An Introduction to Mathematical Analysis for Economic Theory and Econometrics
1st Edition
Published on 16. November 2022
688 pages
978-1-4008-3308-5 (ISBN)
Description
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
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
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Dean Corbae | Maxwell Stinchcombe | Juraj Zeman
An Introduction to Mathematical Analysis for Economic Theory and Econometrics
Book
03/2009
Princeton University Press
€121.50
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Persons
Dean Corbae is the Rex A. and Dorothy B. Sebastian Centennial Professor in Business Administration at the University of Texas at Austin. Maxwell B. Stinchcombe is the E. C. McCarty Centennial Professor of Economics at the University of Texas at Austin. Juraj Zeman is researcher at the National Bank of Slovakia and lecturer in applied mathematics at Comenius University in Bratislava.
Content
- Cover
- Contents
- Preface
- User's Guide
- Notation
- Chapter 1: Logic
- 1.1 Statements, Sets, Subsets, and Implication
- 1.2 Statements and Their Truth Values
- 1.3 Proofs, a First Look
- 1.4 Logical Quantifiers
- 1.5 Taxonomy of Proofs
- Chapter 2: Set Theory
- 2.1 Some Simple Questions
- 2.2 Notation and Other Basics
- 2.3 Products, Relations, Correspondences, and Functions
- 2.4 Equivalence Relations
- 2.5 Optimal Choice for Finite Sets
- 2.6 Direct and Inverse Images, Compositions
- 2.7 Weak and Partial Orders, Lattices
- 2.8 Monotonic Changes in Optima: Supermodularity and Lattices
- 2.9 Tarski's Lattice Fixed-Point Theorem and Stable Matchings
- 2.10 Finite and Infinite Sets
- 2.11 The Axiom of Choice and Some Equivalent Results
- 2.12 Revealed Preference and Rationalizability
- 2.13 Superstructures
- 2.14 Bibliography
- 2.15 End-of-Chapter Problems
- Chapter 3: The Space of Real Numbers
- 3.1 Why We Want More Than the Rationals
- 3.2 Basic Properties of Rationals
- 3.3 Distance, Cauchy Sequences, and the Real Numbers
- 3.4 The Completeness of the Real Numbers
- 3.5 Examples Using Completeness
- 3.6 Supremum and Infimum
- 3.7 Summability
- 3.8 Products of Sequences and ex
- 3.9 Patience, Lim inf, and Lim sup
- 3.10 Some Perspective on Completing the Rationals
- 3.11 Bibliography
- Chapter 4: The Finite-Dimensional Metric Space of Real Vectors
- 4.1 The Basic Definitions for Metric Spaces
- 4.2 Discrete Spaces
- 4.3 . as a Normed Vector Space
- 4.4 Completeness
- 4.5 Closure, Convergence, and Completeness
- 4.6 Separability
- 4.7 Compactness in ...
- 4.8 Continuous Functions on ...
- 4.9 Lipschitz and Uniform Continuity
- 4.10 Correspondences and the Theorem of the Maximum
- 4.11 Banach's Contraction Mapping Theorem
- 4.12 Connectedness
- 4.13 Bibliography
- Chapter 5: Finite-Dimensional Convex Analysis
- 5.1 The Basic Geometry of Convexity
- 5.2 The Dual Space of ...
