Real Analysis with Economic Applications / Edition 1 available in Hardcover, eBook
Real Analysis with Economic Applications / Edition 1
- ISBN-10:
- 0691117683
- ISBN-13:
- 9780691117683
- Pub. Date:
- 01/22/2007
- Publisher:
- Princeton University Press
- ISBN-10:
- 0691117683
- ISBN-13:
- 9780691117683
- Pub. Date:
- 01/22/2007
- Publisher:
- Princeton University Press
Real Analysis with Economic Applications / Edition 1
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Overview
The emphasis throughout is on topics directly relevant to economic theory. In addition to addressing the usual topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. Efe Ok complements the mathematical development with applications that provide concise introductions to various topics from economic theory, including individual decision theory and games, welfare economics, information theory, general equilibrium and finance, and intertemporal economics. Moreover, apart from direct applications to economic theory, his book includes numerous fixed point theorems and applications to functional equations and optimization theory.
The book is rigorous, but accessible to those who are relatively new to the ways of real analysis. The formal exposition is accompanied by discussions that describe the basic ideas in relatively heuristic terms, and by more than 1,000 exercises of varying difficulty.
This book will be an indispensable resource in courses on mathematics for economists and as a reference for graduate students working on economic theory.
Product Details
ISBN-13: | 9780691117683 |
---|---|
Publisher: | Princeton University Press |
Publication date: | 01/22/2007 |
Edition description: | New Edition |
Pages: | 832 |
Product dimensions: | 6.12(w) x 9.25(h) x (d) |
About the Author
Table of Contents
Preface xviiPrerequisites xxviiBasic Conventions xxix
Part I: SET THEORY 1
Chapter A: Preliminaries of Real Analysis 3A.1 Elements of Set Theory 4A.1.1 Sets 4A.1.2 Relations 9A.1.3 Equivalence Relations 11A.1.4 Order Relations 14A.1.5 Functions 20A.1.6 Sequences, Vectors, and Matrices 27A.1.7* A Glimpse of Advanced Set Theory: The Axiom of Choice 29
A.2 Real Numbers 33A.2.1 Ordered Fields 33A.2.2 Natural Numbers, Integers, and Rationals 37A.2.3 Real Numbers 39A.2.4 Intervals and R 44
A.3 Real Sequences 46A.3.1 Convergent Sequences 46A.3.2 Monotonic Sequences 50A.3.3 Subsequential Limits 53A.3.4 Infinite Series 56A.3.5 Rearrangement of Infinite Series 59A.3.6 Infinite Products 61
A.4 Real Functions 62A.4.1 Basic Definitions 62A.4.2 Limits, Continuity, and Differentiation 64A.4.3 Riemann Integration 69A.4.4 Exponential, Logarithmic, and Trigonometric Functions 74A.4.5 Concave and Convex Functions 77A.4.6 Quasiconcave and Quasiconvex Functions 80
Chapter B: Countability 82B.1 Countable and Uncountable Sets 82B.2 Losets and Q 90
B.3 Some More Advanced Set Theory 93B.3.1 The Cardinality Ordering 93B.3.2* The Well-Ordering Principle 98
B.4 Application: Ordinal Utility Theory 99B.4.1 Preference Relations 100B.4.2 Utility Representation of Complete Preference Relations 102B.4.3* Utility Representation of Incomplete Preference Relations 107
Part II: ANALYSIS ON METRIC SPACES 115
Chapter C: Metric Spaces 117C.1 Basic Notions 118C.1.1 Metric Spaces: Definition and Examples 119C.1.2 Open and Closed Sets 127C.1.3 Convergent Sequences 132C.1.4 Sequential Characterization of Closed Sets 134C.1.5 Equivalence of Metrics 136
C.2 Connectedness and Separability 138C.2.1 Connected Metric Spaces 138C.2.2 Separable Metric Spaces 140C.2.3 Applications to Utility Theory 145
