Contents
Introduction xi
A Note to the Reader xvii
1 Sets and Spaces 1
1.1 Sets 1
1.2 Ordered Sets9
1.2.1 Relations 10
1.2.2 Equivalence Relations and Partitions 14
1.2.3 Order Relations 16
1.2.4 Partially Ordered Sets and Lattices 23
1.2.5 Weakly Ordered Sets 32
1.2.6 Aggregation and the Pareto Order 33
1.3 Metric Spaces 45
1.3.1 Open and Closed Sets 49
1.3.2 Convergence: Completeness and Compactness 56
1.4 Linear Spaces66
1.4.1 Subspaces 72
1.4.2 Basis and Dimension 77
1.4.3 A½ne Sets83
1.4.4 Convex Sets88
1.4.5 Convex Cones104
1.4.6 Sperner's Lemma 110
1.4.7 Conclusion 114
1.5 Normed Linear Spaces 114
1.5.1 Convexity in Normed Linear Spaces 125
1.6 Preference Relations 130
1.6.1 Monotonicity and Nonsatiation 131
1.6.2 Continuity132
1.6.3 Convexity136
1.6.4 Interactions137
1.7 Conclusion 141
1.8 Notes142
2 Functions 145
2.1 Functions as Mappings 145
2.1.1 The Vocabulary of Functions 145
2.1.2 Examples of Functions 156
2.1.3 Decomposing Functions 171
2.1.4 Illustrating Functions 174
2.1.5 Correspondences 177
2.1.6 Classes of Functions 186
2.2 Monotone Functions 186
2.2.1 Monotone Correspondences 195
2.2.2 Supermodular Functions 198
2.2.3 The Monotone Maximum Theorem 205
2.3 Continuous Functions 210
2.3.1 Continuous Functionals 213
2.3.2 Semicontinuity 216
2.3.3 Uniform Continuity 217
2.3.4 Continuity of Correspondences 221
2.3.5 The Continuous Maximum Theorem 229
2.4 Fixed Point Theorems 232
2.4.1 Intuition 232
2.4.2 Tarski Fixed Point Theorem 233 2.4.3 Banach Fixed Point Theorem 238
2.4.4 Brouwer Fixed Point Theorem 245
2.4.5 Concluding Remarks 259
2.5 Notes 259
3 Linear Functions 263
3.1 Properties of Linear Functions 269
3.1.1 Continuity of Linear Functions 273
3.2 A½ne Functions276
3.3 Linear Functionals277
3.3.1 The Dual Space 280
3.3.2 Hyperplanes 284
3.4 Bilinear Functions287
3.4.1 Inner Products290 3.5 Linear Operators295
3.5.1 The Determinant 296
3.5.2 Eigenvalues and Eigenvectors 299
3.5.3 Quadratic Forms 302
viii Contents
3.6 Systems of Linear Equations and Inequalities 306
3.6.1 Equations308
3.6.2 Inequalities 314
3.6.3 Input±Output Models 319
3.6.4 Markov Chains320
3.7 Convex Functions 323
3.7.1 Properties of Convex Functions 332
3.7.2 Quasiconcave Functions 336
3.7.3 Convex Maximum Theorems 342
3.7.4 Minimax Theorems 349
3.8 Homogeneous Functions 351
3.8.1 Homothetic Functions 356
3.9 Separation Theorems 358
3.9.1 Hahn-Banach Theorem 371
3.9.2 Duality 377
3.9.3 Theorems of the Alternative 388
3.9.4 Further Applications 398
3.9.5 Concluding Remarks 415
3.10 Notes 415
4 Smooth Functions 417
4.1 Linear Approximation and the Derivative 417
4.2 Partial Derivatives and the Jacobian 429
4.3 Properties of Di¨erentiable Functions 441
4.3.1 Basic Properties and the Derivatives of Elementary
Functions 441
4.3.2 Mean Value Theorem 447
4.4 Polynomial Approximation 457
4.4.1 Higher-Order Derivatives 460
4.4.2 Second-Order Partial Derivatives and the Hessian 461
4.4.3 Taylor's Theorem 467
4.5 Systems of Nonlinear Equations 476
4.5.1 The Inverse Function Theorem 477
4.5.2 The Implicit Function Theorem 479
4.6 Convex and Homogeneous Functions 483
4.6.1 Convex Functions 483
ix Contents
4.6.2 Homogeneous Functions 491
4.7 Notes 496
5 Optimization 497
5.1 Introduction 497
5.2 Unconstrained Optimization 503
5.3 Equality Constraints 516
5.3.1 The Perturbation Approach 516
5.3.2 The Geometric Approach 525
5.3.3 The Implicit Function Theorem Approach 529
5.3.4 The Lagrangean 532
5.3.5 Shadow Prices and the Value Function 542
5.3.6 The Net Bene®t Approach 545
5.3.7 Summary 548
5.4 Inequality Constraints 549
5.4.1 Necessary Conditions 550
5.4.2 Constraint Quali®cation 568
5.4.3 Su½cient Conditions 581
5.4.4 Linear Programming 587
5.4.5 Concave Programming 592
5.5 Notes 598
6 Comparative Statics 601
6.1 The Envelope Theorem 603
6.2 Optimization Models 609
6.2.1 Revealed Preference Approach 610
6.2.2 Value Function Approach 614
6.2.3 The Monotonicity Approach 620
6.3 Equilibrium Models 622
6.4 Conclusion 632
6.5 Notes 632
References 635
Index of Symbols641
General Index 643
