, Rrx,z,t1 ðbr Þ, DrG,z,t1 ðbr Þ, and IDrz,t1 ðbr Þ. Thus, the concavity requirements of the function grb,q,t ðbq,r Þ are satisfied for the values λrD,P,z , λrB,S,z , and λrB,x,z from the corresponding intervals, yielding nonpositive values of the corresponding second partial derivatives. Conditions (iii), (iv), and (v) For the problem 1.1b, these conditions are verified in the same way as for the problem 1q; see above. Conditions (vi) Write these first-order conditions of form (3.21) for the problem 1.1b with i ¼ 0. 1. Differentiation with respect to RrP,z,t ð br Þ yields 

(

)   r r r r r r r λ 1  E τ β  β 1  E τ ð b Þ R t t K,b,tþ1 b  P,z,t K,b,tþ1 b  1  τrK,t Srz,t D,P,z þβrb 1  Et τrK,b,tþ1 RrP,z,t ðbr Þ  1   ¼ μrB,t Et Srz,tþ1 λrD,P,z RrP,z,t ðbr Þ  RrP,z,t ðbr Þ þ 1 , 

ð3:178Þ

where μrB,t is the Lagrange multiplier corresponding to constraint (3.177). 2. Differentiation with respect to RrS,z,t ð br Þ yields ( Srz,t

)   λrB,S,z 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 RrS,z,t ðbr Þ    þβrb 1  Et τrK,b,tþ1 RrS,z,t ðbr Þ  1   r  λB,S,z  1 RrS,z,t ðbr Þ þ 1 : ¼ μrB,t mrB,t Et Srz,tþ1

ð3:179Þ

3. Differentiation with respect to Rrx,z,t ðbr Þ yields ( Srz,t

 

) λrB,x,z 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 Rrx,z,t ðbr Þ    þβrb 1  Et τrK,b,tþ1 Rrx,z,t ðbr Þ  1    r ¼ μrB,t mrB,t Etþi Srz,tþ1 λB,x,z  1 Rrx,z,t ðbr Þ þ 1 :

ð3:180Þ

4. Differentiation with respect to DrCB,z,t ðbr Þ yields

   Srz,t 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 RrCB,z,t   ¼ μrB,t θrD,G Et Srz,tþ1 RrCB,z,t :

ð3:181Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

254

5. Differentiation with respect to IDrz,t ðbr Þ yields

   Srz,t 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 RrID,z,t   ¼ μrB,t θrID Et Srz,tþ1 RrID,z,t :

ð3:182Þ

Transversality (Condition vii) holds under the assumption that the expectations b tþi are bounded in i (see the solution of the problem 1q). Therefore, all conditions Et C of Theorem 3.1 are true for the problem 1.1b, and the first-order conditions above define its solution. r

Problem 1.2b Given t, r, br, and RP,z,t ð br Þ , find the values of the variables 1 C b,tþi ðbr Þi¼0 , where  Cb,tþi ð br Þ ¼ DrP,z,t ðbr Þ, RrS,z,tþi ð br Þ, Rrx,z,tþi ðbr Þ, DrCB,z,tþi ðbr Þ, IDrz,tþi ðbr Þ x2X,z2Z , that maximize the discounted sum of dividends (3.173) with the rate RrP,z,t ðbr Þ ¼ r RP,z,t (3.174), (3.175), i.e.,  XT        r 1 r ifr r Div , Z ð b Þ E β ð b Þ , V rb,t C b,tþi ðbr Þ1 ¼ liminf b,tþi t b,tþi i¼0 i¼0 i¼0 b T!1

ð3:183Þ subject to the following constraint derived from (3.164): !  r  DrP,z,tþi ðbr ÞRrP,z,t þ θrD,G DrCB,z,tþi ðbr ÞRrCB,z,tþi ¼ z2Z r Etþi Sz,tþiþ1 þθrID IDrz,tþi ðbr ÞRrID,z,tþi   P mrB,tþi  z2Z r Etþi Srz,tþiþ1   X r r r r r r B ð b ÞR ð b Þ þ B R BrS,z,tþi ðbr ÞRrS,z,tþi ðbr Þ þ x,z,tþi G,b,z,tþi G,b,z,tþi  0 x2X x,z,tþi

grb,tþi ðbr Þ

X

ð3:184Þ 1 In formula (3.184), Z b,tþi ðbr Þi¼0 is the collection of all uncontrolled variables of this optimization problem that 1 consists of the above-mentioned variables not included in the list of Cb,tþi ð br Þi¼0 . Relationships (3.183) and (3.184) involve equalities (3.168) and (3.169). Following the same approach, first check all conditions of Theorem 3.1 for the problem 1.2b. Condition (i) This condition determines the type of the utility function V and its arguments using relationships (3.183), (3.173), (3.168), and (3.169).

3.1 Conceptual Description of the Global Economy Within Model 2

255

Condition (ii) As easily seen, the parameter values λrD,P,z , λrB,S,z , and λrB,x,z from the corresponding intervals (see the verification procedure of Condition (ii) for the problem 1.1b) also guarantee condition (ii) for the problem 1.2b. Conditions (iii), (iv), and (v) For the problem 1.2b, these conditions are verified in the same way as for the problem 1q; see above. Conditions (vi) Write these first-order conditions of form (3.21) for the problem 1.2b with i ¼ 0. 1. Differentiation with respect to DrP,z,t ðbr Þ yields

  r    r Srz,t 1  Et τrK,b,tþ1 βrb  βrb 1  Et τrK,b,tþ1 RP,z,t ¼ μrB,t Et Srz,tþ1 RP,z,t :

ð3:185Þ

2–5. Differentiation with respect to RrS,z,t ð br Þ, Rrx,z,t ðbr Þ, DrCB,z,t ðbr Þ, and IDrz,t ðbr Þ yields expressions (3.179), (3.180), (3.181), and (3.182), respectively. Transversality (Condition vii) holds like for the problem 1q. Therefore, under the above hypotheses, all conditions of Theorem 3.1 are true for the problem 1.2b, and the first-order conditions define its solution. Solving the next pair of problems, a bank br determines the optimal net loans in  r USD in each Region r 2 R (the problem 2b) from each bank ~b 2 Sr (the problem 3b). Solving the problem 3b, a bank bralso determines the optimal net loans in domestic currency from each bank of domestic Region. Consider the statements and solutions of these problems, which are similar to the problems 3q and 4q solved by households. The aggregate loans collected by a borrowing bank br from all Regions are described in accordance with the following CES function (in the case z ¼ us): r

IDrz,t ðbr Þ ¼

X

γr  r 2R ID,z,r

1 μr ID,z

IDrz,r ,t ðbr Þ

μr 1 ID,z μr ID,z

!μrμID,z1 ID,z

:

ð3:186Þ

Here μrID,z denote the positive elasticities of substitution; γ rID,z,r are positive shares, P r r r r 2R γ ID,z,r ¼ 1. If IDz,t ðb Þ < 0, formula (3.186) involves the magnitudes of the corresponding variables. Problem 2b Given t, r 2 R, br 2 Sr, the net loans IDrz,t ðbr Þ , and the loan rates RrID,us,r ,t in each Region r, find the loans IDrus,r ,t ðbr Þ in all Regions by minimizing the loan service cost

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

256

 X    max  r 2R RrID,us,r ,t  1 IDrus,r ,t ðbr Þ fIDrus,r ,t gr 2R subject to the following constraint derived from (3.186): r

IDrz,t ðbr Þ 

X

1

r γ r  μID,z IDrz,r ,t ðbr Þ r  2R ID,z,r

μr 1 ID,us μr ID,us

!μrμID,us1 ID,us

 0:

ð3:187Þ

In the case z ¼ us, the first-order conditions determining the solution of the problem 2b under μrID,us < 1 and μrID,us 6¼ 0 yield the following optimal demands of a bank br for net loans in each Region r and aggregate loan rates:  IDrus,r ,t ðbr Þ

γ rID,us,r

¼

RrID,us,r ,t  1 RrID,us,t  1

μrID,us

IDrus,t ðbr Þ,

ð3:188Þ

where RrID,us,t ¼

X r  2R

 1μrID,us 1μr1 ID,us γ rID,us,r RrID,us,r ,t  1 þ 1:

ð3:189Þ

If z 6¼ us, z 2 Z, then net loans in domestic currency can be taken up in domestic Region only: IDrr,t ðbr Þ ¼ IDrr,r,t ðbr Þ; IDrr,r ,t ðbr Þ ¼ 0 for r 6¼ r. In the case IDrz,t ðbr Þ > 0, the aggregate loans of a borrowing bank br from other banks of Region r are calculated by the formula r



1 nr 

IDrz,r ,t ðbr Þ ¼

λr1 ð



ID,z

T

r

r IDrz,r ,t br , ~b

r



1 λID,z r λ

ID,z

d~b

r



!λrλID,z1 ID,z

:

ð3:190Þ

(For r 6¼ r, all loans are in USD only: z ¼ us.) Problem 3b Given t, r 2 R, z 2 Zr , br 2 Sr, r 2 R, the net loans IDrz,r ,t ðbr Þ > 0, and  r  r the loan rates RrID,z,r ,t ~b of different banks ~b in Region r, find the loans   r RrID,z,r ,t br , ~ b in each bank of this Region by minimizing the loan service cost 

 max  r

IDrz,r ,t br , ~ b

r

,~ b 2T r



 ð  Tr

   r     r r r r ~r ~ ~ R b , b b  1 ID d b ID,z,r ,t z,r ,t 

subject to the following constraint derived from (3.190):

ð3:191Þ

3.1 Conceptual Description of the Global Economy Within Model 2

0 IDrz,r ,t ðbr Þ  @

257 r



1 nr 

ð

 r1 λ

ID,z

Tr

 λr 1 ID,z  λr ID,z

  r r ~r ID  ,t b , b z,r 

1 rλID,z 

r d~b A

λ 1 ID,z

 0:

ð3:192Þ

(If r 6¼ r, then only the case z ¼ us is considered.) The first-order conditions determining the solution of the problem 3b under   λrID,z < 1 and λrID,z 6¼ 0 yield the following optimal demands of a bank br for net r loans in each bank b~ and aggregate loan rates: 

IDrz,r ,t br , ~b  RrID,z,r ,t ¼

3.1.5

  r  1λrID,z r ~ R b  1 ID,z,r ,t 1 A IDrz,r ,t ðbr Þ; ¼ r @ RrID,z,r ,t  1 n

0

 r

1 nr 

ð



Tr



 r 1   r  1λrID,z λ 1 r ID,z RrID,z,r ,t b~  1 d~b þ 1:

ð3:193Þ

ð3:194Þ

Conceptual Description of State’s Behavior

The State of Region r consists of Government, a continuum of public agents (who perform consumption and investments), and Central Bank. Government Under the hypotheses of Model 2, during period t, the Government of Region r serves state budget (budgetary revenues and expenditures) in accordance with a quarter plan as well as forms and serves public debt. For period t the state budget balance equation of Region r has the form GE rt ¼ GRrt þ GDSrt ,

ð3:195Þ

where GE rt and GRrt are government expenditures and revenues, respectively, and GDSrt indicates state budget surplus (GDSrt < 0) or gap (GDSrt < 0). In case of gap, the state budget is served by issuing public bonds in domestic currency and USD that circulate during period t and also by calling other financial sources (OGDSrt ): GDSrt ¼ BrG,t  BrG,t1 þ OGDSrt , X Sr B r : BrG,t ¼ z2Z r z,t G,z,t

ð3:196Þ ð3:197Þ

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

258

Here BrG,t denotes the aggregate public debt in domestic currency; BrG,z,t is the volume of public bonds in currency z that circulate during period t; finally, Srz,t gives the exchange rate of currencies z to r. In case of surplus and no corrections, the state budget is used for serving public debt in accordance with formula (6.184). For r 6¼ us, row, the public debt of Region r in domestic currency r during period t (BrG,r,t ) consists of the public debts to the Central Bank (BrG,CB,r,t ) and second-level banks (BrG,B,r,t ) of domestic Region: BrG,r,t ¼ BrG,B,r,t þ BrG,CB,r,t :

ð3:198Þ

The public debt of each Region r in USD during period t (BrG,us,t ) consists of the public debt to the Central Bank of domestic Region (BrG,CB,us,t ), the public debts to the second-level banks of all Regions (BrG,B,r ,us,t ), and other debts (BOrG,us,t ): BrG,us,t ¼ BrG,CB,us,t þ

X

Br  r  2R G,B,r ,us,t

þ BOrG,us,t :

ð3:199Þ

The public debts in USD to different second-level banks are described by the autoregressions BrG,B,r ,us,t ¼ BrG,B,r ,us 1ρG,B,r ,us BrG,B,r ,us,t1 ρG,B,r ,us eηG,B,r ,us,t , r

r

r

ð3:200Þ

where BrG,B,r ,us denotes an equilibrium value of the public debt to the second-level banks of Region r; ρrG,B,r ,us is an autoregression coefficient; and finally, ηrG,B,r ,us,t represents a Gaussian white noise. The government expenditures (GE rt) of Region r during period t are defined using the Taylor rule [58]: GE rt

¼

r GE rt1 ρGE

 r rrGE,B,G   r !1ρrGE BG,t GDPrt rGE,Y r GE eηGE,t , BrG GDPr r

ð3:201Þ

where GEr denotes an equilibrium value of government expenditures; ρrGE is an autoregression coefficient; r rGE,B,G and r rGE,Y indicate the elasticities of substitution for public debt and GDP; ηrGE,t represents a Gaussian white noise; BrG,t specifies the aggregate public debt; BrG is an equilibrium value of this debt; GDPrt gives the GDP of Region r (see (3.273)); and finally, GDPr is an equilibrium GDP value of Region r. The structure of government expenditures is represented by consumption (Grt with corresponding prices PrG,t ), investments (I rH,G,t with corresponding prices PrI,H,G,t ), public debt service in two currencies (BrG,r,t1 , BrG,us,t1 ), and other expenditures (GEOrt ):

3.1 Conceptual Description of the Global Economy Within Model 2

GE rt ¼ PrG,t Grt þ PrI,H,G,t I rH,G,t þ

X z2Z

 r

259

  Srz,t BrG,z,t1 RrG,z,t1  1 þ GEOrt : ð3:202Þ

In formula (3.202), RrG,z,t1 denotes the public bonds rate, which is equal to the deposit rate of patient households for z ¼ r and also to the loan rates of foreign banks for z 6¼ r; the government consumption (Grt ) and investments (I rH,G,t ) are defined as the corresponding shares of government expenditures, i.e., Grt ¼ I rH,G,t ¼

1 r ε GE r , PrG,t G,t t

ð3:203Þ

1 εr GE r , PrI,H,G,t GI,t t

ð3:204Þ

where εrG,t and εrGI,t represent shocks described by corresponding autoregressions. Within the framework of Model 2, the government revenues GRrt include net taxes and export and distributed (for the EAEU countries) import taxes as well as land lease revenues, transfers from National Fund and other revenues (GROrt ), i.e.,     GRrt ¼ τrC,t PrC,t C rP,t þ CrS,t þ τrI,t PrH,I,t I rH,P,t þ I rH,S,t r

X X X X

r  τex,x,r ,t r EX τim,x,r ,t IM rx,r ,t þ   6¼r x,r ,t þ r 6¼r x2X∖L 1 þ τr r x2X ex,x,r ,t    r  P r r r þ τSC,t þ τrW,t x,q W q,x,t Lq,x,t þ τW,t hX X i r r r S Div  ,t  ,t þ DivP,B,r  ,t r P,x,r  r 2R x2X X  τrK,t X r r þ Div þ Div r  P,x,r,t P,B,r,t r 2R x2X∖L 1  τK,t X r r þT rest,t þ P Or þ Tr rt þ GROrt : x2X∖L O,G,t x,G,t ð3:205Þ The notations are the following: τrC,t and τrI,t as the effective VAT rates for consumption and investment goods in Region r, respectively; CrP,t and C rS,t as the aggregate consumptions of patient and impatient households of Region r, respectively; I rH,P,t and I rH,S,t as the aggregate purchases of nonfinancial assets by the patient and impatient households of Region r, respectively; PrC,t and PrH,I,t as the prices of consumption and investment goods for the households of Region r; τrex,x,r ,t as the export tax rate for Product х from Region r to Region r; τrim,x,r ,t as the import tax rate for Product х from Region r to Region r; EX rx,r ,t as the exports of Product х from Region r to Region r defined by (3.250); IM rx,r ,t as the imports of Product х from Region r to Region r defined by (3.249); τrSC,t , τrW,t , and τrK,t as the effective rates of the social, individual, and corporate income taxes (IIT, CIT) in Region r;

260

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

W rq,x,t and Lrq,x,t as the aggregate wages and labor supplies for Sector х of Region r; DivrP,x,r ,t and DivrP,B,r ,t as the dividends collected by the patient households of Region r from possessing Production x and second-level banks in Region r;   DivrP,x,r,t and DivrP,B,r,t as the dividends collected by the patient households of Region r from possessing Production x and second-level banks in Region r; T rrest,t as other taxes; PrO,G,t Orx,G,t as the lease rate for state’s land; and finally, Tr rt as the transfers from National Fund. The effective rates of net taxes (duties) are defined using the corresponding expressions for the VAT (τrC,t ), social tax (τrSC,t ), IIT (τrW,t ), CIT (τrK,t ), export duties for goods (τrex,x,r ,t ), and import duties (τrim,x,r ,t ): τrC,t ¼ τrC 1ρτ,C τrC,t1 ρτ,C eητ,C,t ;

ð3:206Þ

τrSC,t ¼ τrSC 1ρτ,SC τrSC,t1 ρτ,SC eητ,SC,t ;

ð3:207Þ

r

r

r

τrW,t ¼ τrW

1ρrτ,W

τrK,t ¼ τrK

1ρrτ,K

τrim,x,r ,t ¼ τrim,x,r

1ρrτ,im,x,r

r

r

ρrτ,W

e

ηrτ,W,t

ρrτ,K

ηrτ,K,t

τrW,t1 τrK,t1

r

e

τrim,x,r ,t1

ð3:208Þ

;

ð3:209Þ

;

ρrτ,im,x,r

ηrτ,im,x,r ,t

;

ð3:210Þ

τrex,x,r ,t ¼ τrex,x,r 1ρτ,ex,x,r τrex,x,r ,t1 ρτ,ex,x,r eητ,ex,x,r ,t :

ð3:211Þ

r

r

e

r

Here τrC , τrSC , τrW , τrK , τrim,x,r , and τrex,x,r denote the equilibrium values of the corresponding tax rates and duties (τrC , τrSC , τrW , τrK 2 ð0, 1Þ ); ρrτ,C , ρrτ,SC , ρrτ,W , ρrτ,K , ρrτ,im,x,r , and ρrτ,ex,x,r are autoregression coefficients; finally, ηrτ,C,t , ηrτ,SC,t , ηrτ,W,t , ηrτ,K,t , ηrτ,im,x,r ,t , and ηrτ,ex,x,r ,t represent corresponding Gaussian white noises. The other net taxes (T rrest,t ) of households are described by the autoregression T rrest,t ¼ T rrest 1ρT,rest T rrest,t1 ρT,rest eηT,rest,t , r

r

r

ð3:212Þ

where T rrest denotes an equilibrium value of the other net taxes (T rrest 2 ð0, 1Þ ); ρrT,rest is an autoregression coefficient (ρrT,rest 2 ½0, 1Þ); and finally, ηrT,rest,t represents a Gaussian white noise. Like in the computable general equilibrium models, the revenues and expenditures of the state budget in Model 2 are reduced against their observed values by the transfers to households and firms (which explains the term “net taxes” covering these transfers). Public Agents State implements consumption and investments through a continuum of optimally acting agents denoted by gr 2 Tr. Each agent gr consumes a given quantity of goods Grt ðgr Þ and also invests a given quantity of goods I rH,G,t ðgr Þ. They consist of the

3.1 Conceptual Description of the Global Economy Within Model 2

261

goods of all Sectors, Regions, and firms aggregated in accordance with CES functions. The following auxiliary problems for minimizing the purchase cost of these goods (the problems 1g–6g) are similar to the problems 2x–4x for calculating the optimal intermediate consumption of firms. For consumptive use, during period t, an agent gr purchases goods of different types Grx,t ðgr Þ aggregated in accordance with the CES function X

Grt ðgr Þ ¼

1 μr G

γ r Grx,t ðgr Þ x2X G,x

! rμrG

μr 1 G μr G

μ 1 G

:

ð3:213Þ

Here γ rG,x > 0 is the preference coefficient for the government consumption of goods P from Sector x ( x2X γ rG,x ¼ 1 ); μrG denotes the elasticity of substitution for the government consumption of goods from different Sectors. Problem 1g Given t, r 2 R, gr 2 Sr, the government consumptions Grt ðgr Þ, and prices PrG,x,t of goods x 2 X, find the consumptions Grx,t ðgr Þ by minimizing the purchase cost  X   x2X PrG,x,t Grx,t ðgr Þ max fGrx,t ðgr Þgx2X subject to the following constraint derived from (3.213):

Grt ðgr Þ 

X

1 μr G

γ r Grx,t ðgr Þ x2X G,x

μr 1 G μr G

! rμrG

μ 1 G

 0:

ð3:214Þ

The first-order conditions determining the solution of the problem 1g under μrG < 1 and μrG 6¼ 0 yield the following optimal demands of a public agent gr for the goods of each Sector x (x 2 X) and government consumer prices PrG,t of these goods:  Grx,t ðgr Þ ¼ γ rG,x PrG,t ¼

X

PrG,x,t PrG,t

μrG

Grt ðgr Þ, r

γ r Pr 1μG x2X G,x G,x,t



1 1μr G

:

ð3:215Þ ð3:216Þ

In turn, the goods Grx,t ðgr Þ of each type x 2 X purchased by an agent gr are aggregated over the Regions of production as follows:

262

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . . r

X

Grx,t ðgr Þ ¼

γr  r 2R G,x,r

1 μr G,x

Grx,r ,t ðgr Þ

μr 1 G,x μr G,x

!μrμG,x1 G,x

:

ð3:217Þ

Here γ rG,x,r > 0 denotes the preference coefficient for the consumption of goods P from Sector x of Region r ( r 2R γ rG,x,r ¼ 1); μrG,x is the elasticity of substitution for the consumption of all goods from Sector x of different Regions. Problem 2g Given t, x 2 X, r 2 R, gr 2 Sr, the consumptions Grx,t ðgr Þ of goods x, and the prices Prx,r ,t, find the consumptions Grx,r ,t ðgr Þ of goods x made in Regions r by minimizing the purchase cost  X   r 2R Prx,r ,t Grx,r ,t ðgr Þ max fGrx,r ,t ðgr Þgr 2R subject to the following constraint derived from (3.217): r

Grx,t ðgr Þ 

X

γr  r 2R G,x,r

1 μr G,x

Grx,r ,t ðgr Þ

μr 1 G,x μr G,x

!μrμG,x1 G,x

 0:

ð3:218Þ

The first-order conditions determining the solution of the problem 2g under μrG,x < 1 and μrG,x 6¼ 0 yield the optimal demands of an agent gr for all goods from Sector x (x 2 X) made in each Region r:  Grx,r ,t ðgr Þ ¼ γ rG,x,r

Prx,r ,t PrG,x,t

μrG,x,q

Grx,t ðgr Þ:

ð3:219Þ

The prices Prx,r ,t of these goods satisfy the relationship PrG,x,t ¼

X

r

γ r  Pr  1μG,x r 2R G,x,r x,r ,t



1 1μr G,x

:

ð3:220Þ

Each of the above aggregate consumptions Grx,r ,t ðgr Þ of each type of goods    consists of a continuum of different goods Grx,r ,t gr , hx,r made by firms hx,r of Region r, which are defined by the integral CES functions 

Grx,r ,t ðgr Þ ¼



1 nr 

 r1 ð



λx

T

r

Grx,r ,t gr , h

! rλrx

r

λx 1 x,r λr x

dhx,r



λx 1

:

ð3:221Þ

3.1 Conceptual Description of the Global Economy Within Model 2

263



Here λrx denotes the elasticity of substitution for different goods from Sector x of  Region r.Here λrx denotes the elasticity of substitution for different goods from Sector x of Region r. The next optimization problem gives the consumptions of goods х made in  Region r by a firm hx,r . Problem 3g Given t, x 2 X, r 2 R, r 2 R, gr 2 Tr, the consumptions Grx,r ,t ðgr Þ, and     prices Prx,r ,t hx,r of different types of goods made by firms hx,r 2 Sr of Region r,   find the quantities of goods Grx,r ,t gr , hx,r from these firms by minimizing the purchase cost max  fGrx,r ,t ðgr , hx,r Þg,



hx,r 2T r

 ð  

Tr

Prx,r ,t 



h

x,r



Grx,r ,t



r

g ,h

 x,r dh

x,r

 ð3:222Þ

subject to the following constraint derived from (3.221): 

Grx,r ,t ðgr Þ 



1 nr 

 r1 ð λx

T

r

  Grx,r ,t gr , hx,r

 λrx 1  λrx

! rλrx dhx,r



λx 1

 0:

ð3:223Þ

The first-order conditions determining the solution of the problem 3g under  < 1 and λrx 6¼ 0 yield the optimal demands of an agent gr for the intermediate    goods x of firms hx,r and prices Prx,r ,t hx,r of these goods:  λrx

  !λrx Prx,r ,t hx,r Grx,r ,t , Prx,r ,t

  1 Grx,r ,t gr , hx,r ¼ r n  Prx,r ,t ¼

1 nr 

ð Tr

 x,r 1λrx x,r r P dh  ,t h x,r 

ð3:224Þ



1  1λrx

:

ð3:225Þ

Solving the next triplet of nested optimization problems, an agent gr determines the optimal investments of products x 2 X (the problem 4g) made in each Region  r 2 R (the problem 5g) by a firm hx,r (the problem 6g). The first-order conditions of these optimization problems are similar to those of the problems 1g, 2g, and 3g. For investment in nonfinancial assets, during period t, an agent gr purchases goods of different types I rH,G,t ðgr Þ aggregated in accordance with the CES function r

I rH,G,t ðgr Þ ¼

X

1

μr 1 H,G μr H,G

μr r H,G I r γr H,G,x,t ðg Þ x2X I,H,G,x

!μrμH,G1 H,G

:

ð3:226Þ

264

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Here γ rI,H,G,x > 0 is the preference coefficient for the purchase of goods from Sector P x as nonfinancial assets ( x2X γ rI,H,G,x ¼ 1); μrH,G denotes the elasticity of substitution for the purchase of goods from different Sectors as nonfinancial assets. Problem 4g Given t, r 2 R, gr 2 Tr, the government investments I rH,G,t ðgr Þ, and the prices PrI,H,G,x,t , find the investments in nonfinancial assets of products I rH,G,x,t ðgr Þ by minimizing the purchase cost  X   x2X PrI,H,G,x,t I rH,G,x,t ðgr Þ max fI rH,G,x,t ðgr Þg, x2X subject to the following constraint derived from (3.226): r

I rH,G,t ðgr Þ 

X

1

μr r H,G I r γr H,G,x,t ðg Þ x2X I,H,G,x

μr 1 H,G μr H,G

!μrμH,G1 H,G

 0:

ð3:227Þ

The first-order conditions determining the solution of the problem 4g under μrH,G < 1 and μrH,G 6¼ 0 yield the following optimal investment demands of an agent gr for the goods of each Sector x (x 2 X) and prices PrI,H,G,t of these goods:  I rH,G,x,t ðgr Þ ¼ γ rI,H,G,x PrI,H,G,t ¼

X

PrI,H,G,x,t PrI,H,G,t

μrI,H,G,q

r

I rH,G,t ðgr Þ,

1μI,H,G,q γr Pr x2X I,H,G,x I,H,G,x,t



1 1μr I,H,G,q

ð3:228Þ

:

ð3:229Þ

In turn, the goods I rH,G,x,r ,t ðgr Þ of each type x 2 X purchased by an agent gr for investment are aggregated over the Regions of production as follows: r

I rH,G,x,t ðgr Þ ¼

X

1

μr 1 I,H,G,x μr I,H,G,x

μr r I,H,G,x I r γr  H,G,x,r ,t ðg Þ r 2R I,H,G,x,r

!μrμI,H,G,x1 I,H,G,x

:

ð3:230Þ

Here γ rI,H,G,x,r > 0 is the preference coefficient for the purchase of goods from P Sector x of Region r as nonfinancial assets ( r 2R γ rI,H,G,x,r ¼ 1); μrI,H,G,x denotes the elasticity of substitution for the purchase of goods from Sector x of different Regions as nonfinancial assets. Problem 5g Given t, r 2 R, x 2 X, gr 2 Sr, the investments Grx,t ðgr Þ, and the prices Prx,r ,t , find the investments I rH,G,x,r ,t ðgr Þ in nonfinancial assets of goods from Sector x made in Regions r 2 R by minimizing the purchase cost

3.1 Conceptual Description of the Global Economy Within Model 2

max fI rH,G,x,r ,t ðgr Þg,

x2X

265

 X   r 2R Prx,r ,t I rH,G,x,r ,t ðgr Þ

subject to the following condition derived from (3.230): r

I rH,G,x,t ðgr Þ 

X

1

μr r I,H,G,x I r γr  H,G,x,r  ,t ðg Þ r 2R I,H,G,x,r

μr 1 I,H,G,x μr I,H,G,x

!μrμI,H,G,x1 I,H,G,x

 0:

ð3:231Þ

The first-order conditions determining the solution of the problem 5g under μrI,H,G,x < 1 and μrI,H,G,x 6¼ 0 yield the following optimal investment demands of an agent gr for the goods of each Sector x (x 2 X) made in each Region r:  I rH,G,x,r ,t ðgr Þ

¼

Prx,r ,t PrI,H,G,x,t

γ rI,H,G,x,r

μrI,H,G,x

I rH,G,x,t ðgr Þ:

ð3:232Þ

The prices Prx,r ,t of these goods satisfy the relationship PrI,H,G,x,t ¼

X

r

r 2R

γ rI,H,G,x,r Prx,r ,t 1μI,H,G,x



1 1μr I,H,G,x

:

ð3:233Þ

Each of the above investments I rH,G,x,r ,t ðgr Þ in nonfinancial assets by a public   agent gr consists of a continuum of different goods I rH,G,x,r ,t gr , hx,r made by firms  hx,r of Region r, which are defined by the integral CES functions: 

I rH,G,x,r ,t ðgr Þ ¼



1 nr 

 r1 ð λx

T

r

! rλrx

r



I rH,G,x,r ,t gr , h

λx 1 x,r λr x

dhx,r



λx 1

:

ð3:234Þ



Here λrx denotes the elasticity of substitution for different goods from Sector x of Region r. Problem 6g Given t, 2X, r 2 R, r 2 R, gr 2 Tr, the investments I rH,G,x,r ,t ðgr Þ, and     prices Prx,r ,t hx,r of different types of goods made by firms hx,r 2 Sr of Region r,   find the investments I rH,G,x,r ,t gr , hx,r in the nonfinancial assets of these firms by minimizing the purchase cost max  fI rH,G,x,r ,t ðgr , hx,r Þg,



hx,r 2T r

 ð  

Tr

Prx,r ,t 



h

x,r



I rH,G,x,r ,t



subject to the following constraint derived from (3.234):

r

g ,h

x,r

 x,r dh

 ð3:235Þ

266

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

I rH,G,x,r ,t ðgr Þ 



1 nr 

λr

λr1 ð x,r

T

r

  I rH,G,x,r ,t gr , hx,r

λr  1 x,r λr  x,r

!λr x,r1 dhx,r

x,r 



 0:

ð3:236Þ

The first-order conditions of the problem 6g under λrx,r < 1 and λrx,r 6¼ 0 yield the following optimal investment demands of an agent gr for the intermediate goods x of    firms hx,r and prices Prx,r ,t hx,r of these goods: I rH,G,x,r ,t

 r x,r  1 ¼ r g ,h n  Prx,r ,t

¼

1 nr 

  !λrx,r Prx,r ,t hx,r I rH,G,x,r ,t ðgr Þ, Prx,r ,t

ð Tr

Prx,r ,t 



h

 r x,r  1λx,r

dh

x,r

ð3:237Þ

 r

1 λ  1 x,r

:

ð3:238Þ

Central Bank For each period t, the Central Bank of Region r determines the benchmark interest rate for issuing loans and providing deposits to second-level banks; serves its part of the public debt; forms and uses the National Fund of domestic Region. For period t, the Central Bank of Region r determines the benchmark interest rate Rrt for domestic currency using the Taylor rule [76] with cofactors Rer rus,t for considering the real exchange rate of USD:  r ρrr  r rrrπ  r r  rr !1ρrr Rrt Rt1 πt GDPrt ry Rer rus,t rrer ¼ Rr Rr π rtarget,t GDPr Rer rus  r rrrdπ  r  rrrdrer π GDPrt rrdy Rer rus,t r  rt eηR : π t1 GDPrt1 Rer rus,t1

ð3:239Þ

The notations are the following: Rr as an equilibrium value of the benchmark interest rate; GDPrt as the GDP of Region r; π rt ¼ PrC,t =PrC,t1 as inflation, the ratio of the consumer price indexes (CPI) for the current and previous periods, PrC,t to PrC,t1 (see (3.42)); and π rtarget,t as the target value of consumer price inflation described by the autoregression π rtarget,t ¼ π rtarget 1ρπ,target π rtarget,t1 ρπ,target eηπ,target,t , r

r

ð3:240Þ

with π rtarget as an equilibrium value of the target consumer price inflation (π rtarget 2 ð0, 1Þ ); ρπ,target as an autoregression coefficient (ρπ,target 2 [0, 1)); ηrπ,target,t and ηrR as Gaussian white noises; and ρrr , r rrπ , r rrrer , r rrdπ , r rrdy , and r rrdrer as positive constants, ρrr 2 ½0, 1Þ.

3.1 Conceptual Description of the Global Economy Within Model 2

267

With this benchmark interest rate Rrt the Central Bank of Region r issues loans and accepts deposits from domestic second-level banks, i.e., Rrt ¼ RrG,r,t . If r 6¼ us, row, then the loan and deposit rates of Central Bank in USD for second-level banks is given by   RrG,us,t ¼ RrG,r,t Srus,t =Et Srus,tþ1 :

ð3:241Þ

This relationship between the yields of the two currencies establishes equal loan conditions for them. The dynamics of the public debt served by Central Bank (BrG,CB,r,t ) are described by an autoregression for the deviations from an equilibrium value BrG,CB,r : BrG,CB,r,t ¼ BrG,CB,r



BrG,CB,r,t1 BrG,CB,r

ρrB,G,CB,r

eηB,G,CB,r : r

ð3:242Þ

Here ρrB,G,CB,r denotes an autoregression coefficient, while ηrB,G,CB,r represents a Gaussian white noise. For period t the National Fund of Region r denominated in USD (NF rus,t ) consists of its carryover from the previous period taking into account the deposit loans in r second-level banks (Rus t NF t1 ); the export taxes of this Region on Products r L calculated at rates τex,L,r ,t (see formula (3.250) below); the taxes on the dividends  (DivrP,L,r,t ) obtained by patient households of Region r from Sector L of Region r; the lease payments of Sector L for the public land (OrL,G,t ); and minus the transfers (Tr rt ) to the state budget of domestic Region taking into account the other revenues/ deductions (NFOrus,t ): NF rus,t

¼

r Rus t NF t1

þ

þ

Sus r,t



X r 6¼r

τrex,L,r ,t EX r  1 þ τrex,L,r ,t L,r ,t



τrK,t X  r r us r r Sus DivrP,L,r,t þ Sus r,t PO,G,t OL,G,t  Sr,t Tr t þ NFOus,t : r  2R r ,t 1  τrK,t ð3:243Þ

The transfers Tr rt from National Fund to the state budget are described by the autoregression ρr r r  Tr rt ¼ ðTr r Þ1ρTr Tr rt1 Tr eηTr,t ,

ð3:244Þ

where Trr denotes an equilibrium value of these transfers; ρrTr is an autoregression coefficient (ρrTr 2 ½0, 1Þ); and finally, ηrTr,t represents a Gaussian white noise.

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

268

3.1.6

Balance and Auxiliary Equations

For period t the aggregate exports EX rt from Region r and imports IM rt to Region r in domestic currency consist of the following sums, where EX rx,t and IM rx,t denote the corresponding components (the exports and imports of Product х): EX rt ¼ IM rt ¼

X x2X

EX rx,t ,

ð3:245Þ

x2X

IM rx,t :

ð3:246Þ

X

In turn, the exports and imports of Product х are written as the sums of the terms (EX rx,r ,t and IM rx,r ,t ) that determine the exports from other Regions r and the imports to other Regions r: EX rx,t ¼ IM rx,t ¼

X r 2R; r6¼r

EX rx,r ,t ,

ð3:247Þ

r 2R; r6¼r

IM rx,r ,t :

ð3:248Þ

X

By assumption, EX rx,r,t ¼ IM rx,r,t ¼ 0: The imports IM rx,r ,t are the aggregate cost of goods х from Region r in Region r used for consumption and investment by both types of households   r r Cq,x,r ,t , I H,q,x,r ,t , for investment and intermediate consumption by each firm x r  as well as for government consumption and investment I , Vr  xr,x,r ,t rx ,x,r ,t  GG,x,r ,t , I H,G,x,r ,t taking into account the domestic price of this product in Region        r Prx,t , the exchange rate Srr ,t , and the export duty in Region r τrex,x,r,t :     IM rx,r ,t ¼ 1 þ τrex,x,r,t Srr ,t Prx,t 2 P



 3 r r C þ I   q,x,r ,t H,q,x,r ,t q2Q 5: 4 X   þ x 2X I rx ,x,r ,t þ V rx ,x,r ,t þ GrG,x,r ,t þ I rH,G,x,r ,t

ð3:249Þ

The exports EX rx,r ,t are the aggregate cost of goods х supplied to Region r in Region r used for consumption and investment by both types of households    Crq,x,r,t , I rH,q,x,r,t , for investment and intermediate consumption by each firm x  r   I , V rx ,x,r,t as well as for government consumption and investment  xr,x,r,t    GG,x,r,t , I rH,G,x,r,t taking into account the domestic price of this product in Region     r Prx,t and the export duty in Region r τrex,x,r ,t :

3.1 Conceptual Description of the Global Economy Within Model 2

269

  EX rx,r ,t ¼ 1 þ τrex,x,r ,t Prx,t hX    X i  r  r r r r r þ  C þ I I  ,x,r,t þ V x ,x,r,t þ Gx,r,t þ I H,G,x,r,t : q,x,r,t H,q,x,r,t  x q2Q x 2X

ð3:250Þ The balance of payments of Region r in domestic currency with the exchange rates (Srr ,t ) is described by the following relationship, which expresses the equality   of the net exports of goods EX rt  IM rt to the net exports of capital: X



r 6¼r





fSrus,t ½IDrr ,us,t þ IDrr ,us,t1 RrID,r ,us,t1 þ IDrr,us,t  IDrr,us,t1 RrID,r,us,t1 





BrG,B,r ,us,t þ BrG,B,r ,us,t1 RrG,r ,us,t1 þ BrG,B,r,us,t  BrG,B,r,us,t1 RrG,r,us,t1 





þNF rus,r ,t  NF rus,r ,t1 RrIB,r ,us,t1  NF rus,r,t þ NF rus,r,t1 RrIB,r,us,t1  hX  i X  r r r r Srr ,t Div Div þ Div þ   P,x,r,t þ DivP,B,r,t g P,x,r ,t P,B,r ,t x2X x2X þ Or ¼ EX rt  IM rt : ð3:251Þ The notations are the following: IDrr ,us,t as the loans taken up by the banks of Region r from those of Region r; RrID,r ,us,t1 as the loan rate; BrG,B,r ,us,t as the public debt (in form of public bonds) to the banks of Region r; RrG,r ,us,t1 as the public bonds rate; NF rus,r ,t as the part of the National Fund stored in the banks of Region r; DivrP,x,r ,t and DivrP,B,r ,t as the dividends obtained by patient households from different Sectors and second-level banks in Region r; and finally, Or as unaccounted factors described by an autoregression. The labor market balance, requiring that for each Sector x the labor supply by the households of type q (Lrq,x,t ) is equal to the corresponding labor demand by production (Lrx,q,t ), is defined by Lrq,x,t ¼

1 nr

Lrx,q,t ¼

1 nr

ð ð

S

r

Sr

Lrq,x,t ð jq,r Þdjq,r ,

ð3:252Þ

Lrx,q,t ðhx,r Þdhx,r ,

ð3:253Þ

Lrq,x,t ¼ Lrx,q,t :

ð3:254Þ

(Here (3.254) is the balance equation itself.) The products market balance, requiring that for each Product x the supply by firms (Y rx,t ) is equal to the corresponding consumer demand, is defined by

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

270

ð 1 Y r ðhx,r Þdhx,r , nr T r x,t ð      1 C rq,x,r,t ¼ r  C rq,x,r,t jq,r djq,r , n Tr ð     1 r I H,q,x,r,t ¼ r  I rH,q,x,r,t jq,r djq,r , n Tr ð      1 V rx ,x,r,t ¼ r  V rx ,x,r,t hx,r dhx,r , n Tr ð     1 I rx ,x,r,t ¼ r  I rx,x ,r ,t hx,r dhx,r , n Tr ð      1 Grx,r,t ¼ r  Grx,r,t gr dgr , n Tr ð      1 I rH,G,x,r,t ¼ r  I rH,G,x,r,t gr dgr , n Tr Y rx,t ¼

Y rx,t ¼

X

X r  2R

 q2Q

ð3:255Þ ð3:256Þ ð3:257Þ ð3:258Þ ð3:259Þ ð3:260Þ ð3:261Þ

 X   r    r r r þ I C rq,x,r,t þ I rH,q,x,r,t þ I þ V þ G   x ,x,r,t H,G,x,r,t x,r,t : x 2X x ,x,r,t

ð3:262Þ 





(Here (3.262) is the balance equation itself.) In addition, Y rx,t , C rq,x,r,t , I rH,q,x,r,t , I rx ,x,r,t ,    V rx ,x,r,t , Grx,r,t , and I rH,G,x,r,t denote the aggregate values corresponding to the integrands of the mentioned agents. The first dividends balance, requiring that the dividends obtained by the patient  households of all Regions (DivrP,x,r,t ) from possessing the Producing Sector x of Region r are equal to the dividends paid in Sector х (Divrx,t ), is defined by ð 1 Divrx,t ðhx,r Þdhx,r , nr T r ð      1 DivrP,x,r,t ¼ r  DivrP,x,r,t jP,r djP,r , n Tr X  Divrx,t ¼ DivrP,x,r,t : r  2R Divrx,t ¼

ð3:263Þ ð3:264Þ ð3:265Þ

(Here (3.265) is the balance equation itself.) The second dividends balance, requiring that the dividends obtained by the  patient households of all Regions (DivrP,B,r,t ) from possessing the second-level banks of Region r are equal to the dividends paid in the Banking Sector of Region r (DivrB,t ), is defined by

3.1 Conceptual Description of the Global Economy Within Model 2

ð 1 DivrB,t ðhx,r Þdhx,r ; nr T r ð     1 ¼ r  DivrP,B,r,t jP,r djP,r ; n Tr X  DivrB,t ¼ DivrP,B,r,t :

271

DivrB,t ¼

ð3:266Þ



ð3:267Þ

DivrP,B,r,t

ð3:268Þ

r  2R

(Here (3.268) is the balance equation itself.) The demand-supply balances for deposits and loans, requiring that (for each Region r and each currency z 2 Z ) the demands for deposits by patient households and also for loans by impatient households and firms are equal to the corresponding supplies, is defined by 1 nr

ð

1 nr 1 nr

T

r

  DrP,z,t jP,r , br djP,r ¼ DrP,z,t ðbr Þ;

ð3:269Þ

  BrS,z,t jS,r , br djS,r ¼ BrS,z,t ðbr Þ;

ð3:270Þ

Brx,z,t ðhx,r , br Þdhx,r ¼ Brx,z,t ðbr Þ:

ð3:271Þ

ð

Tr

ð

Tr

The interbank borrowings balance, requiring that (for each pair of Regions (r, r) and each currency z) the demands for interbank borrowings are equal to the corresponding supplies, is defined by 1 nr

ð Tr

   r   r IDrz,r ,t br , ~b dbr ¼ IDrz,r,t ~b :

ð3:272Þ

(If r 6¼ r, then z ¼ us; otherwise z 2 Zr). The auxiliary equation for calculating the GDP of Region r at constant prices (GDPrt) is written as the sum of the internal final consumptions of domestic Products in Region r (C rq,x,r,t , I rq,x,r,t , I rx ,x,r,t, Grx,r,t , I rH,G,x,r,t ) and the external final consumptions       of these Products in other Regions r (Crq,x,r,t , I rH,q,x,r,t , I rx ,x,r,t , V rx ,x,r,t , Grx,r,t , I rH,G,x,r,t ): 0P

1  P C rq,x,r,t þ I rq,x,r,t þ x 2X I rx ,x,r,t þ Grx,r,t þ I rH,G,x,r,t C X B (P ) B C P GDPrt ¼ r r r r B C ðCq,x,r,t þI H,q,x,r,t Þþ x 2X ðI x ,x,r,t þV x ,x,r,t Þ x2X @ P A q2Q þ r 6¼r r r þGx,r,t þ I H,G,x,r,t 

q2Q

þ OGDPrt , ð3:273Þ where OGDPrt denotes unaccounted factors and statistical errors.

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

272

The GDP of Region r at current prices (PGDPrt ) is obtained from the previous formula using the internal prices (Prx,t ) of Product х: 0P

1  P r r r r r C þ I I þ G þ I þ   x,r,t q,x,r,t H,G,x,r,t C x 2X x ,x,r,t B q2Q q,x,r,t X (P ) C r B P     PGDPrt ¼ P B C ðCrq,x,r,t þI rH,q,x,r,t Þþ x 2X ðI rx ,x,r,t þV rx ,x,r,t Þ x2X x,t @ P A q2Q þ r 6¼r r r þGx,r,t þ I H,G,x,r,t 

þ OPGDPrt : ð3:274Þ (Again, OPGDPrt denotes unaccounted factors and statistical errors.)  The GDP deflator of Region r PrGDP,t ) is defined by the ratio of the above GDP values: PrGDP,t ¼ PGDPrt =GDPrt :

ð3:275Þ



The relationships between the internal (Prx,t) and external (Prx,r ,t) prices of Product х exported from Region r to Region r have the form      Prx,r ,t ¼ 1 þ τrex,x,r,t 1 þ τrim,x,r :t Srr ,t Prx,t , Prx,r,t ¼ Prx,t :

ð3:276Þ



Here τrex,x,r,t and τrim,x,r :t are the export and import duties of this Product in Regions rand r, respectively. For the bilateral trade of EAEU countries, these duties are nullified.

3.2 3.2.1

Nonlinear Modeling and Parameter Estimation Nonlinear Model 2 and Estimation of Its Parameters

The nonlinear Model 2 consists of the equations describing the aggregate first-order conditions of the optimization problems and behavioral rules of all economic agents, the balance and auxiliary equations, as well as the shock rules. This nonlinear model can be written in the vector form:   Et F θ X t1 , X t , X tþ1 , Η Σt Η ¼ 0:

ð3:277Þ

Here Et denotes the conditional expectation given information available at time t (t ¼ 1, 2, . . .); Fθ is a known vector function; θ means the collection of the structural parameters of Model 2 and parameters of the shock autoregressions; the vector Xt consists of the endogenous variables and shocks described by first-order

3.2 Nonlinear Modeling and Parameter Estimation

273

autoregressions with given X0; the vector Η Σt Η consists of the Gaussian white noises; finally, Σ Η represents the collection of the standard deviations of these noises. The nonlinear Model 2 (3.277) was reduced to its nonlinear counterpart using log-linearization with an acceptable coincidence of the volatility indexes of the corresponding endogenous variables of the former and latter models.

