Essential Mathematics for Economic Analysis

Cover......Page 1
Half Title Page......Page 2
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 8
Preface......Page 12
Publisher’s Acknowledgements......Page 18
 1.1 Essentials of Set Theory......Page 20
 1.2 Some Aspects of Logic......Page 26
 1.3 Mathematical Proofs......Page 31
 1.4 Mathematical Induction......Page 33
 Review Exercises......Page 35
 2.1 The Real Numbers......Page 38
 2.2 Integer Powers......Page 41
 2.3 Rules of Algebra......Page 47
 2.4 Fractions......Page 52
 2.5 Fractional Powers......Page 57
 2.6 Inequalities......Page 62
 2.7 Intervals and Absolute Values......Page 68
 2.8 Summation......Page 71
 2.9 Rules for Sums......Page 75
 2.10 Newton’s Binomial Formula......Page 78
 2.11 Double Sums......Page 80
 Review Exercises......Page 81
 3.1 Solving Equations......Page 86
 3.2 Equations and Their Parameters......Page 89
 3.3 Quadratic Equations......Page 92
 3.4 Nonlinear Equations......Page 97
 3.5 Using Implication Arrows......Page 99
 3.6 Two Linear Equations in Two Unknowns......Page 101
 Review Exercises......Page 105
 4.1 Introduction......Page 108
 4.2 Basic Definitions......Page 109
 4.3 Graphs of Functions......Page 115
 4.4 Linear Functions......Page 118
 4.5 Linear Models......Page 125
 4.6 Quadratic Functions......Page 128
 4.7 Polynomials......Page 135
 4.8 Power Functions......Page 142
 4.9 Exponential Functions......Page 145
 4.10 Logarithmic Functions......Page 150
 Review Exercises......Page 155
 5.1 Shifting Graphs......Page 160
 5.2 New Functions from Old......Page 165
 5.3 Inverse Functions......Page 169
 5.4 Graphs of Equations......Page 175
 5.5 Distance in the Plane......Page 179
 5.6 General Functions......Page 182
 Review Exercises......Page 185
 6.1 Slopes of Curves......Page 188
 6.2 Tangents and Derivatives......Page 190
 6.3 Increasing and Decreasing Functions......Page 195
 6.4 Rates of Change......Page 198
 6.5 A Dash of Limits......Page 201
 6.6 Simple Rules for Differentiation......Page 207
 6.7 Sums, Products, and Quotients......Page 211
 6.8 The Chain Rule......Page 217
 6.9 Higher-Order Derivatives......Page 222
 6.10 Exponential Functions......Page 227
 6.11 Logarithmic Functions......Page 231
 Review Exercises......Page 237
 7.1 Implicit Differentiation......Page 240
 7.2 Economic Examples......Page 247
 7.3 Differentiating the Inverse......Page 251
 7.4 Linear Approximations......Page 254
 7.5 Polynomial Approximations......Page 258
 7.6 Taylor’s Formula......Page 262
 7.7 Elasticities......Page 265
 7.8 Continuity......Page 270
 7.9 More on Limits......Page 276
 7.10 The Intermediate Value Theorem and Newton’s Method......Page 285
 7.11 Infinite Sequences......Page 289
 7.12 L’Hˆopital’s Rule......Page 292
 Review Exercises......Page 297
 8.1 Extreme Points......Page 302
 8.2 Simple Tests for Extreme Points......Page 306
 8.3 Economic Examples......Page 309
 8.4 The Extreme Value Theorem......Page 313
 8.5 Further Economic Examples......Page 319
 8.6 Local Extreme Points......Page 324
 8.7 Inflection Points, Concavity, and Convexity......Page 330
 Review Exercises......Page 335
 9.1 Indefinite Integrals......Page 338
 9.2 Area and Definite Integrals......Page 344
 9.3 Properties of Definite Integrals......Page 351
 9.4 Economic Applications......Page 355
 9.5 Integration by Parts......Page 362
 9.6 Integration by Substitution......Page 366
 9.7 Infinite Intervals of Integration......Page 371
 9.8 A Glimpse at Differential Equations......Page 378
 9.9 Separable and Linear Differential Equations......Page 384
