# Further Mathematics for Economic Analysis

Financial Times Prentice Hall, 2008 - Business & Economics - 616 pages
Preface p. ix 1 Topics in Linear Algebra p. 1 1.1 Review of Basic Linear Algebra p. 2 1.2 Linear Independence p. 7 1.3 The Rank of a Matrix p. 11 1.4 Main Results on Linear Systems p. 14 1.5 Eigenvalues p. 19 1.6 Diagonalization p. 25 1.7 Quadratic Forms p. 28 1.8 Quadratic Forms with Linear Constraints p. 35 1.9 Partitioned Matrices and Their Inverses p. 37 2 Multivariable Calculus p. 43 2.1 Gradients and Directional Derivatives p. 44 2.2 Convex Sets p. 50 2.3 Concave and Convex Functions I p. 53 2.4 Concave and Convex Functions II p. 63 2.5 Quasiconcave and Quasiconvex Functions p. 68 2.6 Taylor's Formula p. 77 2.7 Implicit and Inverse Function Theorems p. 80 2.8 Degrees of Freedom and Functional Dependence p. 89 2.9 Differentiability p. 93 2.10 Existence and Uniqueness of Solutions of Systems of Equations p. 98 3 Static Optimization p. 103 3.1 Extreme Points p. 104 3.2 Local Extreme Points p. 110 3.3 Equality Constraints: The Lagrange Problem p. 115 3.4 Local Second-Order Conditions p. 125 3.5 Inequality Constraints: Nonlinear Programming p. 129 3.6 Sufficient Conditions p. 135 3.7 Comparative Statics p. 139 3.8 Nonnegativity Constraints p. 143 3.9 Concave Programming p. 148 3.10 Precise Comparative Statics Results p. 150 3.11 Existence of Lagrange Multipliers p. 153 4 Topics in Integration p. 157 4.1 Review of One-Variable Integration p. 157 4.2 Leibniz's Formula p. 159 4.3 The Gamma Function p. 164 4.4 Multiple Integrals over Product Domains p. 166 4.5 Double Integrals over General Domains p. 171 4.6 The Multiple Riemann Integral p. 175 4.7 Change of Variables p. 178 4.8 Generalized Double Integrals p. 186 5 Differential Equations I: First-Order Equations in One Variable p. 189 5.1 Introduction p. 190 5.2 The Direction is Given: Find the Path! p. 193 5.3 Separable Equations p. 194 5.4 First-Order Linear Equations p. 200 5.5 Exact Equations and Integrating Factors p. 206 5.6 Transformation of Variables p. 208 5.7 Qualitative Theory and Stability p. 211 5.8 Existence and Uniqueness p. 217 6 Differential Equations II: Second-Order Equations and Systems in the Plane p. 223 6.1 Introduction p. 223 6.2 Linear Differential Equations p. 226 6.3 Constant Coefficients p. 228 6.4 Stability for Linear Equations p. 235 6.5 Simultaneous Equations in the Plane p. 237 6.6 Equilibrium Points for Linear Systems p. 243 6.7 Phase Plane Analysis p. 246 6.8 Stability for Nonlinear Systems p. 251 6.9 Saddle Points p. 255 7 Differential Equations III: Higher-Order Equations p. 259 7.1 Linear Differential Equations p. 259 7.2 The Constant Coefficients Case p. 263 7.3 Stability of Linear Differential Equations p. 266 7.4 Systems of Differential Equations p. 269 7.5 Stability for Nonlinear Systems p. 273 7.6 Qualitative Theory p. 278 7.7 A Glimpse at Partial Differential Equations p. 280 8 Calculus of Variations p. 287 8.1 The Simplest Problem p. 288 8.2 The Euler Equation p. 290 8.3 Why the Euler Equation is Necessary p. 293 8.4 Optimal Savings p. 298 8.5 More General Terminal Conditions p. 300 9 Control Theory: Basic Techniques p. 305 9.1 The Basic Problem p. 306 9.2 A Simple Case p. 308 9.3 Regularity Conditions p. 312 9.4 The Standard Problem p. 314 9.5 The Maximum Principle and the Calculus of Variations p. 322 9.6 Adjoint Variables as Shadow Prices p. 324 9.7 Sufficient Conditions p. 330 9.8 Variable Final Time p. 336 9.9 Current Value Formulations p. 338 9.10 Scrap Values p. 341 9.11 Infinite Horizon p. 348 9.12 Phase Diagrams p. 353 10 Control Theory with Many Variables p. 359 10.1 Several Control and State Variables p. 360 10.2 Some Examples p. 366 10.3 Infinite Horizon p. 370 10.4 Existence Theorems and Sensitivity p. 373 10.5 A Heuristic Proof of the Maximum Principle p. 377 10.6 Mixed Constraints p. 380 10.7 Pure State Constraints p. 383 10.8 Generalizations p. 386 11 Difference Equations p. 389 11.1 First-Order Difference Equations p. 390 11.2 Economic Applications p. 396 11.3 Second-Order Difference Equations p. 401 11.4 Linear Equations with Constant Coefficients p. 404 11.5 Higher-Order Equations p. 410 11.6 Systems of Difference Equations p. 415 11.7 Stability of Nonlinear Difference Equations p. 419 12 Discrete Time Optimization p. 423 12.1 Dynamic Programming p. 424 12.2 The Euler Equation p. 433 12.3 Infinite Horizon p. 435 12.4 The Maximum Principle p. 441 12.5 More Variables p. 444 12.6 Stochastic Optimization p. 448 12.7 Infinite Horizon Stationary Problems p. 458 13 Topology and Separation p. 465 13.1 Point Set Topology in R n p. 465 13.2 Topology and Convergence p. 471 13.3 Continuous Functions p. 475 13.4 Maximum Theorems p. 481 13.5 Convex Sets p. 486 13.6 Separation Theorems p. 491 13.7 Productive Economies and Frobenius's Theorem p. 495 14 Correspondences and Fixed Points p. 499 14.1 Correspondences p. 500 14.2 A General Maximum Theorem p. 509 14.3 Fixed Points for Contraction Mappings p. 513 14.4 Brouwer's and Kakutani's Fixed Point Theorems p. 516 14.5 Equilibrium in a Pure Exchange Economy p. 521 Appendix A Sets, Completeness, and Covergence p. 525 A.1 Sets and Functions p. 525 A.2 Least Upper Bound Principle p. 530 A.3 Sequences of Real Numbers p. 533 A.4 Infimum and Supremum of Functions p. 541 Appendix B Trigonometric Functions p. 545 B.1 Basic Definitions and Results p. 545 B.2 Differentiating Trigonometric Functions p. 551 B.3 Complex Numbers p. 555 Answers p. 559 References p. 605 Index.

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