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1 Mathematical Preliminaries1

1.1 Notation and Preliminary Definitions2

1.1.1 Integers, Rationals, Reals, Rn2

1.1.2 Inner Product, Norm, Metric4

1.2 Sets and Sequences in Rn7

1.2.1 Sequences and Limits7

1.2.2 Subsequences and Limit Points10

1.2.3 Cauchy Sequences and Completeness11

1.2.4 Suprema, Infima, Maxima, Minima14

1.2.5 Monotone Sequences in R17

1.2.6 The Lim Sup and Lim Inf18

1.2.7 Open Balls, Open Sets, Closed Sets22

1.2.8 Bounded Sets and Compact Sets23

1.2.9 Convex Combinations and Convex Sets23

1.2.10 Unions, Intersections, and Other Binary Operations24

1.3 Matrices30

1.3.1 Sum, Product, Transpose30

1.3.2 Some Important Classes of Matrices32

1.3.3 Rank of a Matrix33

1.3.4 The Determinant35

1.3.5 The Inverse38

1.3.6 Calculating the Determinant39

1.4 Functions41

1.4.1 Continuous Functions41

1.4.2 Differentiable and Continuously Differentiable Functions43

1.4.3 Partial Derivatives and Differentiability46

1.4.4 Directional Derivatives and Differentiability48

1.4.5 Higher Order Derivatives49

1.5 Quadratic Forms: Definite and Semidefinite Matrices50

1.5.1 Quadratic Forms and Definiteness50

1.5.2 Identifying Definiteness and Semidefiniteness53

1.6 Some Important Results55

1.6.1 Separation Theorems56

1.6.2 The Intermediate and Mean Value Theorems60

1.6.3 The Inverse and Implicit Function Theorems65

1.7 Exercises66

2 Optimization in Rn74

2.1 Optimization Problems in Rn74

2.2 Optimization Problems in Parametric Form77

2.3 Optimization Problems: Some Examples78

2.3.1 Utility Maximization78

2.3.2 Expenditure Minimization79

2.3.3 Profit Maximization80

2.3.4 Cost Minimization80

2.3.5 Consumption-Leisure Choice81

2.3.6 Portfolio Choice81

2.3.7 Identifying Pareto Optima82

2.3.8 Optimal Provision of Public Goods83

2.3.9 Optimal Commodity Taxation84

2.4 Objectives of Optimization Theory85

2.5 A Roadmap86

2.6 Exercises88

3 Existence of Solutions: The Weierstrass Theorem90

3.1 The Weierstrass Theorem90

3.2 The Weierstrass Theorem in Applications92

3.3 A Proof of the Weierstrass Theorem96

3.4 Exercises97

4 Unconstrained Optima100

4.1 “Unconstrained” Optima100

4.2 First-Order Conditions101

4.3 Second-Order Conditions103

4.4 Using the First- and Second-Order Conditions104

4.5 A Proof of the First-Order Conditions106

4.6 A Proof of the Second-Order Conditions108

4.7 Exercises110

5 Equality Constraints and the Theorem of Lagrange112

5.1 Constrained Optimization Problems112

5.2 Equality Constraints and the Theorem of Lagrange113

5.2.1 Statement of the Theorem114

5.2.2 The Constraint Qualification115

5.2.3 The Lagrangean Multipliers116

5.3 Second-Order Conditions117

5.4 Using the Theorem of Lagrange121

5.4.1 A “Cookbook” Procedure121

5.4.2 Why the Procedure Usually Works122

5.4.3 When It Could Fail123

5.4.4 A Numerical Example127

5.5 Two Examples from Economics128

5.5.1 An Illustration from Consumer Theory128

5.5.2 An Illustration from Producer Theory130

5.5.3 Remarks132

5.6 A Proof of the Theorem of Lagrange135

5.7 A Proof of the Second-Order Conditions137

5.8 Exercises142

6 Inequality Constraints and the Theorem of Kuhn and Tucker145

6.1 The Theorem of Kuhn and Tucker145

6.1.1 Statement of the Theorem145

6.1.2 The Constraint Qualification147

6.1.3 The Kuhn-Tucker Multipliers148

6.2 Using the Theorem of Kuhn and Tucker150

6.2.1 A “Cookbook” Procedure150

6.2.2 Why the Procedure Usually Works151

6.2.3 When It Could Fail152

6.2.4 A Numerical Example155

6.3 Illustrations from Economics157

6.3.1 An Illustration from Consumer Theory158

6.3.2 An Illustration from Producer Theory161

6.4 The General Case: Mixed Constraints164

6.5 A Proof of the Theorem of Kuhn and Tucker165

6.6 Exercises168

7 Convex Structures in Optimization Theory172

7.1 Convexity Defined173

7.1.1 Concave and Convex Functions174

7.1.2 Strictly Concave and Strictly Convex Functions176

7.2 Implications of Convexity177

7.2.1 Convexity and Continuity177

7.2.2 Convexity and Differentiability179

