Mathematics for Stability and Optimization of Economic Systems

PrefaceAcknowledgmentsNotation and SymbolsPart 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 Exercises 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 Exercises 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 Exercises 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 Exercises References and Further ReadingPart 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 Exercises References and Further ReadingChapter 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 Exercises References and Further ReadingChapter 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 Exercises References and Further ReadingChapter 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 Exercises References and Further ReadingChapter 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 Exercises References and Further ReadingAuthor IndexSubject Index