3.2.2

Building of Linear Model 2

The linear Model 2 was obtained from the nonlinear Model 2 (3.277) using the loglinearization procedure as follows. Step 1. Cast out the time subscripts and nullify the Gaussian white noises for reducing (3.277) to the nonlinear system of equations: F θ ðX, X, X, 0Þ ¼ 0:

ð3:278Þ

Step 2. Find the roots of the vector equation (3.278) in MATLAB. Step 3. Use the change of variables i

b X it ¼ X i eX t ,

ð3:279Þ i

b t is the coordinate of the where X it denotes the ith coordinate of the vector Xt and X b t characterizing the log-deviation of X it from its equilibrium value Xi, new variable X reducing the nonlinear system (3.277) to   ~θ X b t1 , X bt , X b tþ1 , Η Σt Η ¼ 0: Et F

ð3:280Þ

The equilibrium of model (3.280) is the origin.  b t1 , X bt , X b tþ1 , Η Σt Η into the Taylor series in a ~θ X Step 4. Expand the function F neighborhood of the origin:   ~θ ~θ b t1 , X ~θ X bt , X b tþ1 , ΗΣt Η ¼ F b t1 þ ∂F ð0, 0, 0, 0ÞX bt ~ θ ð0, 0, 0, 0Þ þ ∂F ð0, 0, 0, 0ÞX F b t1 bt ∂X ∂X   ~θ ~θ ∂F b tþ1 þ ∂F ð0, 0, 0, 0ÞΗΣt Η þ Rθ2 X bt , X b tþ1 , ΗΣt Η , b t1 , X þ ð0, 0, 0, 0ÞX Σ b tþ1 ∂Ηt Η ∂X ð3:281Þ   b t1 , X bt , X b tþ1 , Η Σt Η is Taylor’s remainder. where Rθ2 X

274

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Step 5. Cast out this remainder, and apply the conditional expectation operator Et to both sides of expression (3.281) for obtaining the linear Model 2: b tþ1 þ Bθ X b t þ Cθ X b t1 þ Dθ Η Σt Η ¼ 0: Aθ Et X

ð3:282Þ

Here Aθ, Bθ, Cθ, and Dθ are matrices of compatible dimensions. Step 6. Find the solution of the linear Model 2 (3.282) using the Sims algorithm [73] in the form b t ¼ Qθ X b t1 þ F θ Η Σt Η , t ¼ 1, 2, 3, . . . , X

ð3:283Þ

where Qθ and Fθ are matrices of compatible dimensions calculated by the Sims algorithm. b0 ¼ X0, Step 7. Supplement the linear Model 2 (3.283) with the initial condition X where X 0 is given.

3.2.3

Adaptation to the Goals of Research, Parameter Identification and Calibration of Linear Model 2

The accepted hypotheses and the nonlinear and linear Models 2 were adapted to the goals of research in the following ways: • First, by choosing a collection of several Regions of the global economy, namely, Kazakhstan (kaz), Russia (rus), Belarus (blr), Armenia (arm), Kyrgyzstan (kgz), USA (usa), China (chn), the European Union (eu), and the Rest of World (row). Thus, the Regions of Model 2 form the set R ¼ {kaz, rus, blr, arm, kgz, usa, chn, eu, row}. • Second, by choosing the set X ¼ {L, M, N} of Producing Sectors of commodities (L ), tradable non-commodities (M), and non-tradable goods (N). • Third, by estimating the parameters of Model 2 using statistical data. Following the approach [28, 32, 61], the parameter set θ of the linear Model 2 was divided into two subsets (groups). The first group contained the calibrated parameters, which were identified using the equilibrium values of the ratios of basic economic indexes to GDP based on available statistical data for Kazakhstan, Russia, Belarus, Armenia, Kyrgyzstan, China, the USA, and the Rest of World for the period 2000–2015 and also based on the macroeconomic studies [30, 44, 52, 56, 67, 70]. The second group consisted of the estimated parameters. The resulting subset θ0 of the parameter set θ and the components of the set Σ Η were estimated by the two-stage Bayesian algorithm. At Stage 1, the Bayesian algorithm was employed for estimating all parameters only at the level of Region r and at Stage 2, at the level of the whole Model 2.

3.2 Nonlinear Modeling and Parameter Estimation

275

Stage 1 of the Bayesian algorithm included several steps as follows: Step 1. Partition the sets θ, θ0, and Σ Η by Regions so that θr is a subset of θ consisting of the structural parameters of Model 2 and the autoregression parameters of the shocks directly related to Region r θ0, r is a subset of θ0 consisting of the estimated structural parameters of Model 2 and also of the estimated autoregression parameters of the shocks directly related to Region r Σ rΗ is a subset of Σ Η consisting of the standard deviations of the Gaussian white noises directly related to Region r b t by Regions so that Step 2. Partition the endogenous variables vector X r b t is a vector consisting of the endogenous variables and shocks described by the X first-order autoregressions directly related to Region r b t by Regions so that Step 3. Partition the endogenous variables vector H r b t is a vector consisting of the Gaussian white noises directly related to Region r H Step 4. For each Region, write equation (3.283) in the form r

Η b rt ¼ Qθr X b rt1 þ F θr Η r,Σ X , t ¼ 1, 2, 3, . . . t

ð3:284Þ

where Qθ and F θ are the submatrices of the matrices Qθ and Fθ, respectively, that correspond to Region r. r Step 5. Introduce the observable variables vector b St of Region r and supplement the linear model (3.284) with the measurement equation r

r

r b b rt , St ¼ M r X

ð3:285Þ

where each row of a matrix Mr contains a single element equal to 1 and all other elements are 0. The log-deviations of the following macroeconomic indexes of Region r from their HP trends with seasonal correction (in %) were adopted as the measured values r of the observable variables b St : • • • • • • • • • • •

GDP The output of tradable non-commodities The output of non-tradable goods The output of commodities, including oil Consumption The investments of households The investments of firms producing tradable goods The investments of firms producing non-tradable goods The investments of firms producing commodities, including oil The deposits of households The loans of households

276

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

• • • • • • • • • • • • • • • • • • • • • • • • • •

The loans of firms producing tradable goods The loans of firms producing non-tradable goods The loans of firms producing commodities, including oil The deposit rates for households The loan rates for households The loan rates for firms producing tradable goods The loan rates for firms producing non-tradable goods The loan rates for firms producing commodities, including oil The aggregate imports of tradable non-commodities The aggregate imports of commodities, including oil The aggregate exports of tradable non-commodities The aggregate exports of commodities, including oil Government expenditures Government consumption Government investments Public debt Transfers from non-budgetary funds Individual income tax (IIT) revenues Corporate income tax (CIT) revenues Social tax revenues Value added tax (VAT) revenues (or sales tax revenues) Import tax revenues Export tax revenues Extraction tax revenues Exchange rate Government interest rates (the benchmark interest rate of Central Bank or treasury bond rates) • Employment of population Step 6. For each Region r, using the Kalman filter, find the probability distribur tions of the observable variables b St of the linear Model 2 described by (3.284) and (3.285). Step 7. For each Region r, construct the likelihood function based on these probability distributions for the linear Model 2 described by (3.284) and (3.285). Step 8. For each Region r, specify the prior probability distributions of the parameter vectors θ0,r and ΣrΗ in accordance with the following rules [51]: • The standard deviations of the Gaussian white noises are given by the inverse gamma distributions. • The parameters taking values within the range [0, 1] are given by the beta distributions. • The other parameters are given by the normal distribution. Define the probabilistic characteristics of all parameters using the approach [28, 61] taking into account the specific features of regional economies.

3.2 Nonlinear Modeling and Parameter Estimation

277

Step 9. Construct the posterior probability density functions of the parameter vectors θ0,r and Σ rΗ using the Bayes formula. Step 10. Generate a sample of parameter values based on the posterior probability density functions of the parameter vectors θ0,r and Σ rΗ using the Metropolis–Hastings algorithm and calculate the numerical characteristics of the posterior probability density functions. Stage 2 of the Bayesian algorithm includes the following steps: Step 1. Introduce the observable variables vector b St for all Regions simultaneously and supplement the linear model (3.283) with the measurement equation b bt , St ¼ M X

ð3:286Þ

where each row of a matrix M contains a single element equal to 1 and all other elements are 0. The log-deviations of the same macroeconomic indexes (see Stage 1 of the algorithm) aggregated over all Regions r from their HP trends with seasonal correction (in %) were adopted as the measured values of the observable variables b St . Step 2. Using the Kalman filter, find the probability distributions of the observable variables b St of the linear Model 2 described by (3.283) and (3.286). Step 3. Construct the likelihood function based on these probability distributions for the linear Model 2 described by (3.283) and (3.286). Step 4. Specify the prior probability distributions of the parameter vectors θ0 and ΣΗ in accordance with the following rules [51]: • The standard deviations of the Gaussian white noises are given by the inverse gamma distributions. • The parameters taking values within the range [0, 1] are given by the beta distributions. • The other parameters are given by the normal distribution. Define the probabilistic characteristics of all parameters for obeying the posterior probability density functions obtained at Step 9 of Stage 1. Step 5. Construct the posterior probability density function of the parameter vectors θ0 and ΣΗ using the Bayes formula. Step 6. Generate a sample of parameter values based on the posterior probability density functions of the parameter vectors θ0 and ΣΗ using the Metropolis– Hastings algorithm [42, 50], and calculate the resulting estimates in form of the corresponding sample means. This two-stage Bayesian algorithm for estimating the parameters of the linear Model 2 was implemented in Dynare. The estimated parameters of the linear Model 2 were transferred to the nonlinear Model 2 using a special function developed in MATLAB.

278

3.2.4

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Adjustment of Parameter Values for Nonlinear Model 2

For accepting the obtained estimates as the parameter values of the nonlinear Model 2, the volatility characteristics of all endogenous variables (autocorrelation, correlation, and cross-correlation matrices with single-quarter shift) over all Regions were compared, both for the linear and nonlinear setups of Model 2. The results of comparative analysis demonstrated the following facts: • The differences between the autocorrelation coefficients for the linear and nonlinear Models 2 did not exceed 1.9%. • The differences between the correlation coefficients for the linear and nonlinear Models 2 did not exceed 2.2%. • The differences between the values of the cross-correlation matrices for the linear and nonlinear Models 2 did not exceed 2.4%. These facts indicated that the volatilities of the corresponding endogenous variables were sufficiently close to each other and hence the parameter estimates derived for the linear Model 2 could be accepted as the parameter values of its nonlinear counterpart. Finally, note that the nonlinear Model 2 (3.277) with the calculated parameter values was solved using the Fair–Taylor method [37], with implementation in Dynare.

3.3

Applicability Testing of Model 2

For the applicability tests of Model 2, software was developed on the basis of parametric control and theory of dynamic stochastic general equilibrium models. It consists of six program modules with the following functions: Module 1 estimates the stability indexes of different mappings defined by Model 2 (see Sect. 1.3.2). Module 2 is responsible for retrospective forecasting. Module 3 compares the sampling moments (autocorrelations, correlation, and crosscorrelation matrices) calculated using the statistical observations and data obtained by Model 2. Module 4 estimates the impulse responses of Model 2 as well as verifies their consistency with the major postulates of macroeconomic theory. Module 5 estimates the sensitivity coefficients of the output variables of Model 2 using its input parameters as well as verifies their consistency with the major postulates of macroeconomic theory. Module 6 compares Model 2 with alternative models in terms if marginal likelihood. Note that Module 1 involves the theory of parametric control while the other Modules the theory of dynamic stochastic general equilibrium models.

3.3 Applicability Testing of Model 2

279

Table 3.1 Values of stability indexes Period (quarter, year) Index β( p, α) Period (quarter, year) Index β( p, α)

3.3.1

Q1 2016 3.73 Q3 2018 1.28

Q2 2016 3.87 Q4 2018 1.09

Q3 2016 3.83 Q1 2019 0.73

Q4 2016 3.40 Q2 2019 0.27

Q1 2017 3.00 Q3 2019 0.03

Q2 2017 2.63 Q4 2019 0.00

Q3 2017 2.30 Q1 2020 0.00

Q4 2017 2.00 Q2 2020 0.00

Q1 2018 1.73 Q3 2020 0.00

Q2 2018 1.49 Q4 2020 0.00

Testing with Stability Indexes

Module 1 of the software is intended to test the applicability of Model 2 using stability indexes (see Sect. 1.3.2). For computer simulations with this Module, all estimated parameters were treated as the input parameters ( p), while the quarter forecasts of all observable variables for all Regions since Q1 2016 as the output parameters ( y). The values of the stability indexes β(p,α) obtained for the basic value p and α ¼ 0.01 can be seen in Table 3.1. The simulation results in Table 3.1 do not exceed 3.83%, which testifies to a rather high stability of Model 2 (in terms of the stability indexes considered) on the period 2016–2019.

3.3.2

Testing with Retrospective Forecasting

Module 2 of the software gives a retrospective forecast of the observed economic indexes for the period Q1–Q4 2014 on the basis of Model 2. This Module was used for estimating the root-mean-square deviations of the economic forecasts from the corresponding observations. Table 3.2 presents the results of retrospective forecasting and also the observed values of some economic indexes for Kazakhstan and year 2014 (their relative deviations from trend in %), namely, consumption (bckz t ), investkz kz kz d d ments (bxkz ), exports ( e xp ), GDP (b y ), employed population ( emp ), average wage t t t t kz kz ), inflation (b π ), benchmark interest rate (b r ), exchange rate (domestic unit per (b wkz t t t kz kz kz b b ), and government USD, bs ), public debt (b ), government consumption (ge rw,t

t

t

b kz expenditures (gr t ). The simulation results indicated of sufficient accuracy of retrospective forecasting for the endogenous variables of Model 2 (less than 3 percentage points).

3.3.3

Testing with Moments

Module 3 of the software estimates how close are the moments of observed data to those generated by Model 2. The following results were obtained using this Module:

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

280

Table 3.2 Retrospective forecasting for Kazakhstan’s economy, year 2014 Period (quarter, year) Economic index bckz t bckz t bxkz t bxkz t

(observation)

Q1 2014 0.66

Q2 2014 4.18

Q3 2014 2.03

Q4 2014 4.18

0.63

2.78

2.72

2.85

(observation)

5.59

3.17

2.55

6.36

(forecast)

4.71

3.63

2.57

1.68

15.71

7.06

1.40

1.73

(forecast)

kz d exp t (observation) kz d exp t (forecast) bykz t (observation) bykz t (forecast) kz emp d t (observation) d kz emp t (forecast) b kz (observation) w t b kz w t (forecast) b π kz t (observation) b π kz t (forecast) br kz t (observation) br kz t (forecast) bskz rw,t (observation) bskz rw,t (forecast) kz b bt (observation) kz b bt (forecast) ge b kz t (observation) b kz ge t (forecast) b kz gr t (observation) b kz gr t (forecast)

12.31

8.94

2.01

1.33

0.04

0.43

0.10

0.51

1.34

1.75

0.56

0.55

0.73

0.44

0.20

0.92

1.35

1.64

0.93

0.24

2.15

1.40

0.98

2.51 0.62

2.77

1.21

0.91

1.11

0.26

0.34

0.45

0.79

0.36

0.21

0.21

0.11

0.07

0.20

0.15

0.51

0.47

0.44

0.33

0.67

1.51

1.51

1.70

1.03

0.89

0.89

0.50

2.51

0.65

3.89

0.43

4.08

1.61

0.18

0.65

6.79

16.17

4.15

18.69

9.34

14.49

6.24

17.73

2.23

0.61

0.13

0.49

3.42

2.70

2.54

1.52

Root-mean-square deviation from trend (in percentage points) 2.57 2.39 2.59 1.01 0.96 1.37 0.96 0.52 2.19

2.46

1.91 1.87

1. The sample autocorrelation coefficients up to order 5, the sample correlation, and cross-correlation matrices (with single-quarter shift) were estimated for the observable variables based on statistical data. 2. The sample autocorrelation coefficients up to order 5, the sample correlation, and cross-correlation matrices (with single-quarter shift) were estimated for the corresponding data generated by Model 2. 3. The conditions under which the differences between the sample moments of the observed data and those generated by Model 2 belonged to a given range. Tables 3.3, 3.4 and 3.5 present the moment characteristics calculated for Kazakhstan.

3.3 Applicability Testing of Model 2

281

Table 3.3 Differences between the sample autocorrelation coefficients of observed and generated economic indexes (in %) Economic index

First-order autocorrelation

Second-order autocorrelation

Third-order autocorrelation

Fourth-order autocorrelation

Fifth-order autocorrelation

17.97

12.02

11.08

bckz t

10.79

bxkz t bykz t

14.96

6.13

17.35

17.31

19.10

5.86

0.80

0.60

9.87

11.95

18.09

d exp t

kz

6.63

7.24

7.08

2.51

12.20

d kz emp t b kz w t b π kz t brkz t bskz rw,t kz b bt b kz ge t b kz gr t

12.61

16.69

5.40

10.80

9.03

15.46

3.45

3.97

14.49

16.31

14.31

7.70

8.04

2.52

17.37

7.15

19.53

13.88

1.90

18.21

1.57

11.95

15.95

1.69

19.14

10.04

3.96

14.68

12.64

7.78

19.22

6.35

0.76

6.51

7.44

13.58

4.77

4.65

7.02

3.43

Comparative study of the observed and calculated moments indicates that both setups of Model 2 (the original and estimated ones) are of high adequacy.

3.3.4

Testing with Impulse Responses

Module 4 of the software estimates the impulse responses of Model 2—the deviations of different variables from their equilibrium values under the effect of one noise (shock) equal to its standard deviation. (Note that all other noises of Model 2 as well as this noise at all subsequent times are set to 0.) The resulting impulse responses can be analyzed for consistency with economic theory. As an example, Fig. 3.1 demonstrates the impulse responses of Kazakhstan’s GDP and inflation to two shocks of economic policy, namely, fiscal and monetary shocks. Clearly, these impulse responses agreed with the well-known postulates of macroeconomics.

3.3.5

Testing with Local Sensitivity Analysis

Consider some endogenous variable   Z(θ) of Model 2 in a neighborhood of the parameter vector θ ¼ θ1 , . . . , θn , where n denotes the number of parameters, with given t. For estimating its sensitivity to a parameter θi ði ¼ 1, . . . , nÞ, Module 5 of the software calculates the sensitivity coefficient as the limit value of the variable

kz d exp t kz dt emp b kz w t b π kz t brkz t bskz rw,t kz b bt b kz ge t b kz gr t

bckz t bxkz t bykz t

0.00

kz

15.68

4.24

24.72

6.97

1.55 0.34

7.47

0.74

13.66

4.45

16.29

13.69 7.93

21.94 2.60

16.00

5.63

6.64

17.04

15.94

7.70

21.70

23.81

7.93

15.91

6.93

13.39

20.21

16.47

14.19

1.83

11.09

11.18

11.33

15.15

2.80

15.08

8.92

19.21

0.00

16.64

12.89

7.77 0.00

18.18

15.13

10.69

9.55

1.24 5.48 6.85

b π kz t 13.86

b kz w t

6.21

7.51

0.00

4.03

7.90

7.70

kz

dt emp 12.05

6.49

15.25

0.00

9.75

23.06

d exp t 16.42

19.48

20.54

7.22

16.09

24.90

0.89

9.85

10.82

14.84

7.55

1.44

9.83

7.12

0.00

1.79

bykz t

17.43

bxkz t 14.00

bckz t 0.00

8.34

8.23

12.44

7.41

0.00

2.86

1.60

0.60

0.74

8.96

21.92

br kz t 9.93

2.13

3.47

0.90

0.00

16.21

8.82

12.90

11.96

8.10

3.03

0.54

bskz rw,t 16.19

16.37

13.06

0.00

0.31

5.92

12.24

1.02

21.41

12.84

10.89

15.13

kz b bt 10.28

12.49

0.00

13.88

23.43

8.93

24.90

7.16

2.74

11.61

11.86

18.93

b kz ge t 3.99

0.00

19.14

7.24

16.30

10.15

2.06

7.02

12.53

11.89

16.05

23.18

b kz gr t 15.09

Table 3.4 Differences between the corresponding elements of correlation matrices calculated using observed and generated economic indexes (in %)

282 3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

kz d exp t kz dt emp b kz w t b π kz t br kz t bskz rw,t kz b bt b kz ge t b kz gr t

bckz t bxkz t bykz t

12.02

12.60

5.82

15.22

1.65

8.49

9.77

9.58

23.71

13.41

3.06

15.21

9.26

0.38

4.24

13.45

6.90

0.90

16.63 7.94

14.24

11.41

15.35

11.54

14.76

13.44

12.29

1.23

10.33

13.68

13.09

0.00

11.07

12.48

0.00

9.64

1.63

15.62

1.23

20.06

0.77

4.14

9.83

0.00

0.00

kz

d exp t 7.79

bykz t 23.88

bxkz t 12.69

bckz t

12.09

13.09

8.08

6.29

7.55

2.82

1.51

0.00

1.33

5.16

15.76

kz

dt emp 6.61

14.85

9.16

13.23

11.69

0.33

14.71

0.00

3.76

4.53

5.62

6.20

b kz w t 9.42

13.55

13.37

15.09

12.06

12.35

0.00

8.78

3.01

16.64

15.37

0.98

b π kz t 8.74

19.62

3.90

23.92

2.52

0.00

6.57

8.13

14.32

8.56

8.05

9.88

br kz t 12.27

8.40

10.79

0.80

0.00

11.30

6.83

11.04

15.84

10.45

10.70

8.91

bskz rw,t 0.20

12.61

14.07

0.00

13.43

4.59

11.26

15.88

16.63

22.91

14.60

2.96

kz b bt 10.63

12.69

0.00

21.05

18.21

8.43

11.49

13.22

16.71

3.47

15.09

22.01

b kz ge t 9.65

0.00

8.76

1.17

11.35

16.61

16.02

1.12

3.01

7.03

2.41

23.89

b kz gr t 2.09

Table 3.5 Differences between the corresponding elements of cross-correlation matrices calculated using observed and generated economic indexes (in %)

3.3 Applicability Testing of Model 2 283

284

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Fig. 3.1 Impulse responses of Kazakhstan’s GDP (upper graphs) and inflation (lower graphs) to fiscal and monetary shocks in economic policy of Kazakhstan, Russia, and the Rest of World

    Z θ1 , . . . , θi þ δθi , . . . , θn  Z θ1 , . . . , θn   SI i ¼ , δZ θ1 , . . . , θn

ð3:287Þ

3.3 Applicability Testing of Model 2

285

Table 3.6 Estimated sensitivity coefficients Regional parameter Price shock for exported commodities Technological coefficient shock for gross output Share of intermediate products in GDP Share of consumption products in GDP Share of investment products in GDP Share of exported products in GDP Share of government consumption in government expenditures Effective rate of CIT

Region Kazakhstan 0.02198 0.8699 0.3954 0.0084 0.0106 1.8996 0.3482

Russia 0.0457 1.1241 0.5314 0.0173 0.0103 1.2581 0.1459

Belarus 0.0147 0.7026 0.2746 0.0122 0.0085 1.1170 0.5579

0.0009

0.0079

0.0114

as δ ! 0. Table 3.6 contains the sensitivity coefficients of regional GDPs (Kazakhstan, Russia, and Belarus) to several exogenous parameters estimated using Model 2. As easily observed, the estimated sensitivity coefficients were consistent with the well-known postulates of macroeconomics.

3.3.6

Testing with Marginal Likelihood Comparison

For the basic and other (simplified) setups of Model 2 estimated on observed data, Module 6 of the software calculates the so-called marginal likelihoods [42] M¼

ð   obs obs obs P b S1 , b S2 , . . . , b ST jθ Ppr ðθÞdθ:

ð3:288Þ

Ωθ

The notations are the following: Ωθ as the domain of the parameter vector θ;  obs obs obs b P S1 , b S2 , . . . , b ST jθ as the likelihood function obtained by the Kalman filter using statistical data; Ppr(θ) as the prior probability density function of the model obs obs obs parameters; and finally, b S1 , b S2 , . . . , b ST as the observed values. The greater is the value of the marginal likelihood (2.3), the better Model 2 fits the observed data. The marginal likelihoods were estimated in Dynare following the approach [42]. Module 6 was employed for comparing the marginal likelihoods М in the basic and several simplified setups of Model 2, i.e.: • • • •

Model 2 without Banking Sector Model 2 with Banking Sector without capital flows Model 2 without comprehensive taxation system Model 2 without intermediate consumption

These setups of Model 2 were assessed for “viability” using the same collection of statistical data. The estimated marginal likelihoods can be found in Table 3.7.

