Murata / Shell Mathematics for Stability and Optimization of Economic Systems

1. Auflage 2014
ISBN: 978-1-4832-7129-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 438 Seiten, Web PDF

Economic Theory and Mathematical Economics: Mathematics for Stability and Optimization of Economic Systems provides information pertinent to the stability aspects and optimization methods relevant to various economic systems. This book presents relevant mathematical theorems sufficient to develop important economic systems, including Leontief input-output systems, Keynesian dynamic models, the Ramsey optimal accumulation systems, and von Neumann expanding economic systems. Organized into two parts encompassing nine chapters, this book begins with an overview of useful theorems on matrices, eigenvalue problems, and matrices with dominant diagonals and P-matrices. This text then explores the linear transformations on vector spaces. Other chapters consider the Hawkins-Simon theorem concerning non-negative linear systems. This book discusses as well the dual linear relations and optimization methods applicable to inequality economic systems. The final chapter deals with powerful optimal control method for dynamical systems. This book is a valuable resource for mathematicians, economists, research workers, and graduate students.

Autoren/Hrsg.

Murata, Yasuo

Shell, Karl

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematics for Stability and Optimization of Economic Systems;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;Preface;12
7;Acknowledgments;16
8;Notation and Symbols;18
9;Part I: LINEAR STRUCTURE AND STABILITY OF ECONOMIC SYSTEMS;22
9.1;Chapter 1. Fundamentals of Square Matrices;24
9.1.1;1.1 Determinants, Inversion of Matrices, and Partitioned Matrices;24
9.1.2;1.2 Eigenvalues, Eigenvectors, and the Generalized Eigenvalue Problem;34
9.1.3;1.3 Matrices with Dominant Diagonals and P-Matrices;42
9.1.4;Exercises;51
9.1.5;References and Further Reading;52
9.2;Chapter 2. Linear Equations and Related Topics with Reference to Economics;53
9.2.1;2.1 Vector Spaces and Convex Sets;53
9.2.2;2.2 Linear Transformations;60
9.2.3;2.3 Rank and Nullity;64
9.2.4;2.4 Elementary Operations and Hawkins–Simon Conditions;69
9.2.5;2.5 Symmetric Matrices, Stable Matrices, and the Lyapunov Theorem;74
9.2.6;Exercises;85
9.2.7;References and Further Reading;85
9.3;Chapter 3. Linear Dynamic Systems and Stability;87
9.3.1;3.1 Linear Differential Equations;87
9.3.2;3.2 Jordan Form of a Square Matrix;92
9.3.3;3.3 Difference Equations and Dynamic Multipliers;100
9.3.4;3.4 Modified Routh–Hurwitz Conditions for Stability;108
9.3.5;3.5 Sufficient Conditions for the Tinbergenian;117
9.3.6;Exercises;123
9.3.7;References and Further Reading;124
9.4;Chapter 4. Nonnegative Square Matrices and Stability in Economic Systems;126
9.4.1;4.1 Frobenius Theorems;126
9.4.2;4.2 Solow Conditions, Stability, and Comparative Statics in Leontief–Hicks–Metzler Systems;136
9.4.3;4.3 Primitivity, the Kakeya Theorem, and Relative Stability;150
9.4.4;4.4 Price Systems of Leontief Type, the Fundamental Marxian Theorem, and Dual Stability;161
9.4.5;4.5 Generalization of the Hicks–Metzler System and Global Stability;172
9.4.6;Exercises;180
9.4.7;References and Further Reading;181
10;Part II: OPTIMIZATION METHODS FOR ECONOMIC SYSTEMS;184
10.1;Chapter 5. Preliminary Mathematical Concepts;186
10.1.1;5.1 Normed Spaces and Inner Product Spaces;186
10.1.2;5.2 Closedness and Continuity;192
10.1.3;5.3 Banach Spaces and Hilbert Spaces;196
10.1.4;5.4 Separable Sets and Isomorphisms;200
10.1.5;5.5 Bounded Linear Functionals and Dual Spaces;205
10.1.6;5.6 Minkowski Functionals and the Hahn–Banach Theorem;210
10.1.7;Exercises;216
10.1.8;References and Further Reading;217
10.2;Chapter 6. Projection and Generalized Inverse with Reference to Economics;219
10.2.1;6.1 Projection Theorems and the Gauss–Markov Theorem;219
10.2.2;6.2 Adjoint Operators;225
10.2.3;6.3 Generalized Inverse (Pseudoinverse);229
10.2.4;6.4 Generalization of the Gauss–Markov Theorem;239
10.2.5;6.5 Generalized Linear Equation Economic Systems;249
10.2.6;Exercises;256
10.2.7;References and Further Reading;257
10.3;Chapter 7. Optimization under Economic Equation Constraints;259
10.3.1;7.1 Differentials and Extrema;259
10.3.2;7.2 The Euler Equation, the Ramsey Path, and Concave Functionals;264
10.3.3;7.3 Contraction Mappings, the Implicit Function Theorem, Univalence Theorems, and a Nonlinear Price System;273
10.3.4;7.4 The Lagrange Multiplier Theory under Equality Constraints;281
10.3.5;7.5 Second-Order Conditions for Local Maxima and Demand Laws;284
10.3.6;Exercises;292
10.3.7;References and Further Reading;293
10.4;Chapter 8. Optimization in Inequality Economic Systems;294
10.4.1;8.1 Hyperplanes and Separation Theorems;294
10.4.2;8.2 Dual Linear Relations and Gale–Nikaido Theorems;304
10.4.3;8.3 The von Neumann Economic System and Maximal Paths;312
10.4.4;8.4 Kuhn–Tucker Theorems, Concave and Quasi-Concave Programming;327
10.4.5;8.5 Duality in Linear Programming, the Morishima Turnpike Theorem, and Other Related Problems;344
10.4.6;Exercises;358
10.4.7;References and Further Reading;361
10.5;Chapter 9. Optimal Control of Dynamical Economic Systems;364
10.5.1;9.1 Pontryagin Maximum Principle: Necessity and Sufficiency;364
10.5.2;9.2 Optimal Accumulation of Nontransferable Capital;376
10.5.3;9.3 Controllability of Linear Dynamical Economic Systems: Generalization of the Static Tinbergen Theory of Policy;382
10.5.4;9.4 Optimal Stabilization Policy for Linear Dynamical Economic Systems with Quadratic Cost Criteria;392
10.5.5;9.5 Realization of Controllable and Observable Linear Dynamical Systems;411
10.5.6;Exercises;423
10.5.7;References and Further Reading;424
11;Author Index;428
12;Subject Index;432