C.3 Compactness 147C.3.1 Basic Definitions and the Heine-Borel Theorem 148C.3.2 Compactness as a Finite Structure 151C.3.3 Closed and Bounded Sets 154
C.4 Sequential Compactness 157C.5 Completeness 161C.5.1 Cauchy Sequences 161C.5.2 Complete Metric Spaces: Definition and Examples 163C.5.3 Completeness versus Closedness 167C.5.4 Completeness versus Compactness 171
C.6 Fixed Point Theory I 172C.6.1 Contractions 172C.6.2 The Banach Fixed Point Theorem 175C.6.3* Generalizations of the Banach Fixed Point Theorem 179
C.7 Applications to Functional Equations 183C.7.1 Solutions of Functional Equations 183C.7.2 Picard's Existence Theorems 187
C.8 Products of Metric Spaces 192C.8.1 Finite Products 192C.8.2 Countably Infinite Products 193
Chapter D: Continuity I 200D.1 Continuity of Functions 201D.1.1 Definitions and Examples 201D.1.2 Uniform Continuity 208D.1.3 Other Continuity Concepts 210D.1.4* Remarks on the Differentiability of Real Functions 212D.1.5 A Fundamental Characterization of Continuity 213D.1.6 Homeomorphisms 216
D.2 Continuity and Connectedness 218D.3 Continuity and Compactness 222D.3.1 Continuous Image of a Compact Set 222D.3.2 The Local-to-Global Method 223D.3.3 Weierstrass’ Theorem 225D.4 Semicontinuity 229
D.5 Applications 237D.5.1* Caristi's Fixed Point Theorem 238D.5.2 Continuous Representation of a Preference Relation 239D.5.3* Cauchy's Functional Equations: Additivity on Rn 242D.5.4* Representation of Additive Preferences 247
D.6 CB(T) and Uniform Convergence 249D.6.1 The Basic Metric Structure of CB(T) 249D.6.2 Uniform Convergence 250D.6.3* The Stone-Weierstrass Theorem and Separability of C(T) 257D.6.4* The Arzelà-Ascoli Theorem 262D.7* Extension of Continuous Functions 266
D.8 Fixed Point Theory II 272D.8.1 The Fixed Point Property 273D.8.2 Retracts 274D.8.3 The Brouwer Fixed Point Theorem 277D.8.4 Applications 280
Chapter E: Continuity II 283E.1 Correspondences 284E.2 Continuity of Correspondences 287E.2.1 Upper Hemicontinuity 287E.2.2 The Closed Graph Property 294E.2.3 Lower Hemicontinuity 297E.2.4 Continuous Correspondences 300E.2.5* The Hausdorff Metric and Continuity 302E.3 The Maximum Theorem 306
E.4 Application: Stationary Dynamic Programming 311E.4.1 The Standard Dynamic Programming Problem 312E.4.2 The Principle of Optimality 315E.4.3 Existence and Uniqueness of an Optimal Solution 320E.4.4 Application: The Optimal Growth Model 324
E.5 Fixed Point Theory III 330E.5.1 Kakutani's Fixed Point Theorem 331E.5.2* Michael's Selection Theorem 333E.5.3* Proof of Kakutani's Fixed Point Theorem 339E.5.4* Contractive Correspondences 341
E.6 Application: The Nash Equilibrium 343E.6.1 Strategic Games 343E.6.2 The Nash Equilibrium 346E.6.3* Remarks on the Equilibria of Discontinuous Games 351
Part III: ANALYSIS ON LINEAR SPACES 355
Chapter F: Linear Spaces 357F.1 Linear Spaces 358F.1.1 Abelian Groups 358F.1.2 Linear Spaces: Definition and Examples 360F.1.3 Linear Subspaces, Affine Manifolds, and Hyperplanes 364F.1.4 Span and Affine Hull of a Set 368F.1.5 Linear and Affine Independence 370F.1.6 Bases and Dimension 375
F.2 Linear Operators and Functionals 382F.2.1 Definitions and Examples 382F.2.2 Linear and Affine Functions 386F.2.3 Linear Isomorphisms 389F.2.4 Hyperplanes, Revisited 392
F.3 Application: Expected Utility Theory 395F.3.1 The Expected Utility Theorem 395F.3.2 Utility Theory under Uncertainty 403
F.4* Application: Capacities and the Shapley Value 409F.4.1 Capacities and Coalitional Games 410F.4.2 The Linear Space of Capacities 412F.4.3 The Shapley Value 415
