An Introduction to Mathematical Analysis for Economic Theory and Econometrics available in Hardcover, eBook
An Introduction to Mathematical Analysis for Economic Theory and Econometrics
- ISBN-10:
- 0691118671
- ISBN-13:
- 9780691118673
- Pub. Date:
- 03/09/2009
- Publisher:
- Princeton University Press
- ISBN-10:
- 0691118671
- ISBN-13:
- 9780691118673
- Pub. Date:
- 03/09/2009
- Publisher:
- Princeton University Press
An Introduction to Mathematical Analysis for Economic Theory and Econometrics
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Overview
Unlike other mathematics textbooks for economics, An Introduction to Mathematical Analysis for Economic Theory and Econometrics takes a unified approach to understanding basic and advanced spaces through the application of the Metric Completion Theorem. This is the concept by which, for example, the real numbers complete the rational numbers and measure spaces complete fields of measurable sets. Another of the book's unique features is its concentration on the mathematical foundations of econometrics. To illustrate difficult concepts, the authors use simple examples drawn from economic theory and econometrics.
Accessible and rigorous, the book is self-contained, providing proofs of theorems and assuming only an undergraduate background in calculus and linear algebra.
- Begins with mathematical analysis and economic examples accessible to advanced undergraduates in order to build intuition for more complex analysis used by graduate students and researchers
- Takes a unified approach to understanding basic and advanced spaces of numbers through application of the Metric Completion Theorem
- Focuses on examples from econometrics to explain topics in measure theory
Product Details
| ISBN-13: | 9780691118673 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 03/09/2009 |
| Edition description: | New Edition |
| Pages: | 688 |
| Product dimensions: | 7.20(w) x 10.10(h) x 2.00(d) |
About the Author
Table of Contents
Preface xi
User's Guide xiii
Notation xix
Chapter 1 Logic 1
1.1 Statements, Sets, Subsets, and Implication 1
1.2 Statements and Their Truth Values 3
1.3 Proofs, a First Look 6
1.4 Logical Quantifiers 9
1.5 Taxonomy of Proofs 11
Chapter 2 Set Theory 15
2.1 Some Simple Questions 16
2.2 Notation and Other Basics 17
2.3 Products, Relations, Correspondences, and Functions 21
2.4 Equivalence Relations 26
2.5 Optimal Choice for Finite Sets 28
2.6 Direct and Inverse Images, Compositions 33
2.7 Weak and Partial Orders, Lattices 39
2.8 Monotonic Changes in Optima: Supermodularity and Lattices 42
2.9 Tarski's Lattice Fixed-Point Theorem and Stable Matchings 49
2.10 Finite and Infinite Sets 56
2.11 The Axiom of Choice and Some Equivalent Results 62
2.12 Revealed Preference and Rationalizability 64
2.13 Superstructures 68
2.14 Bibliography 69
2.15 End-of-Chapter Problems 70
Chapter 3 The Space of Real Numbers 72
3.1 Why We Want More Than the Rationals 72
3.2 Basic Properties of Rationals 73
3.3 Distance, Cauchy Sequences, and the Real Numbers 75
3.4 The Completeness of the Real Numbers 82
3.5 Examples Using Completeness 87
3.6 Supremum and Infimum 90
3.7 Summability 92
3.8 Products of Sequences and ex 99
3.9 Patience, Lim inf, and Lim sup 101
3.10 Some Perspective on Completing the Rationals 104
3.11 Bibliography 105
Chapter 4 The Finite-Dimensional Metric Space of Real Vectors 106
4.1 The Basic Definitions for Metric Spaces 107
4.2 Discrete Spaces 113
4.3 Rl as a Normed Vector Space 114
4.4 Completeness 120
4.5 Closure, Convergence, and Completeness 124
4.6 Separability 128
4.7Compactness in Rl 129
4.8 Continuous Functions on Rl 136
4.9 Lipschitz and Uniform Continuity 143
4.10 Correspondences and the Theorem of the Maximum 144
4.11 Banach's Contraction Mapping Theorem 154
4.12 Connectedness 167
4.13 Bibliography 171
Chapter 5 Finite-Dimensional Convex Analysis 172
5.1 The Basic Geometry of Convexity 173
5.2 The Dual Space of Rl 181
5.3 The Three Degrees of Convex Separation 184
5.4 Strong Separation and Neoclassical Duality 186
5.5 Boundary Issues 194
5.6 Concave and Convex Functions 199
5.7 Separation and the Hahn-Banach Theorem 209
5.8 Separation and the Kuhn-Tucker Theorem 214
5.9 Interpreting Lagrange Multipliers 228
5.10 Differentiability and Concavity 232
5.11 Fixed-Point Theorems and General Equilibrium Theory 239
5.12 Fixed-Point Theorems for Nash Equilibria and Perfect Equilibria 245
5.13 Bibliography 258
Chapter 6 Metric Spaces 259
6.1 The Space of Compact Sets and the Theorem of the Maximum 260
6.2 Spaces of Continuous Functions 272
6.3 D(R), the Space of Cumulative Distribution Functions 293
6.4 Approximation in C(M) when M Is Compact 297
6.5 Regression Analysis as Approximation Theory 304
6.6 Countable Product Spaces and Sequence Spaces 311
6.7 Defining Functions Implicitly and by Extension 321
6.8 The Metric Completion Theorem 331
6.9 The Lebesgue Measure Space 335
6.10 Bibliography 343
6.11 End-of-Chapter Problems 344
Chapter 7 Measure Spaces and Probability 355
7.1 The Basics of Measure Theory 356
7.2 Four Limit Results 370
7.3 Good Sets Arguments and Measurability 388
7.4 Two 0-1 Laws 397
7.5 Dominated Convergence, Uniform Integrability, and Continuity of the Integral 400
7.6 The Existence of Nonatomic Countably Additive Probabilities 411
7.7 Transition Probabilities, Product Measures, and Fubini's Theorem 423