286

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Table 3.7 Marginal likelihoods estimated for different setups of Model 2 Setup Model 2 without Banking Sector Model 2 with Banking Sector without capital flows Model 2 without comprehensive taxation system Model 2 without intermediate consumption Basic setup of Model 2

Value М 8 124.12 9 478.43 9 962.97 7 419.48 11 572.60

In accordance with Table 3.7, the complete (basic) setup of Model 2 was best in terms of marginal likelihood. The simulation results of the nonlinear Model 2 with the six testing methods proved that it was applicable to real macroeconomic systems.

3.4 3.4.1

Mid-term Forecasting and Macroeconomic Analysis Based on Model 2 Knoware and Software for Forecasting and Macroeconomic Analysis

The developed knoware for mid-term forecasting based on the nonlinear Model 2 (3.277) with estimated parameters includes several aggregate steps as follows. Step 1. Generate all shocks for mid-term period using the vector autoregression Ξt ¼ ρΞ Ξt1 þ H t :

ð3:289Þ

The notations are the following: Ξt as the shocks vector of Model 2 for quarter t, where t ¼ 0 corresponds to the last quarter of the retrospective period (Q4 2014); ρΞ as the matrix of all autoregression coefficients; and Ht as the vector of independent normally distributed noises with zero means and the standard deviations vector Σ Η. Step 1 consists of three substeps described below. Substep 1.1. Specify the values of the shock parameters ρΞ and Σ Η obtained during parameter estimation of Model 2. Substep 1.2. For t ¼ 0 (the last quarter of the retrospective period—Q4 2014), specify the initial conditions in form of the shocks vector Ξ0 obtained during shocks estimation on the retrospective period. Substep 1.3. For t ¼ 1, . . ., T, where T denotes the number of quarters for mid-term forecasting: • Generate the noises vector using the standard normal randomizer. • Perform the elementwise multiplication of the noises vector and the vector Σ Η. • Obtain a realization of the model shocks using expression (3.289).

3.4 Mid-term Forecasting and Macroeconomic Analysis Based on Model 2

287

Step 2. Specify the initial conditions of Model 2 for mid-term forecasting. This step consists of two substeps described below: Substep 2.1. Calculate the values of all endogenous variables on the retrospective period after parameter estimation. Substep 2.2. Accept the values of the endogenous variables for the last quarter of the retrospective period (t ¼ 0) as the initial values for mid-term forecasting. Step 3. Construct a mid-term forecast based on Model 2 with the generated shocks vectors using the Fair–Taylor method for solving nonlinear dynamic models with rational means. Implement this procedure in Dynare. Step 4. Add the values of the endogenous variables obtained at Step 3 to the corresponding forecast trends calculated using the IMF database [80]. The described knoware was adopted to develop the corresponding software for mid-term forecasting based on Model 2. This software calculates all endogenous variables of Model 2 (and their rates of change) in domestic currencies, for the baseline scenario and also for forecasted scenarios with variations of a given collection of its exogenous variables from their values obtained during calibration of Model 2. The variables considered included the following: international trade indexes; international capital flow indexes; and economic indexes associated with the output and consumption of three types of products at the levels of Regions, economic unions, and separate Countries, with specification by three Sectors. In particular, this software can be used to perform scenario analysis (at the levels of the global economy, economic unions, and separate countries, with specification by three Sectors) for estimating • The effects from establishing free-trade zones between the EAEU and the EU • The effects from export price variations for commodities, etc. • The effects from establishing monetary zones within the EAEU, etc.

3.4.2

Solution of Basic Mid-term Forecasting Problems

For each of the nine Regions of Model 2, a series of mid-term forecasting problems up to year 2022 were stated and solved with the following indexes: • • • • • •

Gross domestic product Aggregate investments Households’ expenditures on final consumption Employed population Public authorities’ expenditures on final consumption Financial indexes such as aggregate loans, aggregate deposits, average loan rate, and average deposit rate

(Note that these problems were coordinated with the existing IMF forecasts [80] using the available trends of macroeconomic indexes.)

288

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Figures 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 and 3.9 show the expected values and rootmean-square deviations of the forecasts for Kazakhstan’s economy. In particular, the forecasts demonstrated a 0.9% increase of Kazakhstan’s GDP for year 2016, followed by a 1.9% increase for years 2017 and 2018, with gradual growth to 3.0% by year 2021; a 0.8% drop of Russia’s GDP for year 2016, followed by a 0.1% increase for year 2017, with gradual growth to 1.9% by year 2021; and a 1.3% drop of Belarus’ GDP for year 2016, followed by a 0.9% drop for year 2017 and a 0% increase for year 2018, with gradual growth to 1.5% by year 2022. In addition, the forecasts demonstrated a 1.5% increase of households’ consumption in Kazakhstan for year 2016, with gradual growth to 3.1% by year 2022; a 0.2% drop of households’ consumption in Russia for year 2016, followed by a 0% increase for year 2017, with gradual growth to 1.6% by year 2021; and a 0.5% drop of households’ consumption in Belarus for year 2016, with gradual growth to 2.0% by year 2022.

Fig. 3.2 GDP forecasting for Kazakhstan

Fig. 3.3 Aggregate investments forecasting for Kazakhstan

3.4 Mid-term Forecasting and Macroeconomic Analysis Based on Model 2

289

Fig. 3.4 Final consumption expenditures forecasting for Kazakhstan’s households

Fig. 3.5 Employed population forecasting for Kazakhstan

3.4.3

Solution of Snapshot Macroeconomic Analysis Problems

In the course of snapshot ex post and ex ante macroeconomic analysis, the volatility of macroeconomic indexes—GDP and inflation—was expanded with respect to the effects of internal and external shocks for each of the nine regions. (Note that snapshot analysis, also known as passive analysis, implies no scenario variations of exogenous parameters.) Figures 3.10, 3.11, 3.12 and 3.13 present the volatilities

290

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Fig. 3.6 Final consumption expenditures forecasting for Kazakhstan’s public authorities

Fig. 3.7 Loans forecasting for Kazakhstan

Fig. 3.8 Deposits forecasting for Kazakhstan

3.4 Mid-term Forecasting and Macroeconomic Analysis Based on Model 2

291

Fig. 3.9 Average loan and deposit rates forecasting for Kazakhstan

Shock 15 Shock 14 Shock 13 Shock 12 Shock 11

Shock 10 Shock 9 Shock 8 Shock 7 Shock 6 Shock 5 Shock 4 Shock 3 Shock 2 Shock 1 Stascs

Fig. 3.10 The effect of most influential shocks on Kazakhstan’s GDP on retrospective period (deviation from trend in %)

of Kazakhstan’s GDP and inflation expanded with respect to the effects of most influential shocks on the retrospective and forecasting periods as follows: Shock 1—The productivity shock for the Sector of commodities Shock 2—The productivity shock for the Sector of tradable non-commodities Shock 3—The productivity shock for the Sector of non-tradable goods Shock 4—The consumption preference shock for households Shock 5—The government expenditures shock Shock 6—The labor supply shock Shock 7—The land supply shock for the Sector of commodities Shock 8—The land supply shock for the Sector of tradable non-commodities Shock 9—The land supply shock for the Sector of non-tradable goods

292

3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . . Shock 15 Shock 14 Shock 13 Shock 12 Shock 11 Shock 10 Shock 9 Shock 8 Shock 7 Shock 6 Shock 5 Shock 4 Shock 3 Shock 2 Shock 1 Stascs

Fig. 3.11 The effect of most influential shocks on Kazakhstan’s inflation on retrospective period (deviation from trend in %) Shock 15 Shock 14 Shock 13 Shock 12 Shock 11 Shock 10 Shock 9 Shock 8 Shock 7 Shock 6 Shock 5 Shock 4 Shock 3 Shock 2 Shock 1

Fig. 3.12 The effect of most influential shocks on Kazakhstan’s GDP on forecasting period (deviation from trend in %)

Shock 10—The government rates shock Shock 11—The investments shock for the Sector of commodities Shock 12—The investments shock for the Sector of tradable non-commodities Shock 13—The investments shock for the Sector of non-tradable goods Shock 14—The investments shock for households Shock 15—The world price shock for commodities In accordance with the simulation results, Kazakhstan’s GDP mostly suffered from the following shocks: the world price shock for commodities, the productivity

3.4 Mid-term Forecasting and Macroeconomic Analysis Based on Model 2

293

Shock 15 Shock 14 Shock 13 Shock 12 Shock 11 Shock 10 Shock 9 Shock 8 Shock 7 Shock 6 Shock 5 Shock 4 Shock 3 Shock 2 Shock 1

Fig. 3.13 The effect of most influential shocks on Kazakhstan’s inflation on forecasting period (deviation from trend in %)

shocks, the consumption preference shock for households, the government expenditures shock, and the labor supply shock. On the other hand, Kazakhstan’s inflation was mostly influenced by the following shocks: the government expenditures shock, the consumption preference shock for households, the world price shock for commodities, the labor supply shock, and the productivity shocks.

3.4.4

Solution of Scenario Macroeconomic Analysis Problems

In the course of scenario (active) ex post and ex ante macroeconomic analysis, some economic indexes were estimated on the retrospective period 2000–2014 and also on the mid-term forecasting period 2015–2022. In particular, the following assessments were obtained: • The effects from reducing taxes, increasing public bonds rates, establishing a free-trade zone between the EAEU and EU, establishing a monetary zone, increasing the world prices for commodities, increasing the macroprudential parameter for banks, and increasing the macroprudential parameter for the firms supplying non-tradable products since year 2015 (see Table 3.8) • The measures on reducing the dollarization of developing economics, by the example of Kazakhstan (see Table 3.9) • The influence of public debt on capital flows, by the example of Kazakhstan (see Table 3.10) In accordance with Table 3.8,

Period (quarter, year) Q1 2015 Q2 2015 Q3 2015 Q4 2015 Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018

Baseline scenario 3 236.04 3 236.71 3 237.22 3 238.83 3 245.36 3 256.60 3 271.54 3 286.93 3 301.64 3 317.41 3 334.32 3 351.80 3 367.47 3 381.90

Scenario

10% decrease of IIT and CIT 3 242.16 (0.19%) 3 243.29 (0.20%) 3 244.54 (0.23%) 3 244.30 (0.17%) 3 249.57 (0.13%) 3 257.55 (0.03%) 3 266.04 (0.17%) 3 275.71 (0.34%) 3 288.51 (0.40%) 3 303.54 (0.42%) 3 319.06 (0.46%) 3 333.69 (0.54%) 3 349.20 (0.54%) 3 365.86 (0.47%)

3% increase of benchmark interest rate 3 197.90 (1.18%) 3 219.31 (0.54%) 3 239.58 (0.07%) 3 256.72 (0.55%) 3 268.90 (0.73%) 3 277.01 (0.63%) 3 282.35 (0.33%) 3 286.94 (0.00%) 3 294.85 (0.21%) 3 305.56 (0.36%) 3 317.30 (0.51%) 3 330.80 (0.63%) 3 346.18 (0.63%) 3 363.26 (0.55%) Free-trade zone with EAEU 3 194.84 (1.27%) 3 220.05 (0.51%) 3 244.32 (0.22%) 3 265.09 (0.81%) 3 278.11 (1.01%) 3 284.38 (0.85%) 3 286.34 (0.45%) 3 288.92 (0.06%) 3 296.13 (0.17%) 3 307.05 (0.31%) 3 322.93 (0.34%) 3 342.43 (0.28%) 3 360.90 (0.20%) 3 378.11 (0.11%)

100% increase of oil price 3 297.72 (1.91%) 3 309.88 (2.26%) 3 339.02 (3.14%) 3 371.33 (4.09%) 3 408.87 (5.04%) 3 446.55 (5.83%) 3 485.49 (6.54%) 3 526.36 (7.28%) 3 572.58 (8.21%) 3 620.01 (9.12%) 3 669.08 (10.04%) 3 721.78 (11.04%) 3 777.11 (12.16%) 3 833.49 (13.35%)

Establishment of monetary union 3 239.10 (0.09%) 3 244.08 (0.23%) 3 248.72 (0.36%) 3 253.94 (0.47%) 3 260.39 (0.46%) 3 265.99 (0.29%) 3 271.91 (0.01%) 3 278.98 (0.24%) 3 290.74 (0.33%) 3 306.23 (0.34%) 3 323.40 (0.33%) 3 341.86 (0.30%) 3 358.39 (0.27%) 3 372.98 (0.26%)

Table 3.8 GDP forecasts for Kazakhstan in different scenarios (in billion KZT) and their variations from basic forecast (in %)

Increase of macroprudential parameter for banks 3 240.01 (0.12%) 3 242.72 (0.14%) 3 243.93 (0.16%) 3 246.01 (0.17%) 3 252.78 (0.18%) 3 264.04 (0.18%) 3 278.88 (0.17%) 3 294.14 (0.17%) 3 308.66 (0.16%) 3 324.08 (0.15%) 3 340.33 (0.14%) 3 356.95 (0.12%) 3 371.85 (0.10%) 3 385.76 (0.09%)

Increase of macroprudential parameter for Sector of non-tradable goods 3 237.69 (0.05%) 3 238.47 (0.05%) 3 239.09 (0.06%) 3 240.79 (0.06%) 3 247.40 (0.06%) 3 258.71 (0.06%) 3 273.70 (0.07%) 3 289.13 (0.07%) 3 303.86 (0.07%) 3 319.62 (0.07%) 3 336.49 (0.07%) 3 353.92 (0.06%) 3 369.52 (0.06%) 3 383.89 (0.06%)

Q3 2018 Q4 2018 Q1 2019 Q2 2019 Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021

3 394.91 3 408.04 3 424.70 3 443.32 3 466.42 3 493.86 3 522.68 3 550.53 3 579.29 3 608.59 3 638.14 3 638.14 3 663.52 3 684.44

3 381.97 (0.38%) 3 397.96 (0.30%) 3 415.79 (0.26%) 3 435.84 (0.22%) 3 457.79 (0.25%) 3 483.76 (0.29%) 3 512.15 (0.30%) 3 543.63 (0.19%) 3 577.48 (0.05%) 3 611.86 (0.09%) 3 644.08 (0.16%) 3 644.08 (0.16%) 3 672.84 (0.25%) 3 698.00 (0.37%)

3 380.21 (0.43%) 3 398.73 (0.27%) 3 416.90 (0.23%) 3 433.62 (0.28%) 3 450.24 (0.47%) 3 470.04 (0.68%) 3 496.08 (0.75%) 3 526.66 (0.67%) 3 560.76 (0.52%) 3 595.88 (0.35%) 3 629.16 (0.25%) 3 629.16 (0.25%) 3 658.11 (0.15%) 3 681.24 (0.09%)

3 393.51 (0.04%) 3 409.62 (0.05%) 3 427.39 (0.08%) 3 444.96 (0.05%) 3 465.07 (0.04%) 3 489.10 (0.14%) 3 515.27 (0.21%) 3 541.02 (0.27%) 3 569.15 (0.28%) 3 600.36 (0.23%) 3 633.54 (0.13%) 3 633.54 (0.13%) 3 665.83 (0.06%) 3 695.81 (0.31%)

3 892.13 (14.65%) 3 952.39 (15.97%) 4 010.39 (17.10%) 4 065.39 (18.07%) 4 118.82 (18.82%) 4 175.09 (19.50%) 4 233.22 (20.17%) 4 288.13 (20.77%) 4 341.50 (21.30%) 4 397.74 (21.87%) 4 456.83 (22.50%) 4 456.83 (22.50%) 4 513.18 (23.19%) 4 567.83 (23.98%)

3 387.39 (0.22%) 3 404.26 (0.11%) 3 424.05 (0.02%) 3 446.13 (0.08%) 3 470.91 (0.13%) 3 498.70 (0.14%) 3 527.96 (0.15%) 3 556.69 (0.17%) 3 586.12 (0.19%) 3 614.85 (0.17%) 3 642.51 (0.12%) 3 642.51 (0.12%) 3 663.80 (0.01%) 3 677.57 (0.19%)

3 398.49 (0.08%) 3 411.53 (0.08%) 3 428.20 (0.08%) 3 446.90 (0.08%) 3 470.12 (0.08%) 3 497.64 (0.08%) 3 526.47 (0.08%) 3 554.25 (0.08%) 3 582.99 (0.08%) 3 612.39 (0.08%) 3 642.07 (0.08%) 3 642.07 (0.08%) 3 667.55 (0.08%) 3 688.52 (0.09%)

3 396.84 (0.06%) 3 409.93 (0.06%) 3 426.56 (0.05%) 3 445.15 (0.05%) 3 468.21 (0.05%) 3 495.62 (0.05%) 3 524.39 (0.05%) 3 552.20 (0.05%) 3 580.92 (0.05%) 3 610.20 (0.04%) 3 639.74 (0.04%) 3 639.74 (0.04%) 3 665.11 (0.04%) 3 686.02 (0.04%)

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3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Table 3.9 Forecasted share of USD deposits in aggregate deposits and forecasted USD deposit rate

Period (quarter, year) Q1 2015 Q2 2015 Q3 2015 Q4 2015 Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018 Q3 2018 Q4 2018 Q1 2019 Q2 2019 Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021

Scenario Scenario 1 Share of USD deposits 55.00 55.18 55.37 55.55 55.73 55.91 56.09 56.26 56.41 56.54 56.64 56.70 56.72 56.73 56.73 56.75 56.76 56.78 56.79 56.80 56.80 56.80 56.81 56.81 56.81 56.81 56.81 56.81

Scenario 2 Share of USD deposits 55.00 56.26 57.51 58.71 59.86 60.94 61.96 62.91 63.80 64.63 65.37 66.02 66.60 67.12 67.59 68.00 68.38 68.70 68.97 69.18 69.34 69.44 69.51 69.56 69.58 69.61 69.64 69.67

USD deposit rate 2.96 3.04 3.08 3.16 3.24 3.32 3.4 3.52 3.6 3.72 3.8 3.88 3.96 3.96 4.00 3.96 3.96 3.92 3.92 3.88 3.84 3.84 3.8 3.76 3.76 3.76 3.76 3.72

• By year 2021 Kazakhstan’s GDP was increasing by 0.5% in comparison with the basic forecast under a 10% decrease of the IIT and CIT since year 2015. • By year 2021 Kazakhstan’s GDP was decreasing by 0.06% in comparison with the basic forecast under a 3% increase of the public bonds rate since year 2015. • By year 2021 Kazakhstan’s GDP was increasing by 0.59% in comparison with the basic forecast under establishing a free-trade zone between the EAEU and EU since year 2015. • By year 2021 Kazakhstan’s GDP was increasing by 24.84% in comparison with the basic forecast under a 100% increase of the world oil prices since year 2015.

3.4 Mid-term Forecasting and Macroeconomic Analysis Based on Model 2 Table 3.10 Forecasted ratio of net capital inflow to GDP for Kazakhstan (in %)

Period (quarter, year) 1 Q1 2015 Q2 2015 Q3 2015 Q4 2015 Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018 Q3 2018 Q4 2018 Q1 2019 Q2 2019 Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021

Scenario Scenario 3 2 1.45 1.53 1.58 1.58 1.41 1.35 0.91 0.43 0.05 0.47 0.81 1.03 1.15 1.18 1.13 1.04 0.92 0.79 0.66 0.55 0.45 0.38 0.32 0.29 0.26 0.25 0.24 0.23

297

Scenario 4 3 1.62 1.61 1.58 1.53 1.41 1.22 0.94 0.60 0.26 0.03 0.23 0.34 0.39 0.38 0.35 0.30 0.25 0.21 0.17 0.14 0.11 0.09 0.06 0.04 0.02 0.02 0.02 0.02

• By year 2021 Kazakhstan’s GDP was decreasing by 0.49% in comparison with the basic forecast under establishing a currency zone within the EAEU since year 2015. • During the period 2015–2021, Kazakhstan’s GDP was increasing by 0.10% in comparison with the basic forecast under a 5 percentage points increase of the macroprudential parameter for banks since year 2015 (this is the restriction on the equilibrium value mrB of the bank’s borrowings-loans ratio defined in (3.165)). • During the period 2015–2021, Kazakhstan’s GDP was increasing by 0.06% in comparison with the basic forecast under a 5 percentage points increase of the macroprudential parameter for the firms supplying non-tradable products since year 2015 (this is the restriction on the equilibrium value mrx of the debt–equity capital ratio defined in (3.91)).

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3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

In addition, the scenario-based computer simulations with macroprudential parameters led to several conclusions as follows: • The above increase of the macroprudential parameter mrB for banks was causing the following variations of Kazakhstan’s indexes in comparison with the basic forecast on the period 2015–2021: investments, a 0.36% increase; consumption, a 0.18% increase; employment, a 0.21% increase; government expenditures, a 0.10% increase; transfers, 0.06% decrease; social tax revenues, a 0.05% increase; IIT revenues, a 0.18% increase; CIT revenues, a 0.53% increase; loans, a 0.53% increase; deposits, a 0.29% increase; loan rates, a 0.4% decrease; and deposit rates, a 0.1% increase. • The above increase of the macroprudential parameter mrx for the firms supplying non-tradable products was causing the following variations of Kazakhstan’s indexes in comparison with the basic forecast on the period 2015–2021: investments, a 0.27% increase; consumption, a 0.09% increase; employment, a 0.06% increase; government expenditures, a 0.33% increase; transfers, 0.05% decrease; social tax revenues, a 0.11% increase; IIT revenues, a 0.24% increase; CIT revenues, a 0.63% increase; loans, a 0.34% increase; deposits, a 0.11% increase; loan rates, a 0.11% increase; and deposit rates, a 0.03% increase. Two cases were considered in the course of scenario analysis for assessing the de-dollarization measures of a developing economy (see Table 3.9), namely, 2% deposit rate for USD deposits (scenario 1) and unlimited deposit rate for USD deposits (scenario 2). In accordance with Table 3.9, limited deposit rates for USD deposits actually reduced the dollarization of Kazakhstan’s economy. Moreover, the scenario forecasts of net capital inflow for Kazakhstan up to year 2021 were constructed under the basic value of net public debt (scenario 3) and the net public debt making up 60% of annual GDP (scenario 4); see Table 3.10. Clearly, by year 2021, the ratio of net capital inflow to GDP was increased by 0.23% in scenario 3 and by 0.02% in scenario 4.

