An Introduction to Mathematical Analysis for Economic Theory and Econometrics

By: Dean Corbae, Maxwell Stinchcombe, Juraj Zeman

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Hardcover | 9 March 2009

At a Glance

Hardcover
688 Pages

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25.4 x 17.7 x 5.0

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Providing an introduction to mathematical analysis as it applies to economic theory and econometrics, this book bridges the gap that has separated the teaching of basic mathematics for economics and the increasingly advanced mathematics demanded in economics research today. Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman equip students with the knowledge of real and functional analysis and measure theory they need to read and do research in economic and econometric theory.

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

Industry Reviews

"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

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Preface	p. xi
User's Guide	p. xiii
Notation	p. xix
Logic	p. 1
Statements, Sets, Subsets, and Implication	p. 1
Statements and Their Truth Values	p. 3
Proofs, a First Look	p. 6
Logical Quantifiers	p. 9
Taxonomy of Proofs	p. 11
Set Theory	p. 15
Some Simple Questions	p. 16
Notation and Other Basics	p. 17
Products, Relations, Correspondences, and Functions	p. 21
Equivalence Relations	p. 26
Optimal Choice for Finite Sets	p. 28
Direct and Inverse Images, Compositions	p. 33
Weak and Partial Orders, Lattices	p. 39
Monotonic Changes in Optima: Supermodularity and Lattices	p. 42
Tarski's Lattice Fixed-Point Theorem and Stable Matchings	p. 49
Finite and Infinite Sets	p. 56
The Axiom of Choice and Some Equivalent Results	p. 62
Revealed Preference and Rationalizability	p. 64
Superstructures	p. 68
Bibliography	p. 69
End-of-Chapter Problems	p. 70
The Space of Real Numbers	p. 72
Why We Want More Than the Rationals	p. 72
Basic Properties of Rationals	p. 73
Distance, Cauchy Sequences, and the Real Numbers	p. 75
The Completeness of the Real Numbers	p. 82
Examples Using Completeness	p. 87
Supremum and Infimum	p. 90
Summability	p. 92
Products of Sequences and e^x	p. 99
Patience, Lim inf, and Lim sup	p. 101
Some Perspective on Completing the Rationals	p. 104
Bibliography	p. 105
The Finite-Dimensional Metric Space of Real Vectors	p. 106
The Basic Definitions for Metric Spaces	p. 107
Discrete Spaces	p. 113
R^l as a Normed Vector Space	p. 114
Completeness	p. 120
Closure, Convergence, and Completeness	p. 124
Separability	p. 128
Compactness in R^l	p. 129
Continuous Functions on R^l	p. 136
Lipschitz and Uniform Continuity	p. 143
Correspondences and the Theorem of the Maximum	p. 144
Banach's Contraction Mapping Theorem	p. 154
Connectedness	p. 167
Bibliography	p. 171
Finite-Dimensional Convex Analysis	p. 172
The Basic Geometry of Convexity	p. 173
The Dual Space of R^l	p. 181
The Three Degrees of Convex Separation	p. 184
Strong Separation and Neoclassical Duality	p. 186
Boundary Issues	p. 194
Concave and Convex Functions	p. 199
Separation and the Hahn-Banach Theorem	p. 209
Separation and the Kuhn-Tucker Theorem	p. 214
Interpreting Lagrange Multipliers	p. 228
Differentiability and Concavity	p. 232
Fixed-Point Theorems and General Equilibrium Theory	p. 239
Fixed-Point Theorems for Nash Equilibria and Perfect Equilibria	p. 245
Bibliography	p. 258
Metric Spaces	p. 259
The Space of Compact Sets and the Theorem of the Maximum	p. 260
Spaces of Continuous Functions	p. 272
D(R), the Space of Cumulative Distribution Functions	p. 293
Approximation in C(M) when M Is Compact	p. 297
Regression Analysis as Approximation Theory	p. 304
Countable Product Spaces and Sequence Spaces	p. 311
Defining Functions Implicitly and by Extension	p. 321
The Metric Completion Theorem	p. 331
The Lebesgue Measure Space	p. 335
Bibliography	p. 343
End-of-Chapter Problems	p. 344
Measure Spaces and Probability	p. 355
The Basics of Measure Theory	p. 356
Four Limit Results	p. 370
Good Sets Arguments and Measurability	p. 388
Two 0-1 Laws	p. 397
Dominated Convergence, Uniform Integrability, and Continuity of the Integral	p. 400
The Existence of Nonatomic Countably Additive Probabilities	p. 411
Transition Probabilities, Product Measures, and Fubini's Theorem	p. 423
Seriously Nonmeasurable Sets and Intergenerational Equity	p. 426
Null Sets, Completions of s-Fields, and Measurable Optima	p. 430
Convergence in Distribution and Skorohod's Theorem	p. 436
Complements and Extras	p. 440
Appendix on Lebesgue Integration	p. 448
Bibliography	p. 451
The L^p (ohm; F, P) and l^p Spaces, p E [1, ?]	p. 452
Some Uses in Statistics and Econometrics	p. 453
Some Uses in Economic Theory	p. 456
The Basics of L^p(ohm; F, P) and l^p	p. 458
Regression Analysis	p. 474
Signed Measures, Vector Measures, and Densities	p. 490
Measure Space Exchange Economies	p. 498
Measure Space Games	p. 503
Dual Spaces: Representations and Separation	p. 509
Weak Convergence in L^p (?, F, P), p E [1, ?)	p. 518
Optimization of Nonlinear Operators	p. 522
A Simple Case of Parametric Estimation	p. 528
Complements and Extras	p. 541
Bibliography	p. 550
Probabilities on Metric Spaces	p. 551
Choice under Uncertainty	p. 551
Stochastic Processes	p. 552
The Metric Space (Delta;(M), p)	p. 553
Two Useful Implications	p. 562
Expected Utility Preferences	p. 563
The Riesz Representation Theorem for Delta;(M), M Compact	p. 567
Polish Measure Spaces and Polish Metric Spaces	p. 569
The Riesz Representation Theorem for Polish Metric Spaces	p. 571
Compactness in Delta;(M)	p. 574
An Operator Proof of the Central Limit Theorem	p. 578
Regular Conditional Probabilities	p. 583
Conditional Probabilities from Maximization	p. 589
Nonexistence of rcp's	p. 590
Bibliography	p. 594
Infinite-Dimensional Convex Analysis	p. 595
Topological Spaces	p. 595
Locally Convex Topological Vector Spaces	p. 603
The Dual Space and Separation	p. 606
Filterbases, Filters, and Ultrafilters	p. 610
Bases, Subbases, Nets, and Convergence	p. 612
Compactness	p. 617
Compactness in Topological Vector Spaces	p. 621
Fixed Points	p. 624
Bibliography	p. 626
Expanded Spaces	p. 627
The Basics of ^*R	p. 628
Superstructures, Transfer, Spillover, and Saturation	p. 632
Loeb Spaces	p. 642
Saturation, Star-Finite Maximization Models, and Compactification	p. 649
The Existence of a Purely Finitely Additive {0, 1}-Valued m	p. 652
Problems and Complements	p. 653
Bibliography	p. 654
Index	p. 655
Table of Contents provided by Ingram. All Rights Reserved.

An Introduction to Mathematical Analysis for Economic Theory and Econometrics

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Hardcover

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