Nonlinearities in Economics: An Interdisciplinary Approach to Economic Dynamics, Growth and Cycles

Description

https://www.springer.com/gp/book/9783030709815 This interdisciplinary book argues that the economy has an underlying non-linear structure and that business cycles are endogenous, which allows a greater explanatory power with respect to the traditional assumption that dynamics are stochastic and shocks are exogenous.  The first part of this work is formal-methodological and provides the mathematical background needed for the remainder, while the second part presents the view that signal processing involves construction and deconstruction of information and that the efficacy of this process can be measured. The third part focuses on economics and provides the related background and literature on economic dynamics and the fourth part is devoted to new perspectives in understanding nonlinearities in economic dynamics: growth and cycles. By pursuing this approach, the book seeks to (1) determine whether, and if so where, common features exist, (2) discover some hidden features of economic dynamics, and (3) highlight specific indicators of structural changes in time series. Accordingly, it is a must read for everyone interested in a better understanding of economic dynamics, business cycles, econometrics and complex systems, as well as non-linear dynamics and chaos theory. Reviews Preface Acknowledgments Contents Contributors 1 Introduction 1.1 Nonlinearity and Unpredictability in Economics 1.2 Scheme of This Book References Part I Mathematical Background 2 Dynamical Systems 2.1 Dynamical Systems and Their Classification 2.2 Continuous-Time Dynamical 2.2.1 Continuous-Time Dynamical Systems from Ordinary Differential Equations 2.2.2 Continuous-Time Dynamical Systems from Systems of Ordinary Differential Equations 2.3 Stability of Continuous-Time Systems 2.4 Limit Cycles and Periodicity of Continuous-Time Systems 2.5 Discrete-Time Dynamical Systems 2.5.1 Cobweb Diagram 2.6 Attractors and Repellers 2.7 Existence of Periodic Behaviour 2.7.1 Schwarz Derivative 2.7.2 Singer's Theorem 2.7.3 Sharkovsky's Theorem References 3 An Example of Nonlinear Dynamical System: The Logistic Map 3.1 The Logistic Map 3.1.1 Fixed Points 3.1.2 Feigenbaum's Universal Constants 3.1.3 Schwarz Derivative 3.1.4 Singer's Theorem Applied to the Logistic Map 3.1.5 Closed Formulas References 4 Bifurcations 4.1 Bifurcations in the Parameter Space 4.2 Homoclinic and Heteroclinic Orbits 4.3 Local and Global Bifurcations 4.4 Local Bifurcations in the Continuous Case 4.4.1 Bifurcations with Colliding Equilibria 4.4.1.1 Saddle-Node Bifurcations 4.4.1.2 Transcritical Bifurcations 4.4.1.3 Pitchfork Bifurcations 4.4.2 Hopf Bifurcation 4.4.3 Degenerate Hopf Bifurcation in the van der Pol Equation 4.5 Local Bifurcations for Discrete-Time Systems References 5 From Local Bifurcations to Global Dynamics: Hopf Systems from the Applied Perspective 5.1 Hopf Bifurcation Theorem 5.2 Two Specific Examples: Lorenz and Rössler Systems 5.3 Shilnikov's Theorem 5.4 Delay-Differential Equations References 6 Chaos 6.1 Chaos 6.2 Measuring Sensitive Dependence on Initial Conditions 6.2.1 Stretching and Folding 6.3 Measures on Attractors 6.3.1 Geometry of Attractors 6.3.2 Measures of Information 6.3.2.1 Correlation 6.3.2.2 Entropy 6.3.3 Thermodynamic Formalism: The Ring That Ties (Almost) All References 7 Embedding Dimension and Mutual Information 7.1 Embedding Dimension 7.1.1 Time Lag 7.2 Mutual Information References Part II Signal Analysis and Modelling Tools for Economic Systems 8 Signal Processing 8.1 Signal Processing as a Process of Computation 8.1.1 Signal Observation and Perception 8.1.2 Computation Performed on a Signal 8.1.3 Dynamics Between Symbols 8.1.4 Complexity of the Prediction of Dynamics 8.1.5 Measuring Computation 8.2 Signal Processing by Neural Networks 8.3 Noise-Cleaning of Signals 8.4 Symbolic, Probabilistic, and Metric Characterizations 8.5 Fourier and Power Spectrum Analysis References 9 Applied Spectral Analysis 9.1 Frequency, Spectral Analysis, Phase and Phasor 9.1.1 Fourier Series and Transform 9.1.2 Spectral Density, Power Spectrum and Periodogram 9.1.3 Time–Frequency Representations of Signals: Gabor and Wavelet Transform 9.2 Applications 9.2.1 Power Spectrum of the Logistic Map 9.2.2 State Space and Power Spectrum of a Periodic, Quasiperiodic and Chaotic Signals 9.2.3 Fourier Methods for Finding Frequency Components of a Given Signal 9.2.4 Gabor and Wavelet Analysis of Hénon's Map References 10 Recurrence Quantification Analysis: Theory and Applications 10.1 Motivation 10.2 Recurrence Plot: Introduction 10.3 Recurrence Quantification Analysis 10.3.1 RQA Measures 10.3.2 RQA Epoch by Epoch Correlation Index 10.4 RQA Applications 10.4.1 Principal Component Analysis (PCA) on RQA 10.4.2 RQE Correlation Index on a Sample Signal References Part III Emergence of Cycles and Growth in Economics 11 On Business Cycles and Growth 11.1 Nonlinearities in Economics 11.2 Business Cycles 11.2.1 Background and Definition 11.2.2 Literature on the Causes of Business Cycles 11.3 Growth 11.4 Dynamical Systems and Economic Theory 11.4.1 Historical Foundations of Dynamical System and Chaos Theory from the Economic Perspective 11.5 Mathematical Approaches to Business Cycles 11.5.1 Detecting Nonlinearities in Data References 12 Trade-Cycle Oscillations: The Kaldor Model and the Keynesian Hansen–Samuelson Principle of Acceleration and Multiplier 12.1 Keynesian Multiplier and Hansen–Samuelson Model 12.2 The Kaldor Model 12.2.1 History of the Kaldor Model on Business Cycles 12.3 Formal Description of the Kaldor Model References 13 The Harrod Model 13.1 Introduction 13.2 Domar's Approach to Economic Dynamics 13.3 Harrod's Approach to Economic Dynamics 13.4 A Mathematical Foundation of Harrod's Instability 13.5 Cycles 13.5.1 Harrod's Knife-Edge 13.5.2 Discussion References 14 Growth and Cycles as a Struggle: Lotka–Volterra, Goodwin and Phillips 14.1 Introduction 14.2 The Phillips Curve 14.2.1 Perfectly Competitive Markets 14.2.2 Monopolistic Markets 14.2.3 Calvo Model and New Keynesian Economics 14.3 Lotka–Volterra Model 14.4 The Goodwin Model 14.4.1 Assumptions of Goodwin's Model 14.4.2 Dynamics of Goodwin's Model 14.5 Kolmogorov Prey–Predator Model References 15 Stable Periodic Economic Cycles from Controlling 15.1 Introduction into Control Fundamentals 15.2 Problem Setting 15.3 Controlling Economics by Thresholds 15.4 Application to a Simple Model of Macroeconomics 15.5 Effects of Threshold (Limiter) Control 15.6 Natural vs. Control-Induced Cycles 15.7 Detailed Control Results 15.8 Conclusions References Part IV New Horizons for Understanding Economics 16 Kaldor–Kalecki New Model on Business Cycles 16.1 The Model 16.1.1 Simulating Shocks in the Economy 16.1.2 Consumptions, Savings and Economic Recessions 16.2 Numerical Proof of Chaotic Behaviour of the Model 16.2.1 Tools 16.2.2 Spectral Analysis 16.2.3 Embedding Dimension 16.2.3.1 Cao Embedding Dimension's Estimation 16.2.3.2 Symplectic Geometry Method 16.2.4 Correlation Integral 16.2.5 Correlation Dimension 16.2.6 Lyapunov Exponents 16.2.7 Entropy 16.2.8 Chaotic Attractor 16.3 Conclusions References 17 Recurrence Quantification Analysis of Business Cycles 17.1 Introduction 17.2 Databases and Time Series 17.2.1 Capital 17.2.2 Income, Investment and Saving 17.3 Tools 17.4 Results and Interpretation 17.4.1 Recurrence Quantification Analysis (RQA) 17.4.1.1 Recurrence Plot (RP) on US GDP 17.4.1.2 RQA Applied to Business Cycle Data 17.4.1.3 Recurrence Quantification Epoch (RQE) on Business Cycle Data 17.4.1.4 RQE Correlation Index 17.4.2 Principal Component Analysis (PCA) on RQA 17.4.3 Statistical Analysis on RQA 17.4.3.1 Mann–Whitney Test 17.4.3.2 Kruskal–Wallis Test 17.5 Conclusions References 18 An Empirical Test of Harrod's Model 18.1 Background and Literature 18.2 Material and Methods 18.2.1 Cycles 18.2.2 US Recessions 18.2.3 Empirical Data 18.3 Calibration of Harrod's Model 18.4 Conclusions References 19 Testing a Goodwin's Model with Capacity Utilization to the US Economy 19.1 Introduction 19.2 Adding Capital Utilization to Goodwin's Model 19.3 The Existence and Stability of Equilibria 19.4 Testing the Extended Goodwin Model on the US Economy 19.5 Conclusions References 20 Financial Stress, Regime Switching and Macrodynamics 20.1 Introduction 20.2 Self-exciting Threshold Cointegrated Autoregressive Model 20.2.1 Test of Cointegration Rank 20.2.2 Parameter Estimation 20.2.3 Test of Switching 20.3 Test of the Model on Economic Data 20.3.1 Discrimination of Regimes 20.3.2 Model Specification 20.3.3 Impulse and Response 20.4 Concluding Remarks References Appendix A A.1 The Dataset A.1.1 US Recessions A.1.2 World GDP Data A.1.3 BEA Data A.1.4 Levy and Chen Data—USA A.1.5 OECD Data A.1.6 RQA Tables A.2 Considerations on tanh versus arctan A.3 Test Results of the Goodwin Model on US Economy A.3.1 Unit Root Tests A.3.2 Lag Length Criteria A.3.3 Granger Non-causality Test A.3.4 Autocorrelation Test A.3.5 Stability of VAR A.4 Financial Stress, Regime Switching and Macrodynamics A.4.1 Estimation Results References Back Cover