3.5 3.5.1

Parametric Control Based on Model 2: A Series of Problem Statements and Their Solutions Problem Statements for Parametric Control

A series of optimal parametric control problems were considered on the basis of Model 2 at the levels of separate countries, regional unions, and the global economy. The objective function was defined as the convolution of several criteria toward economic growth, volatility suppression for macroeconomic indexes (GDP and inflation), as well as inflation targeting on the period between Q1 2016 and Q4 2022. The associated constraints were represented by Model 2 itself, the integration conditions within the EAEU (the restrictions on the difference of inflations rates for

3.5 Parametric Control Based on Model 2: A Series of Problem Statements and Their. . .

299

the EAEU members; on the budget gap-GDP ratio for the EAEU countries; on the public debt-GDP ratio for the EAEU countries), inflation growth restrictions, government expenditures growth restrictions, and the admissible range of public bonds rates. The basic public bond rates and government expenditures on the above period (Q1 2016–Q4 2022) were chosen as the government policy tools (control variables). The formal statements of these parametric control problems based on Model 2 are as follows. Problem DSGE Given a quarter t to start parametric control, a subset of Regions S 2 {S1,kaz, S1,rus, S1,blr, S1,arm, S1,kgz, S1,chn, S1,usa, S1,eu, S1,row, S2, S3} (where S1,j indicates the jth Region of Model 2 treated separately, S2 the set of all 5 EAEU countries, and S3 the set of all other Regions of Model 2), for all r 2 S find the values of the government policy tools ηrR,tþ1 , . . . , ηrR,tþn (the public bonds rates), ηrge,tþ1 , . . . , ηrge,tþn (government consumptions), ηrms,tþ1 , . . . , ηrms,tþn , ηrmb,tþ1 , . . . , ηrmb,tþn (the macroprudential parameters for banks), and ηrmX,tþ1 , . . ., ηrmX,tþn (the macroprudential parameters for the firms supplying non-tradable products) that minimize the objective function K r,t ¼ Et

X r2S

0

Xn

βi @Y rtþi i¼1

r

 α1

Y rtþi  Y tþi r Y tþi

!2

1  r  π tþi  π rtþi 2 A  α2 π rtþi ð3:290Þ

based on Model 2 written as   Et F θ X t1 , X t , X tþ1 , Η Σt Η ¼ 0,

ð3:291Þ

subject to the constraints (i ¼ 1, . . ., 40) Et π rtþi  π rtþi þ 0:025, r Rtþi

 0:025  Et Rrtþi  Et GE rtþi  1:05

r Rtþi

þ 0:025,

r GE tþi ,

ð3:292Þ ð3:293Þ ð3:294Þ

and also subject to the following constraints on the economic indexes of the EAEU countries r0 2 S dictated by the integration conditions within the EAEU (the Treaty on the EAEU): Et π r0tþi  Et π r} tþi þ 0:05,

ð3:295Þ

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3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Et

Def r0tþi  0:03, Y r0tþi

ð3:296Þ

Br0g,tþi  0:5: Y r0tþi

ð3:297Þ

Et

The other notations are the following: Et as mathematical expectation given available information for quarter t; Fθ as a known vector function that describes Model 2; θ as a vector consisting of the structural parameters of Model 2 and the autoregression parameters of the shocks; Xt as a vector consisting of the endogenous variables and shocks defined by first-order autoregressions (X0 is given); Η Σt Η as a vector consisting of Gaussian white noises; ΣΗ as a vector consisting of the standard deviations of these noises; Y rt , π rt , Def rt , Brg,t , and RrP,us,t as the GDP, consumer price inflation, budget gap, public debt, and deposit rate for USD deposits of country r, respectively; “  ” as the trend values of these variables (or targeted level in case of r r inflation); α1 ¼ Y t and α2 ¼ 5Y t as the weight coefficients of the objective function associated with country r; β as the discount factor; n ¼ 40 as the foresight parameter of government policy (with a given value); and finally, r0 and r" (r0 6¼ r") as EAEU countries. For macroeconomic analysis based on Model 2, a simplified setup of the problem DSGE was also considered in which the government policy tools included only ηrR,tþ1 , . . . , ηrR,tþn (the public bonds rates) and ηrge,tþ1 , . . . , ηrge,tþn (government consumptions).

3.5.2

Solutions of Parametric Control Problems

These parametric control problems based on Model 2 were solved in MATLAB using an original nonlinear optimization method with confidence domains [62]. To this end, the problem DSGE was reduced to a special form as follows. Problem DSGE1 Given a quarter t to start parametric control, a subset of Regions S 2 {S1,kaz, S1,rus, S1,blr, S1,arm, S1,kgz, S1,chn, S1,usa, S1,eu, S1,row, S2, S3} (where S1, j indicates the jth Region of Model 2 treated separately, S2 the set of all 5 EAEU countries, and S3 the set of all other Regions of Model 2), for all r 2 S find the values of the government policy tools ηrR,tþ1 , . . . , ηrR,tþn (the public bonds rates), ηrge,tþ1 , . . . , ηrge,tþn (government consumptions), ηrms,tþ1 , . . . , ηrms,tþn , r r ηmb,tþ1 , . . . , ηmb,tþn (the macroprudential parameters for banks), and ηrmX,tþ1 , . . ., ηrmX,tþn (the macroprudential parameters for the firms supplying non-tradable products) that minimize the objective function Kr, t (3.290) based on Model 2 written as

3.5 Parametric Control Based on Model 2: A Series of Problem Statements and Their. . .

  X tþi ¼ Qθ ηrR,t , . . . , ηrR,tþn , ηrge,tþ1 , . . . , ηrge,tþn , X t , , ði ¼ 1, . . . , 40Þ,

301

ð3:298Þ

subject to the additional constraints eπ rtþi  π rtþi þ 0:025,

ð3:299Þ

eRrtþi 

r Rtþi

þ 0:025,

ð3:300Þ

eRrtþi 

r Rtþi

 0:025,

ð3:301Þ

r 1:05GE tþi ,

ð3:302Þ

eGE rtþi 

and also to the following constraints on the economic indexes of EAEU countries r0, r00 2 S dictated by the integration conditions within the EAEU (the Treaty on the EAEU): eDef toY rtþi  0:03,

0

ð3:303Þ

0

ð3:304Þ

eπ rtþi  eπ rtþi þ 0:05, r 0 6¼ r 00

ð3:305Þ

eBgtoY rtþi  0:5, 0

00

Here the new variables eπ rtþi , eRrtþi , eGE rtþi , eDeftoY rtþi , and eBgtoY rtþi (i ¼ 1, . . ., 40) have the form eπ rtþi ¼ Et π rtþi , eRrtþi ¼ Et Rrtþi , eGE rtþi ¼ Et GE rtþi , eDeftoY rtþi ¼ Et

Def rtþi Y rtþi ,

and eBgtoY rtþi ¼ Et

Brg,tþi Y rtþi .

The transformed problem DSGE1 was solved in MATLAB using the nonlinear optimization method with confidence domains. The problem DSGE*1 was stated and solved similar to the problem DSGE*. Consider the solutions of the parametric control problems DSGE and DSGE* toward economic growth and volatility suppression for economic indexes based on Model 2 at the levels of Kazakhstan, the EAEU, and the global economy. Table 3.11 presents the solutions of these parametric control problems in form of deviations (in %) from the corresponding basic average values of the following macroeconomic indexes of Kazakhstan on the forecasting period (Q1 2016–Q4 2022): real GDP, the volatility of real GDP, and the volatility of consumer price inflation. Table 3.11 Variations of Kazakhstan’s macroeconomic indexes (in %) Problem DSGE

Problem DSGE* Index Real GDP Volatility of real GDP Volatility of consumer price inflation

Kazakhstan 0.35 14.13 8.30

EAEU 0.53 15.74 12.59

Global economy 0.68 17.62 13.45

Kazakhstan 1.29 18.86 37.79

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Table 3.12 Optimal values of economic policy tools in problem DSGE* at the level of Kazakhstan (% deviations from basic values) Period (quarter, year) Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018 Q3 2018 Q4 2018 Q1 2019 Q2 2019

Tool Public bonds rate 18.83 22.81 18.38 20.00 17.18 19.83 22.42 19.07 22.11 18.68 10.73 10.79 11.37 11.55

Government expenditures 8.50 1.78 3.91 4.17 4.34 2.80 4.18 1.48 0.79 2.44 1.06 2.58 6.07 3.54

Period (quarter, year) Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021 Q1 2022 Q2 2022 Q3 2022 Q4 2022

Tool Public bonds rate 13.59 9.13 8.29 8.02 6.03 8.22 9.40 8.74 8.84 9.25 8.47 9.24 8.79 9.30

Government expenditures 6.68 7.14 8.60 2.28 2.32 2.33 6.00 1.05 6.68 4.54 1.66 5.96 6.41 3.13

Next, Tables 3.12, 3.13 and 3.14 illustrate the solutions of the parametric control problems DSGE* at the levels of Kazakhstan, the EAEU, and the global economy in form of deviations (in %) of the economic policy tools (the public bonds rate and government expenditures) from the corresponding basic values. As indicated by these solutions of the parametric control problems, for each separate country, optimal economic policy had higher efficiency in case of coordination at the level of the global economy than at the level of a regional union; in turn, coordination at the level of a regional union guaranteed higher efficiency of optimal economic policy in comparison with the level of a separate country. Moreover, the macroprudential parameters should be included in the list of government policy tools considered in the parametric control problems in order to improve efficiency; see Table 3.11.

3.5.3

Applicability Testing of Solutions

The solutions of the parametric control problems DSGE based on Model 2 at different levels (Kazakhstan, the EAEU, and the global economy) were tested for their implementability using the two approaches described in Sect. 3.3.1, i.e., (a) by estimating the stability indexes of the mappings defined by Model 2 in the optimal scenarios corresponding to the solutions of the parametric control problems and (b) by estimating the sensitivity of the endogenous variables of Model 2 with respect to its exogenous parameters in these scenarios.

3.5 Parametric Control Based on Model 2: A Series of Problem Statements and Their. . .

303

Table 3.13 Optimal values of economic policy tools in problem DSGE* at the level of the EAEU (% deviations from basic values) Period (quarter, year) Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018 Q3 2018 Q4 2018 Q1 2019 Q2 2019

Tool Public bonds rate 18.75 24.42 17.13 18.48 17.41 20.89 23.56 19.33 23.37 20.02 9.74 11.47 10.78 10.50

Government expenditures 7.94 1.87 3.72 3.84 4.38 2.63 4.48 1.45 0.80 2.23 1.16 2.51 6.56 3.76

Period (quarter, year) Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021 Q1 2022 Q2 2022 Q3 2022 Q4 2022

Tool Public bonds rate 13.61 8.67 8.84 7.97 6.53 7.73 9.75 9.14 8.41 9.26 9.29 9.18 8.21 9.12

Government expenditures 7.23 7.70 7.78 2.17 2.21 2.38 6.26 0.95 6.01 4.85 1.69 5.95 6.68 2.84

Table 3.14 Optimal values of economic policy tools in problem DSGE* at the level of the global economy (% deviations from basic values) Period (quarter, year) Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018 Q3 2018 Q4 2018 Q1 2019 Q2 2019

Tool Public bonds rate 19.02 23.10 16.94 20.05 17.08 18.77 24.44 19.62 20.34 18.63 10.23 10.64 11.11 10.99

Government expenditures 8.16 1.64 3.65 3.78 4.58 2.92 4.34 1.62 0.71 2.33 1.08 2.56 5.60 3.59

Period (quarter, year) Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021 Q1 2022 Q2 2022 Q3 2022 Q4 2022

Tool Public bonds rate 13.40 9.31 7.91 8.07 6.37 7.96 8.87 8.40 8.15 10.17 8.94 9.01 8.53 9.13

Government expenditures 6.24 7.55 8.00 2.41 2.16 2.32 6.10 1.12 6.52 4.47 1.71 6.02 6.86 2.88

In the course of testing with stability indexes and sensitivity, all estimated parameters were treated as the input parameters of mappings while the forecasts of all observable variables for all Regions of Model 2 as the output variables (for a given quarter). Tables 3.15, 3.16 and 3.17 show the values of the stability indexes

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Table 3.15 Values of stability indexes (the level of Kazakhstan) Period (quarter, year) Index β( p, α) Period (quarter, year) Index β( p, α)

Q1 2016 2.96 Q3 2018 1.01

Q2 2016 2.91 Q4 2018 0.86

Q3 2016 3.16 Q1 2019 0.63

Q4 2016 3.09 Q2 2019 0.23

Q1 2017 2.48 Q3 2019 0.02

Q2 2017 2.41 Q4 2019 0.00

Q3 2017 1.81 Q1 2020 0.00

Q4 2017 1.57 Q2 2020 0.00

Q1 2018 1.36 Q3 2020 0.00

Q2 2018 1.34 Q4 2020 0.00

Q3 2017 1.64 Q1 2020 0.00

Q4 2017 1.47 Q2 2020 0.00

Q1 2018 1.47 Q3 2020 0.00

Q2 2018 1.39 Q4 2020 0.00

Q4 2017 1.54 Q2 2020 0.00

Q1 2018 1.37 Q3 2020 0.00

Q2 2018 1.42 Q4 2020 0.00

Table 3.16 Values of stability indexes (the level of the EAEU) Period (quarter, year) Index β( p, α) Period (quarter, year) Index β( p, α)

Q1 2016 3.17 Q3 2018 0.92

Q2 2016 2.90 Q4 2018 0.85

Q3 2016 3.35 Q1 2019 0.66

Q4 2016 3.16 Q2 2019 0.21

Q1 2017 2.44 Q3 2019 0.02

Q2 2017 2.50 Q4 2019 0.00

Table 3.17 Values of stability indexes (the level of the global economy) Period (quarter, year) Index β( p, α) Period (quarter, year) Index β( p, α)

Q1 2016 3.19 Q3 2018 1.01

Q2 2016 2.97 Q4 2018 0.94

Q3 2016 3.24 Q1 2019 0.60

Q4 2016 3.10 Q2 2019 0.22

Q1 2017 2.68 Q3 2019 0.02

Q2 2017 2.62 Q4 2019 0.00

Q3 2017 1.80 Q1 2020 0.00

β( p,α) (the basic value p and α ¼ 0.01) for the solutions of the parametric control problems DSGE at the levels of Kazakhstan, the EAEU, and the global economy, respectively. In accordance with these tables, the stability index β( p, α) had admissible values (not exceeding 3.19) for all solutions of the parametric control problems based on Model 2, which indicated that Model 2 with the calculated optimal values of government economic policy tools up to Q4 2020 was rather stable. Next, Table 3.18 contains some estimates for the sensitivity coefficients (see Sect. 3.3.5) of the average expected GDPs of Kazakhstan, Russia, and Belarus on the same forecasting period (Q1 2016–Q4 2022) using the standard deviations of the exogenous variables of Model 2 for the solutions of the parametric control problems DSGE at the level of Kazakhstan. (In fact, up to the 4th decimal, they coincided with the corresponding values obtained by solving the problems DSGE at the levels of the EAEU and the global economy.) As easily observed, the estimated sensitivity coefficients were consistent with the well-known postulates of macroeconomics.

3.5 Parametric Control Based on Model 2: A Series of Problem Statements and Their. . .

305

Table 3.18 Estimated sensitivity coefficients for average expected GDPs of Kazakhstan, Russia, and Belarus using the standard deviations of exogenous variables Regional exogenous variable Price shock for exported commodities Technological coefficient shock for gross output Share of intermediate products in GDP Share of consumption products in GDP Share of investment products in GDP Share of exported products in GDP Share of government consumption in government expenditures Effective rate of CIT

Region Kazakhstan 0.0202 0.7983 0.3209 0.0076 0.0112 1.6998 0.3384

Russia 0.0604 1.4388 0.5497 0.0106 0.0065 0.7326 0.1085

Belarus 0.0129 0.4996 0.2943 0.0103 0.0102 1.1709 0.4961

0.0009

0.0078

0.0093

The solutions of the parametric control problems based on Model 2 and their testing demonstrated high potential of the parametric control approach for making recommendations on coordinated optimal economic policy at the global level as well as at the levels of the EAEU countries and separate countries (or Regions).

3.5.4

Study of Solutions Dependence on Uncontrolled Parameters

In addition, it was studied how the solutions of the parametric control problem DSGE based on Model 2 at the level of Kazakhstan depended on the uncontrolled parameters. To this effect, the problem was solved in scenarios with 10% and 20% variations of the world oil prices in comparison with the basic forecast. Table 3.19 shows the solutions of the problem DSGE in these scenarios, including

Table 3.19 Expected GDPs and RMSDs of GDP and inflation for year 2022 in different scenarios of oil price dynamics, with and without parametric control Scenario Without parametric control

With parametric control

Index GDP (billion KZT) RMSD of GDP (% deviation from trend) RMSD of inflation (% deviation from trend) GDP (billion KZT) RMSD of GDP (% deviation from trend) RMSD of inflation (% deviation from trend)

World oil price variations (in %) 20 10 0.0

10

20

54082.63 2.43

54673.96 2.52

55271.78 2.54

55871.06 2.65

56469.06 2.74

1.20

1.21

1.25

1.26

1.28

54390.90 2.14

54903.59 2.18

55459.70 2.18

56049.85 2.23

56627.17 2.30

1.12

1.12

1.14

1.14

1.14

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3 Macroeconomic Analysis and Parametric Control Based on Global Dynamic. . .

Table 3.20 Deviations of public bonds rate from basic values for Kazakhstan (in %) Period (quarter, year) Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018 Q3 2018 Q4 2018 Q1 2019 Q2 2019 Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021 Q1 2022 Q2 2022 Q3 2022 Q4 2022

World oil price variations (in %) 20 10 0 19.46 16.98 18.83 24.82 23.78 22.81 19.45 18.85 18.38 18.39 18.73 20.00 18.53 18.21 17.18 19.34 20.60 19.83 24.02 21.64 22.42 20.60 17.98 19.07 20.26 21.45 22.11 18.43 17.06 18.68 10.35 11.02 10.73 10.60 11.48 10.79 10.38 10.61 11.37 11.88 11.01 11.55 12.46 12.25 13.59 9.63 9.27 9.13 7.66 7.72 8.29 8.34 7.61 8.02 6.22 5.76 6.03 8.62 8.40 8.22 9.59 9.87 9.40 8.37 8.44 8.74 9.63 8.90 8.84 9.39 8.87 9.25 9.04 9.16 8.47 8.49 9.45 9.24 8.32 8.38 8.79 9.05 8.71 9.30

10 18.49 24.04 17.07 18.35 16.31 20.08 23.18 17.68 23.85 18.87 11.66 10.41 10.38 11.86 12.53 9.04 8.87 7.31 5.96 8.45 8.77 8.76 9.05 8.52 9.14 9.27 9.12 9.84

20 17.34 21.96 19.72 19.53 18.53 20.69 20.20 19.62 23.68 19.75 11.48 10.90 10.71 12.14 12.83 8.76 7.67 7.97 6.26 7.61 9.76 9.37 8.01 9.47 7.87 9.30 8.76 9.08

the basic solution. The optimal values of the economic policy tools in these scenarios in form of deviations from the corresponding basic values are presented in Tables 3.20 and 3.21. As easily seen, the solutions of the parametric control problems increased Kazakhstan’s GDP within the range 0.28–0.57% and also reduced the volatilities of GDP and inflation within the ranges 11.85–16.13% and 6.25–10.86%, respectively, in comparison with the scenario forecasts without parametric control.

3.5 Parametric Control Based on Model 2: A Series of Problem Statements and Their. . .

307

Table 3.21 Deviations of government expenditures from basic values for Kazakhstan (in %) Period (quarter, year) Q1 2016 Q2 2016 Q3 2016 Q4 2016 Q1 2017 Q2 2017 Q3 2017 Q4 2017 Q1 2018 Q2 2018 Q3 2018 Q4 2018 Q1 2019 Q2 2019 Q3 2019 Q4 2019 Q1 2020 Q2 2020 Q3 2020 Q4 2020 Q1 2021 Q2 2021 Q3 2021 Q4 2021 Q1 2022 Q2 2022 Q3 2022 Q4 2022

World oil price variations (in %) 20 10 0 19.46 16.98 18.83 24.82 23.78 22.81 19.45 18.85 18.38 18.39 18.73 20.00 18.53 18.21 17.18 19.34 20.60 19.83 24.02 21.64 22.42 20.60 17.98 19.07 20.26 21.45 22.11 18.43 17.06 18.68 10.35 11.02 10.73 10.60 11.48 10.79 10.38 10.61 11.37 11.88 11.01 11.55 12.46 12.25 13.59 9.63 9.27 9.13 7.66 7.72 8.29 8.34 7.61 8.02 6.22 5.76 6.03 8.62 8.40 8.22 9.59 9.87 9.40 8.37 8.44 8.74 9.63 8.90 8.84 9.39 8.87 9.25 9.04 9.16 8.47 8.49 9.45 9.24 8.32 8.38 8.79 9.05 8.71 9.30

10 18.49 24.04 17.07 18.35 16.31 20.08 23.18 17.68 23.85 18.87 11.66 10.41 10.38 11.86 12.53 9.04 8.87 7.31 5.96 8.45 8.77 8.76 9.05 8.52 9.14 9.27 9.12 9.84

20 17.34 21.96 19.72 19.53 18.53 20.69 20.20 19.62 23.68 19.75 11.48 10.90 10.71 12.14 12.83 8.76 7.67 7.97 6.26 7.61 9.76 9.37 8.01 9.47 7.87 9.30 8.76 9.08

Chapter 4

Macroeconomic Analysis and Parametric Control Based on Global Multi-country Hybrid Econometric Model (Model 3)

In this chapter, the efficiency of parametric control is illustrated using the framework of the global multi-country hybrid econometric model (hereinafter referred to as Model 3) developed below. Today, single- [40] and multi-country [45, 46, 48] hybrid mathematical models are a widespread tool of macroeconomic analysis and forecasting. (A hybrid model successfully integrates the properties of separate models representing its parts— submodels.) Chapter 4 presents the following results: • Model 3 itself and an original algorithm to build it • The applicability testing of computer simulations with Model 3 to real macroeconomic systems • Mid-term forecasting and macroeconomic analysis for the evolution of economies (countries) • Statements and solutions for a series of parametric control problems • The feasibility testing of parametric control laws • The dependence of optimal criteria values on uncontrolled parameters in parametric control problems

4.1

Building of Model 3

The global multi-country hybrid econometric model is intended for estimating the effect of economic tools in the fields of monetary, currency exchange, and fiscal policy on the index of poverty and poverty convergence. Model 3 was built using an aggregate algorithm that included the following steps: Step 1. Choose basic structures (econometric models) to be used as submodels of Model 3. Step 2. Develop the basic structures. © Springer Nature Switzerland AG 2020 A. A. Ashimov et al., Macroeconomic Analysis and Parametric Control of a Regional Economic Union, https://doi.org/10.1007/978-3-030-32205-2_4

309

310

4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

Step 3. Design interaction interfaces for the developed basic structures. Step 4. Prepare and test time series and panel data for further restoration of the corresponding regression functions in the developed basic structures and also in their interaction interfaces. Step 5. Estimate the parameters of the regression functions of the developed Fair model using the two-stage least squares method (2LSM). Step 6. Estimate the parameters of the panel regression functions of the IMF model using the generalized method of moments (GMM) [25]. Step 7. Estimate the parameters of the regression functions in the first interaction interface (the IMF model output to the Fair model input) to transfer annual effective exchange rates to the quarter Fair model input. Step 8. Estimate the smoothing parameters in the second interaction interface (the quarter Fair model outputs to the annual IMF model input). Step 9. Assemble Model 3 from the submodels. At Step 1, after detailed study of the available literature, the multi-country econometric Fair model [36] and the multi-country econometric IMF model [57] were chosen as the initial basic structures (submodels) of Model 3 in accordance with its purpose. This choice can be explained as follows. The Fair model well matches the purpose of Model 3 in case of adopting the effective equilibrium exchange rates as the exchange rate estimates. The IMF model well matches the purpose of Model 3 in case of adopting initial data in form of the values of the fundamental factors for calculating the effective equilibrium exchange rates that guarantee inner and outer equilibrium [57] based on the Fair model and using the IMF model jointly with the Fair model. Therefore, the hybrid of the developed Fair and IMF models with proper interaction interfaces well matches the purpose of Model 3. At Step 2, the Fair model was developed in the following way. For each country i, the basic identities of this model were supplemented with the description of state budget and public debt formation processes. Also the influence of the world crude oil price on the economies under consideration was taken into account. The descriptions of the Eurasian Economic Union (EAEU) countries—Kazakhstan, Russia, Belarus, Kyrgyzstan, and Armenia—were augmented by their economic integration rules in the field of inflation, state budget gap, and public debt, taking into account their influence on the behavior of different sectors (households, firms, financial and foreign sectors). For higher forecasting accuracy, in the Fair model, the GDP deflator was represented through disaggregation of the following deflators: the consumption of households, capital saving, government consumption, and exports and imports. In addition, the basic identities of the Fair model were supplemented with the corresponding expressions for the fundamental factors of the IMF model [57] on the basis of the former’s endogenous variables. At Step 3, the RU-MIDAS model [39] was adopted as the basic structure of the first interaction interface that transformed the annual effective equilibrium exchange

4.2 Applicability Testing of Model 3

311

rate of the domestic currency of country i into the corresponding observed quarter exchange rate. The basic structure of the first interaction interface was supplemented with the monetary and currency policy indexes and also with the cyclic factor index, all playing the role of explanatory variables. The following moving average filter [24] was adopted as the model of the second interaction interface that transformed the output of the quarter indexes of the developed Fair model into the annual indexes of the IMF model: XUNDi,t ¼

X2

φX j¼2 j i,tþj

ð4:1Þ

In this formula, Xi,t denotes the value of some index of country i at time t; XUNDi,t is the result of filtering the index Xi,t at time t; and finally, φj gives the filter’s smoothing parameter. At Step 4, the preparation and testing of the time series and panel data for further restoration of the corresponding regression functions were performed in EViews. Note that each time series was tested and prepared by estimating the presence of seasonality and overshoots, with subsequent correction. The panel data were prepared by the contraction of time series and tested using the corresponding regression functions. If the testing results of the corresponding regression functions satisfy the necessary requirements, then the panel data have certain properties for further use. At Steps 5 and 6, the parameters of the corresponding regression functions were estimated by the described methods: the regression functions for the Fair model were restored using 2LSM while those for the IMF model using GMM. For restoring the developed structures of the Fair and IMF models, the elasticities of trade balance indexes on nominal exchange rate in both models were set equal to each other. These elasticities were taken from [78]. At Step 7, the model parameters of the first interaction interface were estimated using the standard least squares method. At Step 8, the model parameters φ j of the second interaction interface were estimated using an original method suggested in [24]. At Step 9, Model 3 was assembled from the developed Fair model, the IMF model, and the models of the first and second interaction interfaces, for subsequent computer simulations.

