Symmetry and Economic Invariance (Advances in Japanese Business and Economics Book 1)

__Symmetry and Economic Invariance__ (second enhanced edition) explores how the symmetry and invariance of economic models can provide insights into their properties. Although the professional economist of today is adept at many of the mathematical techniques used in static and dynamic optimization models, group theory is still not among his or her repertoire of tools. The authors aim to show that group theoretic methods form a natural extension of the techniques commonly used in economics and that they can be easily mastered. Part I provides an introduction that minimizes prerequisites including prior knowledge of group theory. Part II discusses recent developments in the field. Preface 6 Contents 10 Part I Introduction 14 Chapter 1: Introduction 15 1.1 Group Theory and Classification of Mathematical Structure 16 1.2 Lie Groups and Invariance 20 1.3 Economic Applications of Lie Groups 24 Chapter 2: Technical Progress and Economies of Scale:Concept of Holotheticity 25 2.1 A Reformulation of the Problem 25 2.2 Lie Groups 34 2.3 Holotheticity 36 2.4 Conclusion 40 Chapter 3: Holothetic Production Functions and Marginal Rate of Technical Substitution 41 3.1 Types of Technical Progress Functions and Holotheticity 42 3.2 Marginal Rate of Transformation and ExtendedTransformation 46 3.3 Holotheticity and Lie Bracket 49 3.4 Conclusion 53 Chapter 4: Utility and Demand 54 4.1 Integrability Conditions 58 4.2 Conclusion 64 Chapter 5: Duality and Self Duality 65 5.1 Duality in Consumer Theory 65 5.2 Separability and Additivity 69 5.3 Self-Duality in Demand Theory 72 5.4 A Method of Deriving Self-Dual Demand Functions 76 5.5 Empirical Estimation of Self-Dual Demand Functions 78 5.6 Implicit Self-Duality of Production and Cost Functions 79 5.7 Conclusion 82 Chapter 6: The Theory of Index Numbers 83 6.1 Statistical Approach 83 6.2 Test Approach 84 6.3 Economic Index Numbers 89 6.4 Divisia Index 92 Chapter 7: Dynamics and Conservation Laws 96 7.1 The Variational Problem and the Ramsey Rule 97 7.2 Steady State and the Golden Rules 102 7.3 The Hamiltonian Formulation and Control Theory 103 7.4 Noether Theorem and Its Implications 107 7.5 Conservation Laws in von Neumann Model 110 7.6 Measurement of National Income and Income-Wealth Ratios 113 7.7 Conclusion 114 References to Part I 115 Part II Recent Developments 118 Chapter 8: The Invariance Principle and Income-Wealth Conservation Laws 119 8.1 Introduction 119 8.2 Brief Summary of the Literature 120 8.3 A Model with Heterogeneous Capital Goods 121 8.4 Noether'S Theorem (Invariance Principle) 122 8.5 Income-Wealth Conservation Laws 125 8.6 Special Cases 130 8.7 Generalized Income/Wealth Conservation Laws 131 8.8 Income-Capital (Wealth) Conservation Lawin the von Neumann Model 134 8.9 The Total Value Conservation Law of the Firm 138 8.10 Empirical Applications 139 8.11 Summary 144 Appendix 145 References 147 Chapter 9: Conservation Laws in Continuous and Discrete Models 149 9.1 Introduction 149 9.2 Continuous Models 150 9.2.1 Review of the Noether Theorem 150 9.2.2 Model 1: Zero Discount Rate 152 9.2.3 Model 2: Fixed Discount Rate 154 9.2.4 Model 3: Variable Discount Rate 156 9.2.5 Model 4: Technical and Taste Change 157 9.2.6 Model 5: ``Local'' Conservation Laws 158 9.2.7 Total Value Conservation Law of the Firm 159 9.3 Discrete Models (2012 Version) by Shigeru Maeda 160 9.3.1 Introduction 160 9.3.2 Model 6: Discrete Growth Models 160 9.3.3 Quadratic Conservatives: A MathematicalDigression 163 9.3.4 Economic Conservation Laws 169 9.4 Summary 173 Appendix 175 References 181 Chapter 10: Quantity or Quality: The Impact of Labour SavingInnovation on US and Japanese Growth Rates, 1960–2004 183 10.1 Introduction 183 10.1.1 Recent Studies 185 10.2 A Model of Biased (Labour Saving) Technical Change 185 10.2.1 Importance of the Elasticity of Factor Substitution 186 10.2.2 Why Do We Need Biased Technical Change? 