Mathematics for Stability and Optimization of Economic Systems

Mathematics for Stability and Optimization of Economic Systems

1st Edition - September 28, 1977

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  • Author: Yasuo Murata
  • eBook ISBN: 9781483271293

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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.

Table of Contents

  • Preface


    Notation and Symbols

    Part I Linear Structure and Stability of Economic Systems

    Chapter 1 Fundamentals of Square Matrices

    1.1 Determinants, Inversion of Matrices, and Partitioned Matrices

    1.2 Eigenvalues, Eigenvectors, and the Generalized Eigenvalue Problem

    1.3 Matrices with Dominant Diagonals and P-Matrices


    References and Further Reading

    Chapter 2 Linear Equations and Related Topics with Reference to Economics

    2.1 Vector Spaces and Convex Sets

    2.2 Linear Transformations

    2.3 Rank and Nullity

    2.4 Elementary Operations and Hawkins-Simon Conditions

    2.5 Symmetric Matrices, Stable Matrices, and the Lyapunov Theorem


    References and Further Reading

    Chapter 3 Linear Dynamic Systems and Stability

    3.1 Linear Differential Equations

    3.2 Jordan Form of a Square Matrix

    3.3 Difference Equations and Dynamic Multipliers

    3.4 Modified Routh-Hurwitz Conditions for Stability

    3.5 Sufficient Conditions for the Tinbergenian


    References and Further Reading

    Chapter 4 Nonnegative Square Matrices and Stability in Economic Systems

    4.1 Frobenius Theorems

    4.2 Solow Conditions, Stability, and Comparative Statics in Leontief-Hicks-Metzler Systems

    4.3 Primitivity, the Kakeya Theorem, and Relative Stability

    4.4 Price Systems of Leontief Type, the Fundamental Marxian Theorem, and Dual Stability

    4.5 Generalization of the Hicks-Metzler System and Global Stability


    References and Further Reading

    Part II Optimization Methods for Economic Systems

    Chapter 5 Preliminary Mathematical Concepts

    5.1 Normed Spaces and Inner Product Spaces

    5.2 Closedness and Continuity

    5.3 Banach Spaces and Hilbert Spaces

    5.4 Separable Sets and Isomorphisms

    5.5 Bounded Linear Functionals and Dual Spaces

    5.6 Minkowski Functionals and the Hahn-Banach Theorem


    References and Further Reading

    Chapter 6 Projection and Generalized Inverse with Reference to Economics

    6.1 Projection Theorems and the Gauss-Markov Theorem

    6.2 Adjoint Operators

    6.3 Generalized Inverse (Pseudoinverse)

    6.4 Generalization of the Gauss-Markov Theorem

    6.5 Generalized Linear Equation Economic Systems


    References and Further Reading

    Chapter 7 Optimization Under Economic Equation Constraints

    7.1 Differentials and Extrema

    7.2 The Euler Equation, the Ramsey Path, and Concave Functionals

    7.3 Contraction Mappings, the Implicit Function Theorem, Uni valence Theorems, and a Nonlinear Price System

    7.4 The Lagrange Multiplier Theory Under Equality Constraints

    7.5 Second-Order Conditions for Local Maxima and Demand Laws


    References and Further Reading

    Chapter 8 Optimization in Inequality Economic Systems

    8.1 Hyperplanes and Separation Theorems

    8.2 Dual Linear Relations and Gale-Nikaido Theorems

    8.3 The von Neumann Economic System and Maximal Paths

    8.4 Kuhn-Tucker Theorems, Concave and Quasi-Concave Programming

    8.5 Duality in Linear Programming, the Morishima Turnpike Theorem, and Other Related Problems


    References and Further Reading

    Chapter 9 Optimal Control of Dynamical Economic Systems

    9.1 Pontryagin Maximum Principle: Necessity and Sufficiency

    9.2 Optimal Accumulation of Nontransferable Capital

    9.3 Controllability of Linear Dynamical Economic Systems: Generalization of the Static Tinbergen Theory of Policy

    9.4 Optimal Stabilization Policy for Linear Dynamical Economic Systems with Quadratic Cost Criteria

    9.5 Realization of Controllable and Observable Linear Dynamical Systems


    References and Further Reading

    Author Index

    Subject Index

Product details

  • No. of pages: 438
  • Language: English
  • Copyright: © Academic Press 1977
  • Published: September 28, 1977
  • Imprint: Academic Press
  • eBook ISBN: 9781483271293

About the Author

Yasuo Murata

About the Editor

Karl Shell

Affiliations and Expertise

Cornell University

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