- 5.3 The Three Degrees of Convex Separation
- 5.4 Strong Separation and Neoclassical Duality
- 5.5 Boundary Issues
- 5.6 Concave and Convex Functions
- 5.7 Separation and the Hahn-Banach Theorem
- 5.8 Separation and the Kuhn-Tucker Theorem
- 5.9 Interpreting Lagrange Multipliers
- 5.10 Differentiability and Concavity
- 5.11 Fixed-Point Theorems and General Equilibrium Theory
- 5.12 Fixed-Point Theorems for Nash Equilibria and Perfect Equilibria
- 5.13 Bibliography
- Chapter 6: Metric Spaces
- 6.1 The Space of Compact Sets and the Theorem of the Maximum
- 6.2 Spaces of Continuous Functions
- 6.3 ..., the Space of Cumulative Distribution Functions
- 6.4 Approximation in C(M) when M Is Compact
- 6.5 Regression Analysis as Approximation Theory
- 6.6 Countable Product Spaces and Sequence Spaces
- 6.7 Defining Functions Implicitly and by Extension
- 6.8 The Metric Completion Theorem
- 6.9 The Lebesgue Measure Space
- 6.10 Bibliography
- 6.11 End-of-Chapter Problems
- Chapter 7: Measure Spaces and Probability
- 7.1 The Basics of Measure Theory
- 7.2 Four Limit Results
- 7.3 Good Sets Arguments and Measurability
- 7.4 Two 0-1 Laws
- 7.5 Dominated Convergence, Uniform Integrability, and Continuity of the Integral
- 7.6 The Existence of Nonatomic Countably Additive Probabilities
- 7.7 Transition Probabilities, Product Measures, and Fubini's Theorem
- 7.8 Seriously Nonmeasurable Sets and Intergenerational Equity
- 7.9 Null Sets, Completions of s-Fields, and Measurable Optima
- 7.10 Convergence in Distribution and Skorohod's Theorem
- 7.11 Complements and Extras
- 7.12 Appendix on Lebesgue Integration
- 7.13 Bibliography
- Chapter 8: The Lp (O, F, P) and lp Spaces, p ... [1, ...]
- 8.1 Some Uses in Statistics and Econometrics
- 8.2 Some Uses in Economic Theory
- 8.3 The Basics of Lp(O, F, P) and lp
- 8.4 Regression Analysis
- 8.5 Signed Measures, Vector Measures, and Densities
- 8.6 Measure Space Exchange Economies
- 8.7 Measure Space Games
- 8.8 Dual Spaces: Representations and Separation
- 8.9 Weak Convergence in Lp(O, F, P), p . (1, .)
- 8.10 Optimization of Nonlinear Operators
- 8.11 A Simple Case of Parametric Estimation
- 8.12 Complements and Extras
- 8.13 Bibliography
- Chapter 9: Probabilities on Metric Spaces
- 9.1 Choice under Uncertainty
- 9.2 Stochastic Processes
- 9.3 The Metric Space (?(M), ?)
- 9.4 Two Useful Implications
- 9.5 Expected Utility Preferences
- 9.6 The Riesz Representation Theorem for ?(M), M Compact
- 9.7 Polish Measure Spaces and Polish Metric Spaces
- 9.8 The Riesz Representation Theorem for Polish Metric Spaces
- 9.9 Compactness in ?(M)
- 9.10 An Operator Proof of the Central Limit Theorem
- 9.11 Regular Conditional Probabilities
- 9.12 Conditional Probabilities from Maximization
- 9.13 Nonexistence of rcp's
- 9.14 Bibliography
- Chapter 10: Infinite-Dimensional Convex Analysis
- 10.1 Topological Spaces
- 10.2 Locally Convex Topological Vector Spaces
- 10.3 The Dual Space and Separation
- 10.4 Filterbases, Filters, and Ultrafilters
- 10.5 Bases, Subbases, Nets, and Convergence
- 10.6 Compactness
- 10.7 Compactness in Topological Vector Spaces
- 10.8 Fixed Points
- 10.9 Bibliography
- Chapter 11: Expanded Spaces
- 11.1 The Basics of ...
- 11.2 Superstructures, Transfer, Spillover, and Saturation
- 11.3 Loeb Spaces
- 11.4 Saturation, Star-Finite Maximization Models, and Compactification
- 11.5 The Existence of a Purely Finitely Additive {0, 1}-Valued µ
- 11.6 Problems and Complements
- 11.7 Bibliography
- Index