189 10.2.3 Equilibrium Growth and Stability Under Biased Technical Change 191 10.3 Applications to the US and Japanese Data 193 10.3.1 Tests of Non-unity of bold0mu mumu σσdottedσσσσ 193 10.3.2 Estimates of Production Functions 195 10.3.3 Simulation Results 199 10.3.4 Biased Technical Change of Japan and the USA 201 10.3.5 Contrast in Response to Oil Crises 203 10.3.6 Economic Performance Revisited 205 10.4 Conclusion 208 Appendices 210 References 213 Chapter 11: A Survey on Recent Developments 215 11.1 Introduction 215 11.2 Extensions of the Income-Wealth Conservation Law 215 11.3 Externalities and Policy Interventions 217 11.4 Stochastic Income and Wealth Conservation Law 219 11.5 Warning 220 11.6 Conservation Laws and Helmholtz Conditions 221 11.7 Comparisons: Three Approaches 224 11.8 Hartwick Rule and Conservation Laws 224 11.9 Factor-Augmenting Technical Changes as the Magnification Type of Lie Group Transformations: Justification for Biased Technical Change 228 11.10 Empirical Estimation of Biased Technical Change and Aggregate Production in Function 231 11.11 More Abstract Applications of Group Theory to Economics and Finance 232 References 234 12 Appendix to Part II. Symmetry: An Overview of Geometric Methods in Economics 236 12.1 Introduction 236 12.2 Toolbox 241 12.2.1 Mapping 241 12.2.2 Charts and Manifolds 242 12.2.3 Curves and Functions 243 12.2.4 Vectors and Tangents 244 12.2.5 1-Forms 247 12.2.6 Tensors 249 12.2.7 Vector Field, Connections and Covariant Derivatives 252 12.2.8 Groups in Differential Equations 254 12.2.9 Calculus of Variation and the Hamiltonian Formulation 258 12.2.10 Conservation Laws and Noether Theorems 260 12.3 Holotheticity: Symmetry of the Isoquant Map 263 12.4 Examples of Conservation Laws in Economics 266 12.5 Conclusion 269 References 270 Biographies 272 Index 274 Front Matter....Pages i-xii Front Matter....Pages 1-1 Introduction....Pages 3-12 Technical Progress and Economies of Scale: Concept of Holotheticity....Pages 13-28 Holothetic Production Functions and Marginal Rate of Technical Substitution....Pages 29-41 Utility and Demand....Pages 43-53 Duality and Self Duality....Pages 55-72 The Theory of Index Numbers....Pages 73-85 Dynamics and Conservation Laws....Pages 87-105 Back Matter....Pages 107-109 Front Matter....Pages 111-111 The Invariance Principle and Income-Wealth Conservation Laws....Pages 113-142 Conservation Laws in Continuous and Discrete Models....Pages 143-176 Quantity or Quality: The Impact of Labour Saving Innovation on US and Japanese Growth Rates, 1960–2004....Pages 177-208 A Survey on Recent Developments....Pages 209-229 Appendix to Part II. Symmetry: An Overview of Geometric Methods in Economics....Pages 231-266 Back Matter....Pages 267-273 Symmetry and Economic Invariance: An Introduction explores how symmetry and invariance of economic models can provide insights into their properties. While the professional economist is nowadays adept at many of the mathematical techniques used in static and dynamic optimization models, group theory is still not among his or her repertoire of tools. The authors aim to show that group theoretic methods form a natural extension of the techniques commonly used in economics and that they can be easily mastered.