4.2

Applicability Testing of Model 3

The developed software for testing of Model 3 includes the following components: • Program modules for intermediate testing at Steps 1–9 of the aggregate model building algorithm

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

• Program modules for testing of Model 3 itself using the Cowles Commission approach [36] • Program modules for testing of Model 3 using the parametric control approach The program modules of the first group are intended: • To test the time series representing the explanatory or explained variables in the corresponding regression functions (see the aggregate model building algorithm) for the presence of seasonality and overshoots as well as to eliminate these effects from the time series • To test the regression functions (in particular, to estimate the significance of their variables and also the significance of the regression functions themselves), to test the autocorrelation of the remainders of the regression functions, to test the normal distribution of the remainders, to verify the consistency of the regression functions with the main postulates of macroeconomics, to estimate the stability of different parameters of the regression functions for the identification and proper consideration of structural shifts, to test the super identifying constraints for the adjustable 2LSM tools (to eliminate biased parameter estimates), and to test the hypothesis of rational expectations The testing procedure with the program modules of the first group yielded the following results. Among the 4955 time series tested, 3235 had seasonality and 564 overshoots. The testing procedure of the regression functions (2118 functions in total) demonstrated that they could be used as the components of Model 3. The program modules of the second group are intended to test Model 3 using the Cowles Commission approach, i.e., to estimate the sensitivity coefficients (elasticities) of the endogenous variables of Model 3 by its exogenous variables. As an example, Table 4.1 presents the estimated sensitivity coefficients (elasticities) of some endogenous variables of Model 3—the GDPs of Kazakhstan, Russia, and Belarus—by the values of its exogenous variables for Q1 2016. The testing procedure with stability indexes of the mappings defined by Model 3 (see Subsection 1.3.2) indicated that the indexes were vanishing as the radius of an associated input parameter ball was going to 0 with the same rate of convergence (this ball was used to produce a normalized vector of input parameters by a uniform randomizer).

Table 4.1 Estimated sensitivity coefficients of real GDP by values of exogenous variables Exogenous variable Government consumption Indirect tax rate (VAT and excises) IIT rate Nominal exchange rate (domestic currency per USD) Short-term interest rate Price of exported products Price of imported products

Country Kazakhstan 0.070 0.004 0.001 0.044 0.030 0.020 0.033

Russia 0.084 0.006 0.003 0.090 0.050 0.030 0.071

Belarus 0.100 0.005 0.002 0.039 0.030 0.116 0.015

4.2 Applicability Testing of Model 3 Table 4.2 Estimated stability indexes for collection 1 of exogenous variables

Country (region) Kazakhstan Kazakhstan Kazakhstan Kazakhstan Russia Russia Russia Russia Belarus Belarus Belarus Belarus Kyrgyzstan Kyrgyzstan Kyrgyzstan Kyrgyzstan Armenia Armenia Armenia Armenia EAEU EAEU EAEU EAEU

313 Radius of ball 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125

Value of stability index 0.005476786 0.002708833 0.001347827 0.000731192 0.046682261 0.02447997 0.011765964 0.006109221 0.003910275 0.002180988 0.001136726 0.000567304 0.002389511 0.001171124 0.000583133 0.000301484 0.00260088 0.001308217 0.000685401 0.000331361 0.003736519 0.001320242 0.000491236 0.000313121

The estimated stability indexes of the mappings defined by Model 3 (see Tables 4.2 and 4.3) corresponded to the collections of exogenous variables for separate member countries and the whole EAEU. The values of the exogenous variables vector were varied one thousand times on the period 2015–2020. More specifically, the values presented in Table 4.2 corresponded to the following collection of exogenous variables (collection 1): government consumption, indirect tax rate (VAT and excises), IIT rate, and the currency reserves of Central Bank. The values presented in Table 4.3 corresponded to the following collection of exogenous variables (collection 2): potential GDP, government consumption, indirect tax rate (VAT and excises), IIT rate, the currency reserves of Central Bank, and all residuals appearing in the equations of a given country. For estimating the stability index of the whole EAEU, the values of exogenous variables from these collections were disturbed simultaneously for all EAEU countries. As indicated by the above tests, Model 3 was applicable for short- and mid-term forecasting, macroeconomic analysis, and assessment of efficient economic policy.

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

Table 4.3 Estimated stability indexes for collection 2 of exogenous variables

4.3 4.3.1

Country (region) Kazakhstan Kazakhstan Kazakhstan Kazakhstan Russia Russia Russia Russia Belarus Belarus Belarus Belarus Kyrgyzstan Kyrgyzstan Kyrgyzstan Kyrgyzstan Armenia Armenia Armenia Armenia EAEU EAEU EAEU EAEU

Radius of ball 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125 1 0.5 0.25 0.125

Value of stability index 0.018506495 0.00539935 0.003811587 0.001539672 0.154506771 0.065205097 0.045376215 0.026463482 0.014717559 0.006854617 0.003981694 0.001871893 0.013577649 0.008187778 0.003529619 0.001897858 0.021500689 0.010439052 0.005664001 0.002811029 0.021644663 0.011919536 0.004005062 0.002502854

Macroeconomic Analysis and Mid-term Forecasting Based on Model 3 Knowhow and Software for Macroeconomic Analysis and Mid-term Forecasting

The knoware for mid-term forecasting based on Model 3 consists of two parts, (1) the knoware for forecasting of the exogenous variables of Model 3 and (2) the knoware for solving Model 3 with the forecasted exogenous variables in the form of the Fair–Taylor algorithm [37] for solving nonlinear dynamic econometric models. The knoware for forecasting of the exogenous variables of Model 3 consists of an algorithm of restoring the autoregression model of a time series and an algorithm of forecasting for an exogenous variable using the restored autoregression models of time series. This algorithm can be described in aggregate form as follows. Step 1. Choose a time series xi and a period [t1, . . ., t2] for estimating the parameters of the autoregression model. Step 2. Derive the logarithmic form zi ¼ ln ðxi Þ of the time series xi .

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

315

Step 3. Choose the maximal lag ( p ¼ 8) for the tested autoregression model with constant term and deterministic trend defined by zit ¼

Xp

αz j¼1 j itj

þ αpþ1 þ αpþ2 t þ εt :

ð4:2Þ

Step 4. Estimate the parameters of the autoregression model using the standard LSM on the estimation period, thereby obtaining the parameter values b α1 ,. . ., b αpþ2 and the estimate bεt for the realization of the random variable on the period [t1 + p,. . .,t2]. Step 5. Test the autocorrelation of the remainder bεt using the Ljung–Box Q-test [59]. If the significance level of the Ljung–Box Q-statistic does not exceed 0.05 for some lag, then assign p:¼ p + 1 and get back to Step 4; otherwise proceed to Step 6. Step 6. Test the significance of each variable of the autoregression model, and sequentially eliminate the insignificant variables (starting from least significant one), with new estimation of all parameters after each elimination and autocorrelation test for the remainder. If a new estimated remainder has nonzero autocorrelation after elimination of a next insignificant variable, then the previous model obtained before this elimination is the desired autoregression model. Step 7. Choose a forecasting period [t2 + 1, . . ., t3]. Step 8. Sequentially calculate the forecasts zitþl for l ¼ t 2 þ 1, . . . , t 3 using the forecasting model: zitþl ¼

Xp

b αz j¼1 j itþlj

þb αpþ1 þ b αpþ2 ðt þ lÞ:

ð4:3Þ

(By assumption, bεtþl takes its mean value equal to 0.) Step 9. Calculate the forecasts xitþl , l ¼ t 2 þ 1, . . . , t 3 , for the original exogenous variable using the forecasts for zitþl : xitþl ¼ ezitþl :

ð4:4Þ

The notations are the following: i as the number of exogenous variable; xi as the corresponding time series of the exogenous variable; t1 and t2 as the left and right limits of the parameter 0 estimation period, respectively; ln as natural logarithm; α ¼ α1 , α2 , . . . , αpþ2 as the parameter vector of the autoregression model; t as deterministic trend; εt as a random variable obeying the normal distribution with zero mean and constant variance; and finally, t2 + 1 and t3 as the left and right limits of the forecasting period. The developed software for mid-term forecasting based on Model 3 consists of: • A program module implementing the knoware for forecasting of the exogenous variables of Model 3 with the corresponding restored autoregression models of the variables

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

• A program module implementing the Fair–Taylor algorithm for solving nonlinear dynamic econometric models with the forecasted exogenous variables in EViews The software for choosing or specifying the exogenous variables of Model 3 and solving the hybrid model with the exogenous variables of Model 3 at the levels of a separate country (region), a regional economic union, and the global economy consists of: • A program module for choosing or specifying the exogenous variables of Model 3 • A program module implementing the Fair–Taylor method for solving nonlinear dynamic econometric models in EViews The developed software calculates the values of all endogenous variables of Model 3 (and their rates of change) in domestic currency and USD, in the baseline scenario and forecasted scenarios with certain variations of the exogenous variables from a given collection. Such endogenous variables include international trade indexes and economic indexes describing the output and consumption of products at the levels of a separate country (region), a regional economic union, and the global economy. In particular, the software can be used for scenario analysis (at the levels of the global economy, economic unions, and separate countries) for estimating: • • • • •

The effects of fiscal policy tools The effects of monetary policy tools The effects of price variations for exported and imported products The effects of world oil prices The effects of international economic sanctions against separate countries and so on

4.3.2

Examples of Mid-term Forecasts Based on Model 3

Model 3 and the developed software were used for solving a series of mid-term forecasting problems up to year 2020 for the following economic indexes: • Real indexes of the System of National Accounts (SNA), namely, GDP and its consumption components (the consumption of households, government consumption); gross capital saving; variations of circulating assets reserves; the exports of goods and services; the imports of goods and services; domestic savings; and GDP deflator and the deflators of its consumption components (in total and also for the nominal indexes of the SNA) • Real trade indexes, namely, the distribution of exported goods by countries; the distribution of imported goods by countries; the exports and imports of goods; and the exports and imports of services • Indexes of the balance of payments, namely, current operations account (including its trade balance, primary and secondary income) and financial operations account

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317

• Prices, namely, consumer prices; the prices of exported products; and the prices of imported products • Labor and employment indexes, namely, the volume of employment; economically active population; unemployment rate; and wages • Financial indexes of a country, namely, public debt; state budget gap/surplus; and state budget revenues and expenditures and their components • Indexes of monetary system, namely, money stock; the gold and foreign exchange reserves of Central Bank; short- and mid-term interest rates; domestic currency rate; and many other economic indexes Note that the solutions of these problems were consistent with the existing IMF forecasts on the same period. As an example, Tables 4.4 and 4.5 present the forecasts based on Model 3 and also the IMF forecasts for the real growth of GDP and inflation for the EAEU countries, the USA, China, the European Union (EU), the Rest of World (RoW), the whole EAEU, the economies of the Shanghai Cooperation Organization (SCO), the economies of the Organization for Economic Cooperation and Development (OECD), and the global economy. The real growth of each group of countries was calculated by summing up the real GDPs of all its countries in USD. The growth of inflation in each group was calculated using the method [60]. Generally speaking, the forecasts based on Model 3 were consistent with the IMF forecasts for all countries under consideration (in particular, in terms of economic growth, nominal capital investments, the real rates of change of exported and imported goods and services). However, there were significant deviations of economic growth forecasts for the EU and the global economy and significant deviations of inflation forecasts for all countries except for Armenia. By the end of the forecasting period, a gradual reduction of growth below 6% was observed for China. For all other countries and unions, economic growth was restored or even accelerated. In the first half of 2015, the main factors decreasing the annual inflation of Kazakhstan were decelerated economic growth, limited money supply, and a considerable weakening of RUB at the juncture of years 2014 and 2015. (The latter factor caused price reduction for Russian goods in the Kazakh market.) In the second half of 2015, the main factor stimulating inflation was a considerable weakening of KZT.

4.3.3

Examples of Comparative Macroeconomic Analysis

In the course of snapshot ex post and ex ante macroeconomic analysis based on Model 3, different economic indexes were estimated and compared with each other on the retrospective period 2010–2014 and also on the mid-term forecasting period 2015–2020, for each of the EAEU countries, the USA, China, and the EU.

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

Table 4.4 Forecasted economic growth on period 2015–2020 (in %) Country Kazakhstan Russia Belarus Kyrgyzstan Armenia USA China EU RoW EAEU SCO OECD Global economy

Forecast Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF

Year 2015 1.181 1.156 3.643 3.746 3.872 3.894 3.441 3.469 2.720 3.005 2.247 2.426 6.958 6.900 1.534 1.986 1.730 – 3.189 – 5.570 – 1.844 – 2.242 3.090

2016 0.261 0.135 1.675 1.849 2.632 2.656 3.617 3.515 1.788 1.886 2.165 2.401 6.512 6.490 1.570 1.841 1.974 – 1.505 – 5.665 – 1.798 – 2.314 3.162

2017 1.121 1.05 0.954 0.812 0.341 0.374 2.653 2.740 2.583 2.500 2.370 2.502 6.191 6.200 1.727 1.949 2.430 – 0.961 – 5.845 – 1.897 – 2.599 3.535

2018 1.899 1.806 0.843 1.000 0.820 0.850 5.294 5.428 3.170 3.000 2.572 2.375 6.044 6.000 1.706 1.871 2.767 – 0.968 – 5.816 – 2.062 – 2.749 3.640

2019 2.174 2.163 1.361 1.500 1.014 1.039 4.973 4.886 3.544 3.500 2.448 2.134 5.891 6.000 1.679 1.868 3.030 – 1.448 – 5.816 – 2.071 – 2.824 3.754

2020 3.413 3.364 1.423 1.500 1.085 1.111 5.712 5.706 3.583 3.500 2.280 1.959 5.993 6.000 1.658 1.834 3.069 – 1.622 – 5.898 – 2.082 – 2.823 3.795

Figures 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, 4.17, 4.18 and 4.19 illustrate the results of this comparative macroeconomic analysis on the period 2010–2020. Since the beginning of this period, China had a smooth drop of economic growth, while the EAEU countries demonstrated dual dynamics of real GDP (see Fig. 4.1). By the end of the forecasting period (year 2020), Kazakhstan did not reach the previous growth rates observed on the retrospective period. The growth rates of all EAEU countries on the forecasting period were still smaller than that of China. In Fig. 4.2, which shows the economic growth in different groups of countries (unions), the Rest of World and the USA, there are no considerable fluctuations on the forecasting period. Higher growth rates in the SCO against other unions and regions were primarily formed by India and China. The Rest of World, mostly consisting of underdeveloped countries, was gradually increasing real economic growth on the forecasting period.

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

319

Table 4.5 Forecasted consumer-price inflation for period 2015–2020 Country Kazakhstan Russia Belarus Kyrgyzstan Armenia USA China EU EAEU

Forecast Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF Model 3 IMF

Year 2015 8.940 6.450 13.540 15.532 10.790 13.523 8.780 6.503 3.660 3.731 0.995 0.118 1.439 1.441 0.669 0.003 13.096 –

2016 9.940 13.088 14.710 8.355 15.510 13.588 11.910 5.481 2.590 2.561 1.240 0.817 0.402 1.800 0.823 0.404 14.245 –

Kaz Kgz

2017 8.080 9.254 13.940 6.513 18.460 12.117 10.240 6.921 4.650 4.000 1.184 1.538 1.155 2.000 0.890 1.344 13.504 –

Rus Arm

2018 7.810 9.000 12.100 4.952 12.290 9.896 8.590 6.125 5.520 4.000 0.997 2.371 1.062 2.200 1.062 1.550 11.909 –

2019 7.670 8.000 10.820 4.000 8.450 9.447 8.480 5.702 4.900 4.000 1.181 2.487 0.854 2.600 1.163 1.696 10.727 –

2020 7.780 8.000 9.770 4.000 7.780 9.419 8.490 5.340 5.060 4.000 1.445 2.337 0.707 3.000 1.228 1.782 9.508 –

Blr Chn

12 10 8 % variation

6 4 2 0 -2 -4 2010

2011

2012

2013

2014

2015 Year

2016

2017

2018

2019

2020

Fig. 4.1 Annual growth rates of economies: EAEU countries and China

In accordance with Fig. 4.3, for year 2016, the per capita GDPs of Russia, Kazakhstan, and Belarus were reduced almost twice against year 2013. Perhaps, this was the negative effect of the oil price shock on Kazakhstan and Russia and international economic sanctions against Russia. In year 2016, China left behind

4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

320

USA RoW

EAEU OECD

EU SCO

10 8

% variation

6 4 2 0 -2 -4 2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

Year

Fig. 4.2 Annual growth rates of economies: the USA, EAEU, EU, Rest of World, OECD, and SCO Kaz Kgz

Rus Arm

Blr Chn

16000 14000 12000

USD

10000 8000 6000 4000 2000 0

2010

2011

2012

2013

2014

2015 Year

2016

2017

2018

2019

2020

Fig. 4.3 Nominal per capita GDPs: EAEU countries and China

Kazakhstan and Russia on this index, occupying its position up to year 2020 inclusively. Figure 4.4 indicates a large gap between the per capita GDPs of developing and developed countries, which was increasing by the end of the forecasting period. As can be seen in Fig. 4.5, the ratios of nominal consumptions of goods and services by the households of Kyrgyzstan and Armenia to their GDPs were close to or greater than 1. In particular, for Kyrgyzstan the consumption exceeded GDP more than by 20% on the forecasting period. The explanation is that most of the

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

EARU OECD

USA RoW

321

EU SCO

70000 60000

USD

50000 40000 30000 20000 10000 0 2010

2011

2012

2013

2014

2015 Year

2016

2017

2018

2020

2019

Fig. 4.4 Nominal per capita GDPs: the USA, EAEU, EU, Rest of World, OECD, and SCO Kaz Kgz

Blr Chn

Rus RA

1.4 1.2

Ratio

1.0 0.8 0.6 0.4 0.2

2010

2011

2012

2013

2014

2015 Year

2016

2017

2018

2019

2020

Fig. 4.5 Ratios of households’ consumption to GDP: EAEU countries and China

employable population was working abroad (in Russia). In accordance with the SNA, their official income in foreign countries was included not in GDP but in GNP. This led to negative values of the savings index for Kyrgyzstan. For the other EAEU countries, the ratios of households’ consumption to GDP were close to each other, lying within the range 0.4–0.6 on the retrospective and forecasting periods. This ratio was smaller for China than for EAEU countries on the whole period 2010–2020.

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Fig. 4.6 Ratios of households’ consumption to GDP: the USA, EAEU, EU, Rest of World, and SCO Kaz Kgz

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Fig. 4.7 Ratios of investments to GDP: EAEU countries and China

Note that the graph for the Rest of World (RoW) in Fig. 4.6 was obtained by calculating the ratio of aggregate consumptions to the aggregate GDP of 23 countries (with available economic data in Model 3), except for the EAEU countries, the EU, the USA, and China. Clearly—see Fig. 4.7—China had higher ratio of capital investments to GDP against the EAEU countries, in contrast to the case of consumption. For China this index was gradually decreasing, like the real growth of economy. For the EAEU countries, capital investments as a share of GDP had inconsiderable variations.

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3 USA OM*

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Fig. 4.8 Ratios of investments to GDP: the USA, EAEU, EU, Rest of World, and SCO Kaz Kgz

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Fig. 4.9 Ratios of government consumption to GDP: EAEU countries and China

The ratios of investments to GDP (see the graphs in Fig. 4.8) indicated a considerable difference between developing and developed countries. In accordance with simulation results, China and India were the largest investors among the countries of the developing world. The graph for the Rest of World (RoW) in Fig. 4.8 was obtained by calculating the ratio of aggregate investments to the aggregate GDP of 23 countries (with available economic data in Model 3), except for the EAEU countries, the EU, the USA, and China.

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Fig. 4.10 Ratios of government consumption to GDP: the USA, EAEU, EU, Rest of World, and SCO Kaz Kgz

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Fig. 4.11 Ratios of exported goods and services to GDP: the EAEU countries and China

Both on the retrospective and forecasting periods, the ratios of government consumption to GDP for Kazakhstan and Russia considerably differed from each other (Fig. 4.9). At the same time, these indexes took close values for other countries. Figure 4.10 demonstrates the difference of budget policy in developed and developing countries. Government consumption as a share of GDP was reduced for all countries on the forecasting period. In accordance with Fig. 4.11, Belarus occupied the leading position among all EAEU countries in the specific weight of exported goods and services in GDP

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

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Fig. 4.13 Ratios of imported goods and services to GDP: EAEU countries and China

(on the retrospective period, this leadership is observed even within the CIS). China had the smallest ratio of exported goods and services to GDP against all EAEU countries. The reduced share of exports in Kazakhstan’s GDP was caused by world price drops for oil and metals. Direct comparison of imports and exports as shares of GDP (see the graphs in Figs. 4.12, 4.13 and 4.14) led to the following conclusion: the USA and the EU were mostly closed and open economies, respectively. Moreover, the EU and the Rest of World were increasing the openness of their economies, as follows from Figs. 4.12, 4.13

4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

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Fig. 4.14 Ratios of imported goods and services to GDP: the USA, EAEU, EU, Rest of World, and SCO

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Fig. 4.15 Percentage variations of nominal domestic currency-to-USD exchange rates in comparison with the previous year: EAEU countries, eurozone, and China

and 4.14. On the other hand, the openness of the SCO’s economy was decreasing on the forecasting period. The graphs in Fig. 4.15 indicate that, on the retrospective period, Belarus had the highest volatility of domestic currency-to-USD exchange rates among all countries considered, while China the lowest. With transition to floating exchange rates, KZT suffered from considerable weakening: by 23.74% in year 2015 and also by 59.05% in year 2016, in comparison with previous year. The testing procedure [27] of

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Fig. 4.17 Ratios of public debt to GDP: the EAEU countries

Kazakhstan’s gold and foreign exchange reserves statistics demonstrated that de facto the country still had controlled exchange rate. On the period 2015–2016, all currencies under study suffered from considerable weakening with respect to USD: for year 2015, RUB by 58.16%, BYN by 55.77%, KGS by 18.7%, AMD by 14.8%, and Euro by 19.57%. On the forecasting period since year 2017, the volatilities of all these currencies did not exceed 10%, which was expected to promote trade within the EAEU.

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Fig. 4.18 Ratios of consolidated budget deficit ()/surplus (+) to GDP: EAEU countries

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Fig. 4.19 Inflation: the EAEU countries

Figure 4.16 shows the difference between the short-term interest rates of the countries and unions under consideration. In addition, note that the interest rates of the EAEU countries responded to (expected or realized) weakening of domestic currency and also to the variations of such rates of large economies (the USA and the EU). The values of the interest rates for the USA, the EU, and China are associated with the right ordinate axis in Fig. 4.16. The outcomes of monetary policies in the EAEU countries can be compared by analyzing their short-term interest rates (see Fig. 4.16). In Armenia, Kyrgyzstan, and

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

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Russia, with inflation targeting mostly implemented through money market tools, short-term interest rates had more stable dynamics on the retrospective period. Next, Figs. 4.17, 4.18 and 4.19 present the dynamics of indexes associated with stable economic growth conditions (the ratio of public debt to GDP, the ratio of consolidated budget gap/surplus to GDP, and inflation) for the EAEU countries. In accordance with Fig. 4.17, the stable economic growth conditions on public debt (this debt must be smaller than 50% GDP) were violated for Kyrgyzstan and Armenia. Next, in accordance with Fig. 4.18, the stable economic growth conditions on consolidated budget gap (this gap must be smaller than 3% GDP) were violated for Russia and Armenia. Finally, in accordance with Fig. 4.19, the stable economic growth conditions on inflation (inflation must be smaller than the minimum among all the EAEU countries plus 5 percentage points) were violated for all member countries, except for Armenia.