Chapter G: Convexity 422G.1 Convex Sets 423G.1.1 Basic Definitions and Examples 423G.1.2 Convex Cones 428G.1.3 Ordered Linear Spaces 432G.1.4 Algebraic and Relative Interior of a Set 436G.1.5 Algebraic Closure of a Set 447G.1.6 Finitely Generated Cones 450
G.2 Separation and Extension in Linear Spaces 454G.2.1 Extension of Linear Functionals 455G.2.2 Extension of Positive Linear Functionals 460G.2.3 Separation of Convex Sets by Hyperplanes 462G.2.4 The External Characterization of Algebraically Closed and Convex Sets 471G.2.5 Supporting Hyperplanes 473G.2.6* Superlinear Maps 476
G.3 Reflections on Rn 480G.3.1 Separation in Rn 480G.3.2 Support in Rn 486G.3.3 The Cauchy-Schwarz Inequality 488G.3.4 Best Approximation from a Convex Set in Rn 489G.3.5 Orthogonal Complements 492G.3.6 Extension of Positive Linear Functionals, Revisited 496
Chapter H: Economic Applications 498H.1 Applications to Expected Utility Theory 499H.1.1 The Expected Multi-Utility Theorem 499H.1.2* Knightian Uncertainty 505H.1.3* The Gilboa-Schmeidler Multi-Prior Model 509
H.2 Applications to Welfare Economics 521H.2.1 The Second Fundamental Theorem of Welfare Economics 521H.2.2 Characterization of Pareto Optima 525H.2.3* Harsanyi's Utilitarianism Theorem 526
H.3 An Application to Information Theory 528H.4 Applications to Financial Economics 535H.4.1 Viability and Arbitrage-Free Price Functionals 535H.4.2 The No-Arbitrage Theorem 539H.5 Applications to Cooperative Games 542H.5.1 The Nash Bargaining Solution 542H.5.2* Coalitional Games without Side Payments 546
Part IV: ANALYSIS ON METRIC/NORMED LINEAR SPACES 551
Chapter I: Metric Linear Spaces 553I.1 Metric Linear Spaces 554I.2 Continuous Linear Operators and Functionals 561I.2.1 Examples of (Dis-)Continuous Linear Operators 561I.2.2 Continuity of Positive Linear Functionals 567I.2.3 Closed versus Dense Hyperplanes 569I.2.4 Digression: On the Continuity of Concave Functions 573I.3 Finite-Dimensional Metric Linear Spaces 577I.4* Compact Sets in Metric Linear Spaces 582I.5 Convex Analysis in Metric Linear Spaces 587I.5.1 Closure and Interior of a Convex Set 587I.5.2 Interior versus Algebraic Interior of a Convex Set 590I.5.3 Extension of Positive Linear Functionals, Revisited 594I.5.4 Separation by Closed Hyperplanes 594I.5.5* Interior versus Algebraic Interior of a Closed and Convex Set 597
Chapter J: Normed Linear Spaces 601J.1 Normed Linear Spaces 602J.1.1 A Geometric Motivation 602J.1.2 Normed Linear Spaces 605J.1.3 Examples of Normed Linear Spaces 607J.1.4 Metric versus Normed Linear Spaces 611J.1.5 Digression: The Lipschitz Continuity of Concave Maps 614
J.2 Banach Spaces 616J.2.1 Definition and Examples 616J.2.2 Infinite Series in Banach Spaces 618J.2.3* On the "Size" of Banach Spaces 620
J.3 Fixed Point Theory IV 623J.3.1 The Glicksberg-Fan Fixed Point Theorem 623J.3.2 Application: Existence of the Nash Equilibrium, Revisited 625J.3.3* The Schauder Fixed Point Theorems 626J.3.4* Some Consequences of Schauder's Theorems 630J.3.5* Applications to Functional Equations 634
J.4 Bounded Linear Operators and Functionals 638J.4.1 Definitions and Examples 638J.4.2 Linear Homeomorphisms, Revisited 642J.4.3 The Operator Norm 644J.4.4 Dual Spaces 648J.4.5* Discontinuous Linear Functionals, Revisited 649
J.5 Convex Analysis in Normed Linear Spaces 650J.5.1 Separation by Closed Hyperplanes, Revisited 650J.5.2* Best Approximation from a Convex Set 652J.5.3 Extreme Points 654
J.6 Extension in Normed Linear Spaces 661J.6.1 Extension of Continuous Linear Functionals 661J.6.2* Infinite-Dimensional Normed Linear Spaces 663J.7* The Uniform Boundedness Principle 665