7.8 Seriously Nonmeasurable Sets and Intergenerational Equity 426
7.9 Null Sets, Completions of s-Fields, and Measurable Optima 430
7.10 Convergence in Distribution and Skorohod's Theorem 436
7.11 Complements and Extras 440
7.12 Appendix on Lebesgue Integration 448
7.13 Bibliography 451
Chapter 8 The Lp (ohm; F, P) and lp Spaces, p E [1, ?] 452
8.1 Some Uses in Statistics and Econometrics 453
8.2 Some Uses in Economic Theory 456
8.3 The Basics of Lp(ohm; F, P) and lp 458
8.4 Regression Analysis 474
8.5 Signed Measures, Vector Measures, and Densities 490
8.6 Measure Space Exchange Economies 498
8.7 Measure Space Games 503
8.8 Dual Spaces: Representations and Separation 509
8.9 Weak Convergence in Lp (?, F, P), p E [1, ?) 518
8.10 Optimization of Nonlinear Operators 522
8.11 A Simple Case of Parametric Estimation 528
8.12 Complements and Extras 541
8.13 Bibliography 550
Chapter 9 Probabilities on Metric Spaces 551
9.1 Choice under Uncertainty 551
9.2 Stochastic Processes 552
9.3 The Metric Space (Delta;(M), p) 553
9.4 Two Useful Implications 562
9.5 Expected Utility Preferences 563
9.6 The Riesz Representation Theorem for Delta;(M), M Compact 567
9.7 Polish Measure Spaces and Polish Metric Spaces 569
9.8 The Riesz Representation Theorem for Polish Metric Spaces 571
9.9 Compactness in Delta;(M) 574
9.10 An Operator Proof of the Central Limit Theorem 578
9.11 Regular Conditional Probabilities 583
9.12 Conditional Probabilities from Maximization 589
9.13 Nonexistence of rcp's 590
9.14 Bibliography 594
Chapter 10 Infinite-Dimensional Convex Analysis 595
10.1 Topological Spaces 595
10.2 Locally Convex Topological Vector Spaces 603
10.3 The Dual Space and Separation 606
10.4 Filterbases, Filters, and Ultrafilters 610
10.5 Bases, Subbases, Nets, and Convergence 612
10.6 Compactness 617
10.7 Compactness in Topological Vector Spaces 621
10.8 Fixed Points 624
10.9 Bibliography 626
Chapter 11 Expanded Spaces 627
11.1 The Basics of *R 628
11.2 Superstructures, Transfer, Spillover, and Saturation 632
11.3 Loeb Spaces 642
11.4 Saturation, Star-Finite Maximization Models, and Compactification 649
11.5 The Existence of a Purely Finitely Additive {0, 1}-Valued m 652
11.6 Problems and Complements 653
11.7 Bibliography 654
Index 655
What People are Saying About This
A much-needed textbook for graduate students and a useful desk reference for researchers, this book is of tremendous value to the economics profession because it bridges abstract mathematics and concrete economic applications. Given the current technical level required in research, knowledge of materials covered in this book is indispensable for graduate students.
— Han Hong, Stanford University
This book makes accessible an extraordinary amount of mathematics used in economics and carries it to a high level. By means of illustrative examples, the authors succeed in explaining most of the main ideas of economic theory. This is an important resource for economists and an excellent text for mathematics courses for economic graduate students.
— Truman F. Bewley, Yale University
This book will prove extremely useful for anyone who wants to learn mathematical economics in an accessible and intuitive fashion, while still tackling advanced concepts. The range of topics is impressive, with many illuminating examples. An excellent text!
— Jaksa Cvitanic, California Institute of Technology
"I've struggled in teaching a math for economics course for several years without an appropriate text. This book will remedy this problem and, more generally, will fill a gap that has existed in the profession for at least a decade."—L. Joe Moffitt, University of Massachusetts"This book will prove extremely useful for anyone who wants to learn mathematical economics in an accessible and intuitive fashion, while still tackling advanced concepts. The range of topics is impressive, with many illuminating examples. An excellent text!"—Jaksa Cvitanic, California Institute of Technology"This book makes accessible an extraordinary amount of mathematics used in economics and carries it to a high level. By means of illustrative examples, the authors succeed in explaining most of the main ideas of economic theory. This is an important resource for economists and an excellent text for mathematics courses for economic graduate students."—Truman F. Bewley, Yale University"A much-needed textbook for graduate students and a useful desk reference for researchers, this book is of tremendous value to the economics profession because it bridges abstract mathematics and concrete economic applications. Given the current technical level required in research, knowledge of materials covered in this book is indispensable for graduate students."—Han Hong, Stanford University"Without ever sacrificing rigor, the authors have a style that will help students trying to decipher arcane mathematical ideas. I recommend this book to students."—Richard P. McLean, Rutgers University
Without ever sacrificing rigor, the authors have a style that will help students trying to decipher arcane mathematical ideas. I recommend this book to students.
— Richard P. McLean, Rutgers University
I've struggled in teaching a math for economics course for several years without an appropriate text. This book will remedy this problem and, more generally, will fill a gap that has existed in the profession for at least a decade.
— L. Joe Moffitt, University of Massachusetts