4.3.4

Examples of Scenario Macroeconomic Analysis

In the course of scenario (active) ex post and ex ante macroeconomic analysis, some economic indexes were estimated on the retrospective period 2010–2014 and also on the mid-term forecasting period 2015–2020. In particular, the following assessments were obtained for each of the EAEU countries: • The effects on GDP and other indexes from fiscal policy tools (reducing/increasing indirect tax (VAT and excises), IIT and import duties, and government consumption by 10% in comparison with the baseline scenario for the EAEU countries for the period 2015–2020) • The effects on GDP and other indexes from monetary policy tools (reducing/ increasing the short-term interest rate of Central Bank by 10% in comparison with the baseline scenario for the EAEU countries for the period 2015–2020) • The effects on GDP and other indexes from 10% price variations for exported and imported products • The effects on GDP and other indexes from 10% exchange rate variations (domestic currency to USD) Consider some examples of scenario macroeconomic analysis with short-term forecasting for separate EAEU countries (Kazakhstan, Russia, and Belarus). The 10% government consumption increase scenario on the forecasting period 2015–2020 demonstrated that the aggregate real GDP was increasing by 1.489% for Kazakhstan, by 0.941% for Russia, and by 1.162% for Belarus (all figures mentioned in comparison with the baseline scenario). Figures 4.20, 4.21 and 4.22 present the graphs of the quarter real GDPs (at 2005 prices) for the three EAEU countries on the forecasting period, in the baseline and 10% government consumption increase scenarios.

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Fig. 4.20 10% government consumption increase scenario: effects on real GDP of Kazakhstan on forecasting period (at 2005 prices)

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Fig. 4.21 10% government consumption increase scenario: effects on real GDP of Russia on forecasting period (at 2005 prices)

The effects of real government consumption on the economic growth of a given country that were estimated using Model 3 can be summarized as follows. The growth of real government consumption was directly increasing the real domestic sales of firms and hence domestic production; in view of saved circulating assets, a subsequent increase of firms’ proceeds and profits as well as of households’ disposable income were followed by a gradual increase of domestic investments and consumption and hence by an additional growth of real GDP. However, the

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

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Fig. 4.22 10% government consumption increase scenario: effects on real GDP of Belarus on forecasting period (at 2005 prices)

described positive (direct and indirect) effects of government consumption were accompanied by some negative secondary effects through other channels. In particular, the growth of real GDP was causing disruption of output and hence increasing domestic prices. In turn, higher domestic prices and real domestic demand were increasing the real imports of a given country, negatively affecting real GDP. The 10% indirect tax rate (VAT and excises) increase scenario on the forecasting period 2015–2020 demonstrated that the aggregate real GDP was decreasing in comparison with the baseline scenario: by 0.331% for Kazakhstan, by 0.434% for Russia, and by 0.159% for Belarus. Figures 4.23, 4.24 and 4.25 present the graphs of the quarter real GDPs (at 2005 prices) for the three EAEU countries on the forecasting period, in the baseline and 10% indirect tax rate increase scenarios. The effects of indirect tax rates on the economic growth of a given country that were estimated using Model 3 were opposite to the case of government consumption and can be summarized as follows. Indirect tax rates were directly influencing the profits of firms and the disposable income of households. However, their influence on nominal GDP was dual. Higher indirect tax rates were simultaneously reflected in consumer prices, increasing the latter. In real terms the consumption of households was reducing due to lower real disposable income of households. The 10% currency exchange rate increase scenario on the forecasting period 2015–2020 demonstrated that the aggregate real GDP was increasing in comparison with the baseline scenario: by 2.103% for Kazakhstan, by 1.892% for Russia, and by 0.147% for Belarus. Figures 4.26, 4.27 and 4.28 present the graphs of the quarter real GDPs (at 2005 prices) for the three EAEU countries on the forecasting period, in the baseline and 10% currency exchange rate increase scenarios.

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

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Fig. 4.23 10% indirect tax rate increase scenario: effects on real GDP of Kazakhstan on forecasting period (at 2005 prices) Baseline scenario

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Fig. 4.24 10% indirect tax rate increase scenario: effects on real GDP of Russia on forecasting period (at 2005 prices)

The effects of nominal exchange rate on the economic growth of a given country that were estimated using Model 3 can be summarized as follows. The growth of nominal exchange rate was directly increasing the import prices in domestic currency, which had negative influence on the country’s real imports (the main shortterm effect increasing real GDP). Subsequently, as the result of incomplete transfer of the exchange rate increase effect, the country’s export prices in USD were decreasing, stimulating exports and hence causing higher real GDP. However, the described positive influence of nominal exchange rate was accompanied by some

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Fig. 4.25 10% indirect tax rate increase scenario: effects on real GDP of Belarus on forecasting period (at 2005 prices)

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Fig. 4.26 10% currency exchange rate increase scenario: effects on real GDP of Kazakhstan on forecasting period (at 2005 prices)

negative effects through other channels. In particular, domestic interest rates were responding with sharp increase, reducing investments and consumption in the country and hence decelerating the real growth of economy. The 10% short-term interest rate increase scenario on the forecasting period 2015–2020 demonstrated that the aggregate real GDP was decreasing in comparison with the baseline scenario: by 1.035% for Kazakhstan, by 1.022% for Russia, and by 0.273% for Belarus.

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

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Fig. 4.27 10% currency exchange rate increase scenario: effects on real GDP of Russia on forecasting period (at 2005 prices)

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Fig. 4.28 10% currency exchange rate increase scenario: effects on real GDP of Belarus on forecasting period (at 2005 prices)

Figures 4.29, 4.30 and 4.31 present the graphs of the quarter real GDPs (at 2005 prices) for the three EAEU countries on the forecasting period, in the baseline and 10% short-term interest rate increase scenarios. The effects of short-term interest rate on the economic growth of a given country that were estimated using Model 3 can be summarized as follows. Through the interest rate channel (the corresponding growth of mid-term interest rates), the growth of short-term interest rate was directly reducing the borrowing capabilities of firms for further investment and of households for further consumption in a given

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Fig. 4.29 10% short-term interest rate increase scenario: effects on real GDP of Kazakhstan on forecasting period (at 2005 prices) Baseline scenario

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Fig. 4.30 10% short-term interest rate increase scenario: effects on real GDP of Russia on forecasting period (at 2005 prices)

country, which had negative influence on real GDP. (Subsequently the secondary effects were occurring: smaller domestic sales and production were decreasing the profits of firms and the disposable income of households, causing negative influence on further investments and consumption and hence real GDP.) The described primary negative effects of short-term interest rate were accompanied by secondary negative effects through another channel: higher interest rate was possibly

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

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Fig. 4.31 10% short-term interest rate increase scenario: effects on real GDP of Belarus on forecasting period (at 2005 prices)

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Fig. 4.32 10% import price increase scenario: effects on real GDP of Kazakhstan on forecasting period (at 2005 prices)

strengthening domestic currency, also causing negative influence on real GDP through trade indexes. The 10% import price increase scenario on the forecasting period 2015–2020 demonstrated that the aggregate real GDP was decreasing in comparison with the baseline scenario: by 0.2% for Kazakhstan, by 0.914% for Russia, and by 0.749% for Belarus.

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

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Fig. 4.33 10% import price increase scenario: effects on real GDP of Russia on forecasting period (at 2005 prices)

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Fig. 4.34 10% import price increase scenario: effects on real GDP of Belarus on forecasting period (at 2005 prices)

Figures 4.32, 4.33 and 4.34 present the graphs of the quarter real GDPs (at 2005 prices) for the three EAEU countries on the forecasting period, in the baseline and 10% import price increase scenarios. The effects of import price on the economic growth of a given country that were estimated using Model 3 were similar to the case of nominal exchange but considerably smaller due to the indirect influence of import price on interest rates. A model can be used for forecasting two types of exchange rates, which determines one of its features. The first type represents the so-called actual exchange rates

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4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

in a specific country (the ones actually observed on a retrospective period). They are formed under the influence of long-term fundamental factors (the country’s structural fiscal positions, net foreign assets, economic growth, relative income in comparison with a reference country, trade conditions, oil balance, aged-to-employed population ratio, population growth), cyclic factors (output gap and inflation), and temporal factors (short-term interest rates and interventions of Central Bank). The second type represents the so-called effective equilibrium exchange rates, which are formed under the influence of fundamental factors only as defined by the IMF [53]. This feature of Model 3 was exploited in scenario analysis for estimating the efficiency of the calculated exchange rates of the second type in comparison with the forecasted exchange rates of the first type, with real GDP taken as the efficiency criterion. Such a scenario (further referred to as experimental) was prepared and realized in the form of the following algorithm: Step 1. Choose a country to be analyzed. Step 2. Choose a simulation period for Model 3. Step 3. Calculate the baseline scenario of Model 3 on this period, and fix the values of the efficiency criterion under the exchange rates of the first type. Step 4. Using the method [29] and a deterministic trend as proxy variable, disaggregate the annual exchange rate of the second type calculated in the baseline scenario for obtaining the quarter exchange rate of the second type. Step 5. Write the obtained quarter exchange rates into the corresponding time series of the exogenous variable of Model 3 associated with the exchange rate of the second type. Step 6. Deactivate in Model 3 the expression describing the exchange rate of the second type. Step 7. Deactivate in Model 3 the regression function describing the behavior of the exchange rate of the first type, and use the equal exchange rates of the first and second types. Step 8. Choose a year from the period 2010–2020 so that the estimated exchange rates of both types are close to each other as much as possible. Step 9. Launch the modified Model 3 starting from Q1 of the chosen year till the end of year 2020 with the disaggregated effective exchange rates, in order to generate the values of all endogenous variables in the experimental scenario. Step 10. Compare the real GDP of a given country in the baseline and experimental scenarios on the period chosen at Step 9. The year choice criterion adopted at Step 8 guarantees that scenario simulations have almost the same conditions at the beginning of the launch period. This scenario analysis was performed for Kazakhstan on the period 2010–2020. Figure 4.35 presents the dynamics of the actual and effective exchange rates obtained in the baseline scenario of Model 3. Clearly, the actual and effective exchange rates obtained in the baseline scenario were closest to each other for year 2010. Thus 2010 was chosen as the initial year for the experimental scenario. As a result, Kazakhstan’s GDP in the experimental scenario turned out to be by 3.779% greater against its basic value. Expectedly, this improvement was achieved

4.3 Macroeconomic Analysis and Mid-term Forecasting Based on Model 3

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mostly with an effective compromise between domestic interest rates (affecting investments, consumption, and savings) and the exchange rate of domestic currency (affecting exports and imports), which was the consequence of efficient balance of savings-investments and the current account of the balance of payments. However, further research is required for proving the theoretical hypothesis that the economy of a given country demonstrates more stable growth with the effective equilibrium exchange rate (as defined by the IMF [53]) than with the actual exchange rate. In particular, it is necessary to develop and apply scenario analysis algorithms based on Model 3 with disaggregation of annual effective equilibrium exchange rates into quarterly effective equilibrium exchange rates. The results of mid-term forecasting, passive ex post and ex ante macroeconomic analysis on the retrospective (2010–2014) and forecasting (2015–2020) periods, and scenario ex post and ex ante macroeconomic analysis on these periods led to the following conclusions: 1. Model 3 could be successfully used for short- and mid-term forecasting (the forecasts based on Model 3 were consistent with the IMF forecasts for all countries considered in terms of economic growth, nominal capital investments, and real rates of change for exported and imported goods and services). 2. Model 3 could be successfully used for passive and scenario ex post and ex ante macroeconomic analysis on retrospective and mid-term forecasting intervals (the response of the endogenous variables of Model 3 to the variations of exogenous parameters well agrees with main postulates of macroeconomics). 3. Model 3 described many important economic processes, including formation of effective and actual exchange rates, formation of interest rates, production, employment, domestic pricing (consumer prices, producer prices, import and export prices), and global pricing (including world oil prices).

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4. The scenario analysis based on Model 3 for Kazakhstan confirmed the theoretical hypothesis that the economy demonstrated more stable growth (production, consumption, and investments) with the effective exchange rate as defined by the IMF [53] than with the actual exchange rates influenced by additional cyclic and temporal factors.

4.4 4.4.1

Parametric Control Based on Model 3: A Series of Problem Statements and Their Solutions Problem Statements for Parametric Control

A series of optimal parametric control problems were formulated on the basis of Model 3 at the levels of a separate country, the EAEU, and the global economy as follows. Given a collection of L countries and currency policies of all countries within the framework of Model 3, find the values of monetary policy tools (short-term interest rates) and fiscal policy tools (real government consumptions; effective rates of CIT, indirect tax, and IIT) for each country on the period t ¼ 2017, . . . , 2022 that maximize a chosen control criterion—the GDP (PPP) of these countries—subject to existing constraints on the admissible values of these tools (deviations from basic values not exceeding 10%) and also subject to the additional stable growth constraints determined by the Treaty on the EAEU. For L ¼ 1, this parametric control problem corresponds to the level of a separate country. The optimal parametric control problem at the level of L countries indicated by i ¼ 1, . . . , L has the following formal statement. Problem GEA Given a collection A of L countries and currency policies of all countries within the framework of Model 3, find the values of monetary policy tools (RSi(t), i ¼ 1, . . . , L ) and fiscal policy tools (Gi(t), TaxR1i(t), TaxR2i(t), and TaxR3i(t), i ¼ 1, . . . , L) for these countries on the period t ¼ 2017, . . . , 2022 that maximize the criterion KA ¼

1 XL X2022 YPPPi ðt Þ i¼1 t¼2017 YPPP ðt Þ 6 i

ð4:5Þ

(the average ratio of per capita GDP (PPP) to corresponding basic values on this period) subject to the constraints j RSi ðtÞ  RSi ðtÞ j 0:1RSi ðtÞ, jGi ðtÞ  Gi ðtÞj  0:1Gi ðtÞ,

ð4:6Þ ð4:7Þ

4.4 Parametric Control Based on Model 3: A Series of Problem Statements. . .

j TaxR1i ðtÞ  TaxR1i ðtÞ j 0:1TaxR1i ðtÞ, j TaxR2i ðtÞ  TaxR2i ðtÞ j 0:1TaxR2i ðtÞ, j TaxR3i ðtÞ  TaxR3i ðtÞ j 0:1TaxR3i ðtÞ,

341

ð4:8Þ ð4:9Þ ð4:10Þ

where t ¼ 2017, . . . , 2022 and i ¼ 1, . . . , L, and also subject to the following additional stable growth constraints in which i designates one of the EAEU countries: PI i ðtÞ  PI r ðtÞ þ 0:05,

ð4:11Þ

GDEF i ðtÞ  0:03Y i ðtÞPY i ðtÞ,

ð4:12Þ

GDEBT i ðtÞ  0:5Y i ðtÞPY i ðtÞ:

ð4:13Þ

The notations are the following: i as country number (see subscripts of variables below); t as year number; YPPPi ðt Þ as the per capita GDP (PPP) of country i; RSi ðt Þ as the short-term interest rate (annual average %); Gi ðt Þ as the government consumption at 2005 prices; TaxR1i ðt Þ as the effective tax rate of CIT; TaxR2i ðt Þ as the effective tax rate of IIT; TaxR3i ðt Þ as the effective tax rate of indirect tax; PI i ðt Þ as annual average inflation; GDEF i ðt Þ as state budget gap; GDEBT i ðt Þ as public debt; r as the number of an EAEU country with minimum inflation among all other EAEU members; Y i ðt Þ as GDP at 2005 prices; and, finally, PY i ðt Þ as GDP deflator. Note that the symbol “” indicates of the basic value of an appropriate variable. In the case A ¼ i (L ¼ 1), the problem GE i is the parametric control problem at the level of a separate ith country. If A ¼ EA (the set of the five EAEU countries) or A ¼ W (the set of all 62 countries of Model 3), then the problems GE EA and GE W are the coordinated parametric control problems at the levels of the EAEU and the world economy, respectively.

4.4.2

Solutions of Parametric Control Problems

The discrete optimal control problems based on Model 3 (see the previous subsection) were solved using an original algorithm with combination of penalty functions and the Nelder–Mead method. The algorithm was implemented in EViews. To illustrate this algorithm for solving parametric control problems based on Model 3, first represent its equations as the system yit ¼ φi ðytþk , ytþk1 , . . . , ytþ1 , yt1 , . . . , ytp , ut , vt Þ, i ¼ 1, . . . , n, t ¼ t 0 þ p, . . . , t T  k, with the initial conditions

ð4:14Þ

342

4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

yt0 ¼ yt0 , . . . , yt0 þp1 ¼ yt0 þp1 ,

ð4:15Þ

and the terminal conditions ytT kþ1 ¼ ytT k þ c, . . . , ytT ¼ ytT 1 þ c:

ð4:16Þ

  The notations are the following: yt ¼ y1,t , y2,t , . . . , yn,t 2 Rn as the endogenous as the time-invariant endogenous vector; ut ¼ variables vector;c , u , . . . , u as the vector of controlled exogenous variables; vt ¼ u 1,t 2,t q ,t 1   v1,t , v2,t , . . . , vq2 ,t as the vector of uncontrolled exogenous variables and parameters, which includes the estimated coefficients and the remainders of the regression equations; k as the maximal time advance for the endogenous variables vector; p as the maximal lag for the endogenous variables vector; φi as given functions; and, finally, yt0 , yt0 þ1 , . . . , yt0 þp1 as the collection of initial values for the endogenous variables vectors. System (4.14), (4.15) and (4.16) is solved by the Fair–Taylor algorithm [37]. The algorithm for solving parametric control problems based on Model 3 can be described as follows: Step 1. Write the parametric control problem GE A (see the previous subsection) as the following discrete optimal control (DOC) problem based on Model 3 with constraints (4.18) and (4.19) imposed on the control vector and the endogenous variables vector, respectively. This problem will be referred to as the problem DOC. Problem DOC Given the exogenous variables vectors vt , t ¼ t1, . . ., t2, and the initial conditions, find the vectors ut , t ¼ t1,. . ., t2 that maximize the criterion Z¼

Xt2 t¼t 1

K t ð yt Þ

ð4:17Þ

subject to constraints (4.14), (4.15) and (4.16), the control constraints u0jt  ujt  u00jt , j ¼ 1, . . . , l, t ¼ t 1 , . . . , t 2 ,

ð4:18Þ

and the state-space constraints gr ðyt Þ  0, r ¼ 1, . . . , s, t ¼ t 1 , . . . , t 2 :

ð4:19Þ

Here Z denotes the optimality criterion; K t ðyt Þ are given functions of the endogenous variables yt ; u0jt and u00jt specify the lower and upper limits of the admissible range of the tool ujt ; and, finally, gr is a vector function that describes additional constraints on the variables yt .

4.4 Parametric Control Based on Model 3: A Series of Problem Statements. . .

343

Step 2. Using an appropriate change of variables, transform the problem DOC to a mathematical programming problem (the problem MP) with inequality constraints on the control variables ut for all t from the interval [t1, t2]. Problem MP For the given uncontrolled parameter vector v, find the vector u that maximizes the criterion Z ¼ F ðGðU, V ÞÞ

ð4:20Þ

U 0i  U i  U 00i , i ¼ 1, . . . , lðt 2  t 1 þ 1Þ,

ð4:21Þ

gj ðGðU, VÞÞ  0, j ¼ 1, . . . , sðt 2  t 1 þ 1Þ:

ð4:22Þ

subject to the control constraints

Here GðU, V Þ denotes the solution of system (4.14), (4.15) and (4.16) in y by the Fair–Taylor method [20]; U is the controlled parameter vector; V gives the uncontrolled parameter vector; function (4.20) expresses criterion (4.17) in the new variables; and, finally, relationships (4.21) and (4.22) correspond to constraints (4.18) and (4.19) in the new variables. Step 3. Using the method of penalty functions, transform the problem MP to an unconstrained maximization problem (further referred to as the problem MPU) by incorporating the inequality constraints into the control criterion with the penalty function:  ΦðU Þ ¼

99999 if constraints ð4:21Þ and ð4:22Þ are violated; 0 otherwise:

ð4:23Þ

This leads to the following unconstrained maximization problem: Problem MPU For the given uncontrolled parameter vector v, find the vector u that maximizes the criterion: Z ¼ F ðGðU, V ÞÞ þ ΦðU Þ:

ð4:24Þ

Step 4. Solve the problem MPU using the standard Nelder–Mead method. As an example consider (a) the solutions of nine parametric control problems with per capita GDP (PPP) maximization during the period 2017–2022 for separate countries (the problems GE i, i ¼ 1,. . ., 9) and (b) the solutions of parametric control problems of coordinated economic growth at the levels of the EAEU and the global economy (the problems GE EA and GE W , respectively), both within the framework of Model 3.

344

4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

Table 4.6 % variations of per capita GDP (PPP) for period 2017–2022

Problem GE i GE EA GE W

Country Kaz Rus 3.38 0.71 3.74 1.75 5.92 3.66

Blr 0.74 1.17 1.97

Kgz 1.33 2.11 4.62

Arm 0.69 0.90 2.47

Pak 0.45 – 2.41

Ind 1.27 – 4.30

Table 4.6 below presents the solutions of these problems in the form of average % deviations of per capita GDPs (PPP) during the period 2017–2022 in comparison with the baseline scenarios. Here the problems GE i correspond to separate countries with the following notations: Kaz, Kazakhstan; Rus, Russia; Blr, Belarus; Kgz, Kyrgyzstan; Arm, Armenia; Pak, Pakistan; and Ind, India. The solutions of all nine parametric control problems (the calculated values of the economic policy tools) can be summarized as follows. For any quarter on the period 2017–2022, in all nine problems, the short-term interest rate (monetary policy tool) took the minimum admissible value under the corresponding constraint; the government consumption at 2005 prices (budget policy tool) took the maximum admissible value under the corresponding constraint; the effective rates of indirect tax, CIT and IIT (tax policy tools), took the minimum admissible values under the corresponding constraints. As indicated by the data in Table 4.6, coordinated macroeconomic policy measures had higher efficiency at the level of the EAEU than at the levels of a separate member country. For Kazakhstan, the coordinated monetary and fiscal policies at the level of all EAEU countries were increasing the per capita GDP (PPP) by 3.74% on average during the period 2017–2022; at the same time, the coordinated monetary and fiscal policies at the level of Kazakhstan were increasing its per capita GDP (PPP) only by 3.38% on average during the period 2017–2022. Moreover, the same analysis of economic indexes for India and Pakistan also demonstrated higher efficiency of coordinated macroeconomic policy at the level of the global economy than at the level of a separate country. For the EAEU countries, the efficiency of coordinated optimization measures at the level of the global economy was also greater than at the level of the EAEU.

4.4.3

Applicability Testing of Solutions

The developed knoware and software were used to test the implementability of the optimal economic policy tools calculated on the basis of Model 3 (see the testing methods in Sect. 4.3). As an example Table 4.7 presents the estimated stability indexes of the mappings defined by Model 3 with the solution of the coordinated parametric control problem at the level of five EAEU countries. The following collection of input factors was adopted for the five EAEU countries: potential GDP, government consumption, indirect tax rate (VAT and excises), IIT rate, and the currency reserves of Central

4.4 Parametric Control Based on Model 3: A Series of Problem Statements. . . Table 4.7 Estimated stability indexes for different radii of ball

Radius (in %) 1 0.5 0.25 0.125

345

Value of stability index 0.024 0.012 0.006 0.003

Table 4.8 Sensitivity coefficients of real GDP to exogenous variables for the three EAEU countries Exogenous variable Real government consumption Effective rate of indirect tax Nominal exchange rate (USD to domestic currency) Short-term interest rate Price of exported products Price of imported products

Country Kazakhstan 0.130 0.023 0.0001 0.020 0.020 0.033

Russia 0.087 0.025 0.054 0.050 0.030 0.071

Belarus 0.039 0.009 0.0001 0.030 0.116 0.015

Bank. The role of the output variables of the mapping was played by 45 macroeconomic indexes. A normalized vector of input parameters was produced within a ball of given radius by a uniform randomizer. The radius was associated with the maximal % deviation of all chosen input factors from their basic values. In accordance with Table 4.7, the stability indexes were vanishing as the radius of the input parameter ball was going to 0 almost with the same rate of convergence. Next, for the three EAEU countries (Kazakhstan, Russia, and Belarus), Table 4.8 presents the sensitivity coefficients of the real GDP to the exogenous variables during the period 2017–2022 under the optimal economic policy tools calculated by solving the coordinated parametric control problem on the basis of Model 3 at the level of the five EAEU countries. The signs of all data in this table well match the main postulates of macroeconomics. As indicated by the testing results, the optimal values of economic policy tools could be implemented.