7.2.3 Convexity and the Properties of the Derivative183

7.3 Convexity and Optimization185

7.3.1 Some General Observations185

7.3.2 Convexity and Unconstrained Optimization187

7.3.3 Convexity and the Theorem of Kuhn and Tucker187

7.4 Using Convexity in Optimization189

7.5 A Proof of the First-Derivative Characterization of Convexity190

7.6 A Proof of the Second-Derivative Characterization of Convexity191

7.7 A Proof of the Theorem of Kuhn and Tucker under Convexity194

7.8 Exercises198

8 Quasi-Convexity and Optimization203

8.1 Quasi-Concave and Quasi-Convex Functions204

8.2 Quasi-Convexity as a Generalization of Convexity205

8.3 Implications of Quasi-Convexity209

8.4 Quasi-Convexity and Optimization213

8.5 Using Quasi-Convexity in Optimization Problems215

8.6 A Proof of the First-Derivative Characterization of Quasi-Convexity216

8.7 A Proof of the Second-Derivative Characterization ofQuasi-Convexity217

8.8 A Proof of the Theorem of Kuhn and Tucker under Quasi-Convexity220

8.9 Exercises221

9 Parametric Continuity: The Maximum Theorem224

9.1 Correspondences225

9.1.1 Upper- and Lower-Semicontinuous Correspondences225

9.1.2 Additional Definitions228

9.1.3 A Characterization of Semicontinuous Correspondences229

9.1.4 Semicontinuous Functions and Semicontinuous Correspondences233

9.2 Parametric Continuity:The Maximum Theorem235

9.2.1 The Maximum Theorem235

9.2.2 The Maximum Theorem under Convexity237

9.3 An Application to Consumer Theory240

9.3.1 Continuity of the Budget Correspondence240

9.3.2 The Indirect Utility Function and Demand Correspondence242

9.4 An Application to Nash Equilibrium243

9.4.1 Normal-Form Games243

9.4.2 The Brouwer/Kakutani Fixed Point Theorem244

9.4.3 Existence of Nash Equilibrium246

9.5 Exercises247

10 Supermodularity and Parametric Monotonicity253

10.1 Lattices and Supermodularity254

10.1.1 Lattices254

10.1.2 Supermodularity and Increasing Differences255

10.2 Parametric Monotonicity258

10.3 An Application to Supermodular Games262

10.3.1 Supermodular Games262

10.3.2 The Tarski Fixed Point Theorem263

10.3.3 Existence of Nash Equilibrium263

10.4 A Proof of the Second-Derivative Characterization of Supermodularity264

10.5 Exercises266

11 Finite-Horizon Dynamic Programming268

11.1 Dynamic Programming Problems268

11.2 Finite-Horizon Dynamic Programming268

11.3 Histories, Strategies, and the Value Function269

11.4 Markovian Strategies271

11.5 Existence of an Optimal Strategy272

11.6 An Example: The Consumption-Savings Problem276

11.7 Exercises278

12 Stationary Discounted Dynamic Programming281

12.1 Description of the Framework281

12.2 Histories, Strategies, and the Value Function282

12.3 The Bellman Equation283

12.4 A Technical Digression286

12.4.1 Complete Metric Spaces and Cauchy Sequences286

12.4.2 Contraction Mappings287

12.4.3 Uniform Convergence289

12.5 Existence of an Optimal Strategy291

12.5.1 A Preliminary Result292

12.5.2 Stationary Strategies294

12.5.3 Existence of an Optimal Strategy295

12.6 An Example: The Optimal Growth Model298

12.6.1 The Model299

12.6.2 Existence of Optimal Strategies300

12.6.3 Characterization of Optimal Strategies301

12.7 Exercises309

Appendix A Set Theory and Logic: An Introduction315

A.1 Sets, Unions, Intersections315

A.2 Propositions: Contrapositives and Converses316

A.3 Quantifiers and Negation318

A.4 Necessary and Sufficient Conditions320

Appendix B The Real Line323

B.1 Construction of the Real Line323

B.2 Properties of the Real Line326

Appendix C Structures on Vector Spaces330

C.1 Vector Spaces330

C.2 Inner Product Spaces332

C.3 Normed Spaces333

C.4 Metric Spaces336

C.4.1 Definitions336

C.4.2 Sets and Sequences in Metric Spaces337

C.4.3 Continuous Functions on Metric Spaces339

C.4.4 Separable Metric Spaces340

C.4.5 Subspaces341

C.5 Topological Spaces342

C.5.1 Definitions342

C.5.2 Sets and Sequences in Topological Spaces343

C.5.3 Continuous Functions on Topological Spaces343

C.5.4 Bases343

C.6 Exercises345

Bibliography349

Index351