Analytical Methods in Economics

Preface vii Contents xi Introduction xvii Part 1 Preliminaries 1(72) Some Basic Mathematics 3(70) Preliminary Concepts and Notations 4(9) Sets 4(4) Logical Statements 8(2) Functions 10(3) Real Numbers 13(1) Euclidian Space 13(12) A Little Topology 25(11) Convergence 25(3) Continuity 28(4) Some Topological Concepts and Results 32(4) Quadratic Forms 36(4) Differentiation, Concave and Quasi-Concave Functions 40(33) Differentiation 40(5) Some Important Results 45(4) Higher Order Derivatives and Hessians 49(3) Concave and Convex Functions 52(8) Quasi-Concave Functions 60(5) Exercises 65(5) References 70(3) Part 2 Nonlinear Programming and Microeconomics 73(182) Nonlinear Programming 75(50) Introduction 75(10) Characterization of Optimum by First-Order Condition 85(18) Saddle-Point Characterization of Optimum 103(8) Classical Optimization and Nonlinear Programming 111(14) Classical Optimizaion 111(6) Comparison to Nonlinear Programming 117(3) Exercises 120(2) References 122(3) Sensitivity Analysis and Elements of Microeconomics 125(62) Comparative Statics: Conventional Procedure 125(6) The Envelope Theorem 131(8) Elements of Microeconomic Theory 139(12) Elasticity of Factor Substitution and Its Estimation 151(6) On the Concept of Returns to Scale 157(6) The Le Chatelier-Samuelson Principle 163(24) Exercises 171(10) References 181(6) Other Topics in Microeconomic Theory 187(68) Inferior Inputs, Marginal and Average Costs 187(9) Inferior Inputs 187(6) Factor Prices and Shifts of the Average Cost Curve 193(3) Marginal Cost Pricing 196(5) Factor Prices and the Shape of Average Cost Curves 201(5) Supply of Labor: Income-Leisure Choice 206(6) The Case without Nonwage Income 206(4) The Case with Nonwage Income 210(2) Behavior of the Firm under Regulatory Constraint 212(7) The Peak-Load Problem 219(16) Introduction 219(2) The Case of a Welfare-Maximizing Monopoly 221(9) The Case of a Profit-Maximizing Monopoly 230(5) On the Coase Theorem 235(20) Introduction 235(2) Externality and the Pigouvian Scheme 237(5) The Coase Theorem 242(5) References 247(8) Part 3 Economics of Uncertainty 255(68) Economics of Uncertainty 257(66) The Expected Utility Hypothesis 257(14) Expected Utility and Behavior toward Risk 271(10) Behavior toward Risk 271(2) Arrow-Pratt Measures of Risk Aversion 273(5) Examples 278(2) Indifference Curves 280(1) Applications 281(24) Insurance 281(5) The Theory of the Firm 286(4) Portfolio Choice 290(6) Consumption and Savings Decision 296(9) Economics of Information 305(10) Asymmetry of Information, the Lemon Principle, and Signaling 305(2) Preventive Activity and Asymmetry of Information in Insurance 307(5) Adverse Selection 312(3) Concluding Remarks 315(8) References 316(7) Part 4 Differential Equations and Economic Analysis 323(126) Elements of Differential Equations and Economic Applications 325(58) Basic Concepts and Existence of a Solution 325(8) Stability 333(12) Economic Applications 345(25) The Stability of Competitive Equilibrium: the Walrasian Process 345(5) The Stability of Macroeconomic Equilibrium 350(5) Neoclassical Growth Model 355(3) On the Phillips Curve 358(7) ``Tobin's Walras-Keynes-Phillips'' Model: An Application of the Routh-Hurwitz Theorem 365(5) Competitive Equilibrium for the Three-Commodity Case 370(13) References 376(7) Linear Differential Equations on the Plane and Elements of Nonlinear Systems 383(36) Linear Systems 383(2) Homogeneous Linear Systems on the Plane 385(10) Dynamic Behavior of the Solution on the Plane 395(12) Nonlinear Systems 407(12) Local Behavior of the Trajectories on the Plane 407(4) Stability of the Nonlinear System: Liapunov's Direct Method 411(3) Local Asymptotic Stability 414(1) Exercises 415(1) References 416(3) Macroequilibrium and Neoclassical Growth Models 419(30) Static Macroequilibrium and Its Stability 419(4) Money and Growth-Part 1 423(13) Model 423(4) Steady State 427(3) Stability and Instability 430(4) The Source of Instability 434(2) Money and Growth-Part 2 436(13) Introduction 436(2) Model and Momentary Equilibrium 438(2) Long-Run Analysis 440(4) The Source of Stability 444(1) Appendix to Section 8.3 445(2) References 447(2) Part 5 Optimal Control Theory and Applications 449(122) Elements of Optimal Control Theory and Applications 451(42) Pontryagin's Maximum Principle 451(10) Various Cases 461(7) Two Illustrations 468(7) Calculus of Variations 468(4) An Illustrative Problem by Pontryagin et al. 472(3) Optimal Growth Problem 475(18) References 488(5) Infinite Horizon Optimization Control Problem and Applications 493(48) A General Theorem 493(5) Optimal Monetary Policy 498(6) Savings: Permanent Income Hypothesis 504(10) Investment 514(27) The ``Neoclassical'' Theory 515(7) The Adjustment Cost Approach 522(3) Tobin's q, the Value of the Firm, and Keynes's Rule of Marginal Efficiency of Capital 525(10) References 535(6) Extensions of Optimal Control Themes 541(30) The Main Theorem 541(6) Consumer's Lifetime Allocation Process: Finite Horizon Case 547(6) Isoperimetric Problem 553(4) Spatial Pricing Problem 557(14) Appendix to Section 11.4: Spatial Pricing Problem and Nonlinear Programming 564(4) References 568(3) Appendices 571(78) A Elements of Linear Algebra 573(32) A.1 Materices and Basic Operations 573(3) A.2 Transposes, Vectors, Simultaneous Equations, and Nonsingular Matrices 576(5) A.3 Linear Independence, Rank, and Linear Subspaces 581(1) Linear Subspaces and Ranks 582(3) A.4 Matrices as Linear Transformations 585(5) A.5 Determinants and Partitioned Matrices 590(1) A.5.1 Determinants, Laplace Expansion, and Cramer's Rule 590(4) A.5.2 Partitioned Matrices and Determinants 594(1) A.6 Eigenvalues, Eigenvectors, and Symmetric Matrices 595(5) A.7 Negative or Positive Definite Matrices 600(3) References 603(2) B Seven Kinds of Concavity and Quasi-Concave Programming: A Survey of Some Results and Extensions 605(16) B.1 Seven Kinds of Concavity 605(6) B.2 The Role of Nonvanishing Gradients 611(4) B.3 Quasi-Concave Programming 615(4) References 619(2) C Consumer's Surplus 621(28) C.1 Introduction 621(5) C.2 Some Basics 626(1) C.3 Computing the Hicksian Measures 627(3) C.4 The Correct Welfare Measure 630(10) C.5 Hicksian Measures under Constancy of Marginal Utility of Income 640(3) Concluding Remarks 643(1) References 644(5) Indices 649(2) Name 651(6) Subject 657