4.4.4

Study of Solutions Dependence on Uncontrolled Parameters

The results of sensitivity analysis for the criteria of the parametric control problems demonstrated that the endogenous variables of Model 3 were mostly sensitive to the variations of the exogenous variable known as the Brent crude oil price. Therefore, it was studied how the optimal criteria values of all stated parametric control problems on the basis of Model 3 depended on this exogenous parameter. To this end, the

346

4 Macroeconomic Analysis and Parametric Control Based on Global Multi-country. . .

Table 4.9 Optimal criterion values for parametric control problem GE i at the level of Kazakhstan depending on values of coefficient kp oil Criterion K kz

Values of coefficient kp oil 0.80 0.90 1.0382 1.0375

1.00 1.0368

1.10 1.0360

1.20 1.0354

multiplicative coefficient kp oil was placed at the exogenous variable (Brent crude oil price), with the following admissible values: 0.80, 0.90, 1.00, 1.10, and 1.20. For kp oil ¼1.00, the Brent crude oil price in Model 3 on the forecasting period was equal to the basic value. As an example Table 4.9 presents the optimal values of the criterion in the parametric control problem GE i at the level of Kazakhstan (the corresponding values without parametric control were 1). Clearly, for the criterion K kz (4.5) of the problem GE kz , the average efficiency of economic policy tools was about 3.67%. As follows from the optimal criterion values for the parametric control problem at the level of Kazakhstan, the economic policy tools demonstrated higher efficiency in terms of (4.5) under smaller Brent crude oil price. Moreover, the efficiencies of the economic policy tools in terms of criterion (4.5) were monotonically decreasing with each 10% increase of the Brent crude oil price. The solutions of the parametric control problem GE kz (the calculated values of the economic policy tools) for the five scenarios of the Brent crude oil price can be summarized as follows. For any quarter on the period 2017–2022, in all five scenarios, the short-term interest rate (monetary policy tool) took the minimum admissible value under the corresponding constraint; the government consumption at 2005 prices (budget policy tool) took the maximum admissible value under the corresponding constraint; and the effective rates of indirect tax, CIT, and IIT (tax policy tools) took the minimum admissible values under the corresponding constraints.

Conclusions

This book has described the theory of parametric control, in particular, testing methods for the applicability conditions of computer simulations to real macroeconomic systems. The theory consists of seven basic components, which have been used: • To develop a parameter identification algorithm for large-scale macroeconomic models • To suggest numerical estimation methods for the structural stability of dynamic models and the stability of mappings, including their stability indexes • To formulate and prove sufficient conditions for the existence of solutions of optimal parametric control design and choice problems • To prove sufficient conditions for a continuous dependence of optimal criteria values on uncontrolled functions in optimal parametric control design and choice problems, including an analysis method for such dependencies In particular, the definition of a bifurcation point for the extremals of the optimal parametric control choice problem has been introduced as well as sufficient conditions for the existence of a bifurcation point have been established. Note that the efficiency of parametric control has been illustrated on Kondratiev’s cycle model by proving its weak structural stability with a calculated optimal parametric control law and estimating the set of all bifurcation points for the extremals of an associated parametric control problem. Also the efficiency of this theory has been illustrated on the Lorenz model by suppressing its chaotic attractor with a chosen parametric control law. In addition, this book has presented results on the forecasting, macroeconomic analysis, and parametric control of macroeconomic systems using the theory of parametric control and the developed global multi-country dynamic models of the following classes: the computable general equilibrium (CGE) model; the dynamic stochastic general equilibrium (DSGE) model; and, finally, the hybrid econometric model.

© Springer Nature Switzerland AG 2020 A. A. Ashimov et al., Macroeconomic Analysis and Parametric Control of a Regional Economic Union, https://doi.org/10.1007/978-3-030-32205-2

347

348

Conclusions

The CGE model (Model 1) has been built on the basis of a conceptual description of the global economy. This model has been calibrated using the sets of social accounting matrices (SAMs) for nine global regions, which involve national statistical data (input–output tables) as well as statistical and forecasting data on basic macroeconomic indexes and international trade. After calibration Model 1 has been tested for applicability to real macroeconomic systems using the parametric control approach. After testing Model 1 has been employed for mid-term forecasting and macroeconomic (snapshot and scenario) analysis in order to study different scenarios as follows: an increase of effective rates for some taxes and duties; the tightening of international economic sanctions; the establishment of a new monetary union; the collapse of the Eurasian Economic Union (EAEU) or the Commonwealth of Independent States (CIS); and some others. In addition, the 2009 and 2016 economic crises in Kazakhstan have been thoroughly examined in terms of USD. Within the framework of this model, 12 parametric control problems focusing on economic growth, food security, the reduction of trade gap, and regional development disproportions as well as with the economic structural adjustment of the Republic of Kazakhstan have been solved, and their solutions have been tested for applicability to real macroeconomic systems. As demonstrated in the book, the solutions of these dynamic optimization problems with parametric control methods yield a greater economic effect at the levels of a regional economic union and the global economy than at the level of separate countries. Moreover, the parametric control approach has allowed to derive the conditions of solutions’ implementability in real macroeconomic systems as well as to establish the dependence of optimal criteria values on uncontrolled factors in the optimal parametric control design and choice problems. Next, this book has developed the global multi-country and multi-sector DSGE model (Model 2), which includes a conceptual description of the global economy. This model has been comprehensively tested for applicability to real macroeconomic systems with six methods, in particular, parametric control methods. The snapshot ex post and ex ante analysis procedures have allowed to decompose the volatility of basic macroeconomic indexes (GDP and inflation rate) into the impacts of endogenous and exogenous shocks for each of the nine global regions of Model 2. For a wide range of scenario analysis problems, this model has been adopted to consider different effects as follows: an increase of public interest rates; an establishment of a monetary zone; an increase of product prices; an increase of the macroprudential parameter for banks; an increase of the macroprudential parameter for producers of non-tradable goods. Some measures on reducing the dollarization of a national economy and the impact of public debt on capital flows have been assessed in quantitative terms. The statements and optimal solutions of the parametric control problems within Model 2, at the levels of a separate country and the global economy, are focusing on economic growth and smaller volatility of macroeconomic indexes. The parametric control approach has been used to establish the implementability conditions of solutions of the optimal parametric control problems as well as to study the dependence of optimal criteria values on uncontrolled factors in these problems. Moreover, the efficiency of parametric control has been illustrated on the global multi-country econometric model developed on the basis of the Fair model and also

Conclusions

349

on the Global Economy Model of the International Monetary Fund. This multicountry econometric model (Model 3) has been tested for applicability to real macroeconomic systems using parametric control. After testing Model 3 has been employed for mid-term forecasting and snapshot (ex post and ex ante) analysis. In addition, the impact of several economic tools has been estimated using scenario analysis, namely, the ones of fiscal and monetary policy, pricing for export and import products, and domestic currency exchange. Also a series of optimal parametric control problems have been stated and solved within the framework of Model 3, at the levels of a separate country, the EAEU, and the global economy, which is focusing on the growth of the GDP at Purchasing Power Parity (PPP). The parametric control approach has been used to establish the implementability conditions of solutions of the optimal parametric control problems as well as to study the dependence of optimal criteria values on uncontrolled factors in these problems. The implementable solutions of all the parametric control problems within the three large-scale economic models considered in this book have demonstrated greater economic effects in case of coordinated optimal economic policy at higher levels. This conclusion fully applies to each level, from separate countries to the global economy. Therefore, the research results presented in this book form a modern paradigm of macroeconomic analysis and optimal coordinated decision-making on economic policy using the theory of parametric control.

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Index

A Absolute stability index, 10 Applicability testing impulse responses, 281, 284 local sensitivity analysis, 281, 285 marginal likelihoods, 285, 286 moments, 279, 281 retrospective forecasting, 279, 280 stability indexes, 279 Autonomous dynamic model, 1 Autonomous stochastic dynamic systems, 48 Autoregression models, 314 Auxiliary equation, 94, 96, 97

B Balance equation, 98, 99, 101–103 Bayes formula, 277 Bayesian algorithm, 274, 277 Bifurcation point for extremal bisection method, 61 finite collection of algorithms, 59 non-empty closed sets, 60 variational calculus problem, 58 Bolzano–Weierstrass theorem, 38 Brent crude oil price, 345, 346

C Caratheodory function, 34 Cauchy problem, 33, 37 Commonwealth of Independent States (CIS), 348 Comparative macroeconomic analysis annual growth rates of economies, 318–320

domestic currency-to-USD exchange rates, 326 ex post and ex ante, 317 inflation, 328, 329 nominal per capita GDPs, 319–321 ratios of consolidated budget deficit (-)/ surplus (+) to GDP, 328, 329 ratios of exported goods and services to GDP, 324, 325 ratios of government consumption to GDP, 323, 324 ratios of households’ consumption to GDP, 320–322 ratios of imported goods and services to GDP, 325, 326 ratios of investments to GDP, 322, 323 ratios of public debt to GDP, 327, 329 short-term interest rates, 327, 328 Comparative snapshot analysis fiscal policy tools import tax, 140, 141 net capital income tax, 135 net households’ income tax, 136 net labor income tax, 135 net sales tax, 136–139 GDP ratio of aggregate investments, 128 ratio of exports, 129 ratio of government consumption, 129 ratio of Households’ consumption, 129 ratio of imports, 130 ratio of trade gap, 130 GVA share of sector agriculture, forestry and fishery, 163

© Springer Nature Switzerland AG 2020 A. A. Ashimov et al., Macroeconomic Analysis and Parametric Control of a Regional Economic Union, https://doi.org/10.1007/978-3-030-32205-2

355

356 Comparative snapshot analysis (cont.) chemical and petrochemical industry, 165 construction, 163 education, public health and public administration, 160 financial services, 164 food industry, 161 hydrocarbon production and natural gas extraction, 158 metal industry, 159 metalworking and machine building, 159 mining, 158 other industries, 162 other services, 162 production and supply, 160 professional, scientific, and technical activities, 161 textiles, clothes, leather and associated goods, 164 transportation, 165 macroeconomic indexes EAEU countries, 131–134 ratio of sectoral Households’ consumption agriculture, forestry and fishery, 155 chemical and petrochemical industry, 157 construction, 155 education, public health and public administration, 152 financial services, 156 food industry, 153 hydrocarbon production and natural gas extraction, 150 metal industry, 151 metalworking and machine building, 151 mining, 150 production and supply, 152 production of textiles, clothes, leather and associated goods, 156 professional, scientific, and technical activities, 153 transportation, 157 ratios of households’ consumption to outputs of sectors, 154 shares of exports agriculture, forestry and fishery, 147 chemical and petrochemical industry, 149 construction, 147

Index education, public health and public administration, 144 financial services, 148 food industry, 145 hydrocarbon production and natural gas extraction, 142 metal industry, 143 metalworking and machine building, 143 mining, 142 production and supply, 144 production of textiles, clothes, leather and associated goods, 148 professional, scientific and technical activities, 145 transportation, 149 Computable general equilibrium (CGE) model, 73, 347, 348 auxiliary equation, 94, 96, 97 balance equation, 98, 99, 101–103 economic regions and sectors, 106, 107 elementary agents of sectors, households and government, 91–93 Globe’s Behavior, 93 initial database and calibration auxiliary algorithm, 111, 112, 115 auxiliary operations and algorithms, 107 balanced international trade data, 113, 114 collections of SAMs, 107 elements, matrix, 116, 119 input–output tables (IOT), 110, 117, 118 macroeconomic indicators, 120 matrices, 116 Model 1, 125 nonlinear programming problem, 121 parameter estimation procedure, 107 Payment for Labor, 119 profits, mixed income, 120 Regions, 122 SAM, 120 simulation modeling, GAMS, 125 transportation services, 122–124 Model 1 applicability testing, 126, 127 macroeconomic analysis, 127 Model 1 and solution algorithm, 104, 106 retrospective and forecasting periods, 106, 107 solutions dependence, uncontrolled parameters, 200, 201 statement of problem SP, 198, 199

Index structural adjustment problem, 196, 197 testing of solutions implementability, 199 trade gap and regional development disproportions, 192, 193, 195 Computer simulations, 1, 309 Consumer price indexes (CPI), 266 Continuous-time dynamic systems dynamic optimization problem, 31–36 finite collection of algorithms, 36–39 Corporate income tax (CIT), 232, 248, 259 Cowles Commission approach, 312

D Discrete optimal control (DOC), 342 Discrete-time dynamic systems dynamic optimization problem, 40, 41 finite collection of algorithms, 41, 42 Discrete-time stochastic dynamic systems dynamic optimization problem, 42–45 finite collection of algorithms, 45–48 Dynamic stochastic general equilibrium (DSGE) model, 347, 348 applicability tests (see Applicability testing) economic policy tools, 302, 303 estimated sensitivity coefficients, 305 expected GDPs and RMSDs, 305 feasibility testing, 302, 304, 305 GDP forecasts, 294 global economy (see Global economy) government expenditures, 307 influential shocks GDP, forecasting period, 292 GDP, retrospective period, 291 inflation, forecasting period, 293 inflation, retrospective period, 292 knoware and software, 286, 287 linear (see Linear model) macroeconomic models, 203 mid-term forecasting aggregate investments, 288 average loan and deposit rates, 288, 291 deposits, 288, 290 employed population, 288, 289 final consumption expenditures, households, 288, 289 final consumption expenditures, public authorities, 288, 290 GDP, 288 loans, 288, 290 net capital inflow, 297 nonlinear (see Nonlinear model)

357 parametric control problem solutions, 300–302 statements, 298–300 public bonds rate, 306 scenario macroeconomic analysis, 293, 297, 298 snapshot macroeconomic analysis, 289, 293 stability indexes level of EAEU, 304 level of global economy, 304 level of Kazakhstan, 304 uncontrolled parameters, 305, 306 USD deposits, 296 variations, Kazakhstan’s macroeconomic indexes, 301

E Economic growth, 192, 196, 199 Elementary agents, 74 Euler equation, 211, 219, 223 Eurasian Economic Union (EAEU), 310, 348 Exogenous variables estimated sensitivity coefficients, 312 estimated stability indexes, 313, 314

F Fair model, 310 Fair–Taylor algorithm, 314, 316, 342 Fair–Taylor method, 278, 343 Firm’s behavior aggregate loans, 232 autoregression, 231 CES function, 237, 240, 243 Cobb–Douglas production function, 230, 234 debt–equity capital ratio, 230 elasticity of substitution, 241, 244 expenditures, 231 flexible product prices, 233 inflexible product prices, 233, 234 intermediate goods, 238, 239, 241 Lagrange multipliers, 235, 236 lease rates, 245 notations, 230 optimal consumer demands, 237 optimal labor hired, 242 second-level banks, 230, 245 transversality, 237 utility function, 233, 234 wages, 243, 244

358 G Generalized method of moments (GMM), 310 Global economy, 317 balance and auxiliary equations, 268–272 computable general equilibrium model, 74 economic interaction, 74 endogenous variables, 77, 79 exogenous parameters, 77 exogenous variables, 75, 76 firm (see Firm’s behavior) household (see Household’s behavior) hypotheses, 74 prerequisites, 204 second-level banks (see Second-level bank’s behavior) state (see State’s behavior) Government’s behavior, 87, 88, 90, 91 Grönwall’s inequality, 54, 57 Gross value added (GVA), 75

H Hessian matrix, 211, 213, 214 Household’s behavior, 84, 86, 87 budget balance impatient, 205 patient, 205 Calvo inflexible wages, 207 CES function, 207 Dynare converge, 214 Hessian matrix, 211, 213, 214 impatient, 208, 209 integral CES functions, 221, 227 Kuhn–Tucker conditions, 218–220, 223, 224 Lagrange multipliers, 215–217 nonfinancial assets, 205, 206, 225, 226 Riemann sums, 222 second-level banks, 205, 206, 228, 229 transversality, 217 Hybrid econometric model, 347 aggregate algorithm, 309 applicability testing, 311–313 Fair model, 310 IMF model, 310 macroeconomic analysis and mid-term forecasting (see Mid-term forecasting) parametric control (see Parametric control) regression functions, 311 RU-MIDAS model, 310 single- and multi-country, 309 Hypotheses of Mather’s theorem, 11

Index I Individual income taxes (IIT), 259 Interaction interface for the models, 310, 311 Inverse function theorem, 7

K Kondratiev’s cycle model, 2, 61, 347 capital productivity ratio, 64 chain-recurrent set, 62, 63 cyclic phase trajectory, 62 economic systems, 62–64 efficiency of innovations, 65 exogenous parameters, 65, 66 notations, 61 parameter identification procedure, 62 robustness, 62 structural stability, 64 Kuhn–Tucker theorem, 210, 217, 218, 234

L Linear model Bayesian algorithm, 275 calibrated parameters, 274 endogenous variables, 275 estimated parameters, 274 global economy, 274 likelihood function, 276 log-deviations, 275, 277 log-linearization procedure, 273 probability density functions, 277 probability distributions, 276 Lipschitz constants, 44, 47 Lorenz model, 2, 347 asymptotically stable singularity, 69, 71, 72 chaotic attractor, 67, 68, 70 chaotic trajectories (singularities), 66 computer simulations, 66 limit cycle, 68, 70 nonlinear dynamic systems, 65 optimal parametric control law, 67 strange attractor, 69, 71

M Macroeconomic analysis, 127 comparative (see Comparative macroeconomic analysis) mid-term forecasting (see Mid-term forecasting) scenario (see Scenario macroeconomic analysis) snapshot, 289, 293

Index Macroeconomic indexes, 348 Macroprudential parameter, 293, 297, 298 Maximum absolute stability index, 10 Metropolis–Hastings algorithm, 277 Mid-term forecasting conclusions, 339 DSGE (see Dynamic stochastic general equilibrium (DSGE) model) economic indexes, 316 IMF, 317 knoware, 314, 315 software, 315, 316 2015–2020 consumer-price inflation, 319 2015–2020 economic growth, 318 Model 2. See Dynamic stochastic general equilibrium (DSGE) model Model 3. See Hybrid econometric model Moren’s theorem, 25, 29

N Nelder–Mead method, 341, 343 Nelder–Mead simplex direct search method, 4 Nonlinear model parameter estimation, 272 parameter values, 278 Nonlocal injectivity, 14, 15, 19 Numerical estimation methods differentiable model mappings (see Stability of differentiable model mappings; Stability of nodes of differentiable model mappings) stability indexes of model mappings, 8–10 weak structural stability of dynamic models, 5–8

O Observed vs. generated economic indexes autocorrelation coefficients, 281 correlation matrices, 282 cross-correlation matrices, 283 Organization for Economic Cooperation and Development (OECD), 317 Orlov’s definition, 8

P Parametric control DSGE (see Dynamic stochastic general equilibrium (DSGE) model) problem statements, 340, 341

359 sensitivity coefficients, 345 solutions of economic indexes, 344 estimated stability indexes, 345 EViews, 341 implementability testing, 344, 345 maximum and minimum admissible values, 344 problem DOC, 343 problem MP, 343 problem MPU, 344 uncontrolled parameters, 345, 346 Parametric control theory, 347 bifurcation point (see Bifurcation point for extremal) components, 2 continuous dependence of optimal criterion values, 48–50 continuous dependence on uncontrolled functions, 55, 57 continuous-time dynamic systems (see Continuous-time dynamic systems) discrete-time dynamic systems (see Discrete-time dynamic systems) discrete-time stochastic dynamic systems (see Discrete-time stochastic dynamic systems) estimation (see Numerical estimation methods) finite number of continuous functions, 53 Kondratiev’s cycle (see Kondratiev’s cycle model) large-scale macroeconomic models, 3–5 Lorenz model (see Lorenz model) nonautonomous continuous-time dynamic system, 53–55 nonautonomous discrete-time dynamic system, 51–53 stochastic dynamic systems, 58 structure of, 2, 3 Per capita GDP (PPP), 341, 343, 344 Purchasing Power Parity (PPP), 349

R Rest of World (RoW), 317, 322, 323 Robinson’s theorem, 5, 7 RU-MIDAS model, 310 Runge–Kutta method, 67

360 S Scenario analysis effective rates, import taxes GDP variations, regions, 166 GVA variations, sectors, 166 price variations, products, 167 exchange rates in EAEU countries, 182 exchange rates, EAEU countries GDP variations, 183 GVA variations, 183–185 price variations, 186, 187 free-trade zone GDP variations, regions, 172 GVA variations, sectors, 172–175, 177 GDP variations, 188 GVA variations, 188–192 international sanctions GDP variations, regions, 177 GVA variations, sectors, 178–182 macroeconomic indexes, 163 sales and sectoral taxes GDP variations, regions, 168, 171 GVA variations, 170 GVA variations, 170 GVA variations, sectors, 167, 169 price variations, products, 168 Scenario macroeconomic analysis, 293, 297, 298 10% currency exchange rate increase, 331, 333, 334 10% government consumption increase, 329–331 10% import price increase, 336, 337 10% indirect tax rate increase, 331–333 10% short-term interest rate increase, 333, 335, 336 actual and effective exchange rates, 338, 339 assessments, 329 ex post and ex ante, 329 Second-level bank’s behavior aggregate loan rates, 256, 257 borrowing-loan ratio, 247 economy’s de-dollarization, 249 equilibrium (minimal admissible) value, 249 Lagrange multiplier, 253 loans and borrowed finances, 248 nonzero second partial derivatives, 250, 252 patient households, 246 public bonds, 246

Index reduced dividends with variables, 248 transversality, 254, 255 zero second partial derivatives, 252, 253 Sector’s behavior, 79–82, 84 Shanghai Cooperation Organization (SCO), 317 Sims algorithm, 274 Singularities, 12–14 Snapshot macroeconomic analysis, 289, 293 Social accounting matrices (SAM), 107–109, 115, 117, 119, 123, 126, 348 Social income taxes, 259 Stability indexes of model mappings econometric models, 8–10 Euclidean metric, 9 monograph, 8 parametric identification problem, 8 Stability of differentiable model mappings economic indexes, 11 immersion, 11 nonlocal injectivity, 14, 15 set of singularities, 12–14 smooth mappings, 11 submersion, 11 submersion with fold, 12 elementary parallelepiped, 20 injectivity, 19 least squares method, 18 linear homogeneous system, 19 orthonormal basis vectors, 16, 17 singularities, 15, 16 transversality, 17 Whitney mapping, 20, 21 Stability of nodes of differential model mappings germs genotype order k, 29, 30 grid functions, 24 infinitesimal V-stability, 22, 23 linear approximation, 25 local case of Mather’s theorem, 23, 24 notations, 22 plane П spanned estimation, 27, 28 Standard least squares method, 311 State’s behavior Central Bank, 266, 267 government, 257–260 public agents, 257, 260–266 System of National Accounts (SNA), 316

Index T Taylor rule, 266 Taylor series, 273 Two-stage least squares method (2LSM), 310

V Variational calculus problems, 3

W Weak structural stability of dynamic models, 1 aggregate algorithm, 7, 8 chain-recurrent set, 6

361 computational algorithm, 5 discrete-time dynamic system, 7 invertibility of differentiable mapping, 7 Jacobians estimation, 7 national economic systems, 5 symbolic image algorithm, 6 Weierstrass extreme value theorem, 34, 43, 45 Whitney mapping, 20, 21