Chapter K: Differential Calculus 670K.1 Fréchet Differentiation 671K.1.1 Limits of Functions and Tangency 671K.1.2 What Is a Derivative? 672K.1.3 The Fréchet Derivative 675K.1.4 Examples 679K.1.5 Rules of Differentiation 686K.1.6 The Second Fréchet Derivative of a Real Function 690K.1.7 Differentiation on Relatively Open Sets 694
K.2 Generalizations of the Mean Value Theorem 698K.2.1 The Generalized Mean Value Theorem 698K.2.2* The Mean Value Inequality 701
K.3 Fréchet Differentiation and Concave Maps 704K.3.1 Remarks on the Differentiability of Concave Maps 704K.3.2 Fréchet Differentiable Concave Maps 706
K.4 Optimization 712K.4.1 Local Extrema of Real Maps 712K.4.2 Optimization of Concave Maps 716
K.5 Calculus of Variations 718K.5.1 Finite-Horizon Variational Problems 718K.5.2 The Euler-Lagrange Equation 721K.5.3* More on the Sufficiency of the Euler-Lagrange Equation 733K.5.4 Infinite-Horizon Variational Problems 736K.5.5 Application: The Optimal Investment Problem 738K.5.6 Application: The Optimal Growth Problem 740K.5.7* Application: The Poincaré-Wirtinger Inequality 743
Hints for Selected Exercises 747References 777Glossary of Selected Symbols 789Index 793
What People are Saying About This
The idea of doing such a math book directed toward graduate students of economics and finance is an excellent one. There are many students who are interested in this topic, anduntil nowthe existing math books have not directed their examples and exercises toward an economics approach.
Salih Neftci, City University of New York
Because of its comprehensive coverage of the basic topics of real analysis that are of primary interest to economists, this is a much-needed contribution to the current selection of mathematics textbooks for students of economics, and it will be a good addition to any economist's library. It includes a large number of economics applications that will motivate students to learn the math, and its number and variety of exercisesforty to fifty in each chapteris a further asset.
Susan Elmes, Columbia University
This very well written book displays its author's engaging style, and offers interesting questions between topics that make them entertaining to read through.
Darrell Duffie, Stanford University, author of "Dynamic Asset Pricing Theory"
"Because of its comprehensive coverage of the basic topics of real analysis that are of primary interest to economists, this is a much-needed contribution to the current selection of mathematics textbooks for students of economics, and it will be a good addition to any economist's library. It includes a large number of economics applications that will motivate students to learn the math, and its number and variety of exercises—forty to fifty in each chapter—is a further asset."—Susan Elmes, Columbia University"This book is poised to be a standard reference. Its author gets high marks for care of execution and obvious devotion to, and command of, the topics."—Wei Xiong, Princeton University"This very well written book displays its author's engaging style, and offers interesting questions between topics that make them entertaining to read through."—Darrell Duffie, Stanford University, author of Dynamic Asset Pricing Theory"The idea of doing such a math book directed toward graduate students of economics and finance is an excellent one. There are many students who are interested in this topic, and—until now—the existing math books have not directed their examples and exercises toward an economics approach."—Salih Neftci, City University of New York
This book is poised to be a standard reference. Its author gets high marks for care of execution and obvious devotion to, and command of, the topics.
Wei Xiong, Princeton University