Mathematical Economics

1. Introduction 1.1 Mathematical Economics 1.2 Outline of the Book 1.3 Notes on the Literature Part I: Optimizing Theory 2. The General Optimizing Problem 2.1 Introduction 2.2 The General Structure 2.3 Constraints and the Feasible Set 2.4 The General Optimizing Problem 2.5 The General Solution Principle 2.6 Conditions for a Global Optimum 2.7 Important Special Cases 2.8 Direct Solutions or Optimal Condtions? Exercises 3. The Theory of Linear Programming 3.1 Introduction 3.2 The Feasible Set 3.3 Duality 3.4 The Optimum Conditions 3.5 Basic Solutions 3.6 The Basis Theorem 3.7 Interpretation of the Dual Variables Further Reading Exercises 4. Classical Calculus Methods 4.1 Introduction 4.2 The Lagrangean Function 4.3 Interpretation of the Lagrange Multipliers 4.4 A Geometrical Note 4.5 Second Order Conditions for the Classical Case 4.6 The Substitution Effect of Neoclassical Demand Theory 4.7 Global Optimum Condtions in the Classical Problem Exercises 5. Advanced Optimizing Theory 5.1 Introduction 5.2 Nonnegative Variables 5.3 Inequality Constraints 5.4 Saddle Points and Duality 5.5 The Dual Variables 5.6 The Minimax Theorem 5.7 Exercises of Optimal Solutions Further Reading Exercises Part II: Static Economic Models 6. Input-Output and Related Models 6.1 Input-Output Models 6.2 The Closed Model 6.3 The Leontief Open Model 6.4 Direct and Indirect Input Requirements 6.5 Factor Intensity in the Leontief Model 6.6 A Labor Theory of Value 6.7 The Substitution Theorem 6.8 Matrix Multipliers Further Reading Exercises 7. Linear Optimizing Models 7.1 Activity Analysis of Production 7.2 The Production Set 7.3 Efficient Production 7.4 Constrained Production 7.5 Consumption as an Activity Further Reading Exercises 8. Nonlinear Optimizing Models 8.1 Introduction 8.2 Neoclassical Demand Theory 8.3 Convexity Proof of the Substitution Theorem 8.4 The Neoclassical Transformation Surface 8.5 Returns to Scale 8.6 Relative Factor Intensity 8.7 Generalized Production Theory Further Reading 9. General Equilibrium 9.1 Equilibrium in a Market Economy 9.2 Walras' Law and the Budget Constraint 9.3 The Excess Demand Theorem 9.4 The Walras-Wald Model 9.5 The Arrow-Debreu-McKenzie Model Further Reading Part III: Dynamic Economic Models 10. Balanced Growth 10.1 Introduction 10.2 A Leontief-Type Model 10.3 The Von Neumann Growth Model 10.4 The Von Neumann-Leontief Model 10.5 General Balanced Growth Models Further Reading 11. Efficient and Optimal Growth 11.1 Efficiency and Optimality in Dynamic Models 11.2 The Principle of Optimality 11.3 Efficient Growth 11.4 Properties of Efficient Paths 11.5 A Turnpike Theorem 11.6 An Explicit Turnpike Example Further Reading 12. Stability 12.1 The Concept of Stability 12.2 Stability Analysis 12.3 Market Stability 12.4 Stability of Decentralized Economic Policy Part IV: MATHEMATICAL REVIEWS RI. Fundamental Ideas R1.1 Sets R1.2 Ordered and Quasi-Ordered Sets R1.3 Cartesian Products and Spaces R1.4 "Functions, Transformations, Mappings, Correspondences" R1.5 Closedness and Boundedness R1.6 Complex Numbers Exercises R2. Linear Algebra R2.1 Vectors R2.2 Fundamental Theorem of Vector Spaces R2.3 Basis and Rank R2.4 Sums and Direct Sums R2.5 Scalar Products R2.6 Complex Vectors R2.7 Matrices R2.8 Matrix Algebra R2.9 Matrix-Vector Products and Linear Transformations R2.10 Partitioned Matrices R2.11 Vector Sets Exercises R3. Linear Equations and Inequalities R3.1 Introduction R3.2 The Rank of Matrix R3.3 Homogeneous Equations R3.4 Nonhomogeneous Equations R3.5 Nonnegative Vectors and Vector Inequalities R3.6 Fundamental Theorem on Linear Inequalities R3.7 Results on Linear Equations and Inequalities Exercises R4. Convex Sets and Cones R4.1 Geometric Ideas R4.2 Convex Sets R4.3 Separating and Supporting Hyperplanes R4.4 Extreme Points R4.5 Convex Cones R4.6 Finite Cones and Homogeneous Inequalities R4.7 The Dual Cone Exercises R5. Square Matrices and Characteristic Roots R5.1 Introduction R5.2 Determinants and Cramer's Rule R5.3 The Inverse of a Square Matrix R5.4 Charateristic Roots and Vectors   R5.5 Diagonalization R5.6 Convergence of Matrix Series R5.7 Charateristic Row Vectors R5.8 Numerical Examples Exercises R6. Symmetric Matrices and Quadratic Forms R6.1 Symmetric Matrices R6.2 Quadratic Forms R6.3 Constrained Quadratic Forms Exercises R7. Semipositive and Dominant Diagonal Matrices R7.1 Introduction R7.2 Indecomposability R7.3 Properties of Semipositive Square Matrices R7.4 Properties of Dominant Diagonal Matrices R7.5 Proofs Exercises R8. Continuous Functions R8.1 Introduction R8.2 Derivatives and Differentials R8.3 Some Mapping Relationships R8.4 Maxima and Minima R8.5 Convex and Concave Functions R8.6 Homogeneous and Homothetic Functions R8.7 The Brouwer Fixed Point Theorem R8.8 Linear Homogeneous Vector-Valued Functions Exercises R9. Point-to-Set Mappings R9.1 Introduction R9.2 The Graph of a Mapping R9.3 Continuity &