 Review Exercises......Page 390
 10.1 Interest Periods and Effective Rates......Page 394
 10.2 Continuous Compounding......Page 398
 10.3 Present Value......Page 400
 10.4 Geometric Series......Page 402
 10.5 Total Present Value......Page 409
 10.6 Mortgage Repayments......Page 414
 10.7 Internal Rate of Return......Page 418
 10.8 A Glimpse at Difference Equations......Page 420
 Review Exercises......Page 423
 11.1 Functions of Two Variables......Page 426
 11.2 Partial Derivatives with Two Variables......Page 430
 11.3 Geometric Representation......Page 436
 11.4 Surfaces and Distance......Page 443
 11.5 Functions of More Variables......Page 446
 11.6 Partial Derivatives with More Variables......Page 450
 11.7 Economic Applications......Page 454
 11.8 Partial Elasticities......Page 456
 Review Exercises......Page 458
 12.1 A Simple Chain Rule......Page 462
 12.2 Chain Rules for Many Variables......Page 467
 12.3 Implicit Differentiation along a Level Curve......Page 471
 12.4 More General Cases......Page 476
 12.5 Elasticity of Substitution......Page 479
 12.6 Homogeneous Functions of Two Variables......Page 482
 12.7 Homogeneous and Homothetic Functions......Page 487
 12.8 Linear Approximations......Page 493
 12.9 Differentials......Page 496
 12.10 Systems of Equations......Page 501
 12.11 Differentiating Systems of Equations......Page 505
 Review Exercises......Page 511
 13.1 Two Choice Variables: Necessary Conditions......Page 514
 13.2 Two Choice Variables: Sufficient Conditions......Page 519
 13.3 Local Extreme Points......Page 523
 13.4 Linear Models with Quadratic Objectives......Page 528
 13.5 The Extreme Value Theorem......Page 535
 13.6 The General Case......Page 540
 13.7 Comparative Statics and the Envelope Theorem......Page 544
 Review Exercises......Page 548
 14.1 The Lagrange Multiplier Method......Page 552
 14.2 Interpreting the Lagrange Multiplier......Page 559
 14.3 Multiple Solution Candidates......Page 562
 14.4 Why the Lagrange Method Works......Page 564
 14.5 Sufficient Conditions......Page 568
 14.6 Additional Variables and Constraints......Page 571
 14.7 Comparative Statics......Page 577
 14.8 Nonlinear Programming: A Simple Case......Page 582
 14.9 Multiple Inequality Constraints......Page 588
 14.10 Nonnegativity Constraints......Page 593
 Review Exercises......Page 597
 15.1 Systems of Linear Equations......Page 600
 15.2 Matrices and Matrix Operations......Page 603
 15.3 Matrix Multiplication......Page 607
 15.4 Rules for Matrix Multiplication......Page 611
 15.5 The Transpose......Page 618
 15.6 Gaussian Elimination......Page 621
 15.7 Vectors......Page 627
 15.8 Geometric Interpretation of Vectors......Page 630
 15.9 Lines and Planes......Page 636
 Review Exercises......Page 639
 16.1 Determinants of Order 2......Page 642
 16.2 Determinants of Order 3......Page 646
 16.3 Determinants in General......Page 651
 16.4 Basic Rules for Determinants......Page 655
 16.5 Expansion by Cofactors......Page 659
 16.6 The Inverse of a Matrix......Page 663
 16.7 A General Formula for the Inverse......Page 669
 16.8 Cramer’s Rule......Page 672
 16.9 The Leontief Model......Page 676
 Review Exercises......Page 680
17 Linear Programming 665......Page 0
 17.1 A Graphical Approach......Page 685
 17.2 Introduction to Duality Theory......Page 691
 17.3 The Duality Theorem......Page 694
 17.4 A General Economic Interpretation......Page 698
 17.5 Complementary Slackness......Page 700
 Review Exercises......Page 705
Appendix......Page 708
Solutions to the Exercises......Page 712
Index......Page 820