An Introduction to Mathematical Analysis for Economic Theory and Econometrics

306 Pages·2009·1.91 MB·English

by  Dean Corbae

An Introduction to Mathematical Analysis
in Economics1
Dean Corbae and Juraj Zeman
December 2002
1Still Preliminary. Not to be photocopied or distributed without permission of
the authors.
2
Contents
1 Introduction 13
1.1 Rules of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Taxonomy of Proofs . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Bibliography for Chapter 1 . . . . . . . . . . . . . . . . . . . . 19
2 Set Theory 21
2.1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Algebraic properties of set operations . . . . . . . . . . 24
2.2 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Equivalence relations . . . . . . . . . . . . . . . . . . . 25
2.3.2 Order relations . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Correspondences and Functions . . . . . . . . . . . . . . . . . 30
2.4.1 Restrictions and extensions . . . . . . . . . . . . . . . 32
2.4.2 Composition of functions . . . . . . . . . . . . . . . . . 32
2.4.3 Injections and inverses . . . . . . . . . . . . . . . . . . 33
2.4.4 Surjections and bijections . . . . . . . . . . . . . . . . 33
2.5 Finite and In(cid:222)nite Sets . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Algebras of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 Bibliography for Chapter 2 . . . . . . . . . . . . . . . . . . . . 43
2.8 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 44
3 The Space of Real Numbers 45
3.1 The Field Axioms . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 The Order Axioms . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 The Completeness Axiom . . . . . . . . . . . . . . . . . . . . 50
3.4 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . 53
3.5 Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
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3.6 Bibilography for Chapter 3 . . . . . . . . . . . . . . . . . . . . 63
3.7 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 64
4 Metric Spaces 65
4.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.1 Convergence of functions . . . . . . . . . . . . . . . . . 75
4.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Completion of a metric space. . . . . . . . . . . . . . . 80
4.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 88
4.5.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5.2 A (cid:222)nite dimensional vector space: Rn . . . . . . . . . . 93
4.5.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.4 An in(cid:222)nite dimensional vector space: ! . . . . . . . . . 99
p
4.6 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . 105
4.6.1 Intermediate value theorem . . . . . . . . . . . . . . . 108
4.6.2 Extreme value theorem . . . . . . . . . . . . . . . . . . 110
4.6.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . 111
4.7 Hemicontinuous Correspondences . . . . . . . . . . . . . . . . 113
4.7.1 Theorem of the Maximum . . . . . . . . . . . . . . . . 122
4.8 Fixed Points and Contraction Mappings . . . . . . . . . . . . 127
4.8.1 Fixed points of functions . . . . . . . . . . . . . . . . . 127
4.8.2 Contractions . . . . . . . . . . . . . . . . . . . . . . . . 130
4.8.3 Fixed points of correspondences . . . . . . . . . . . . . 132
4.9 Appendix - Proofs in Chapter 4 . . . . . . . . . . . . . . . . . 138
4.10 Bibilography for Chapter 4 . . . . . . . . . . . . . . . . . . . . 144
4.11 End of Chapter Problems . . . . . . . . . . . . . . . . . . . . 145
5 Measure Spaces 149
5.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.1.1 Outer measure . . . . . . . . . . . . . . . . . . . . . . 151
5.1.2 measurable sets . . . . . . . . . . . . . . . . . . . . 154
L−
5.1.3 Lebesgue meets borel . . . . . . . . . . . . . . . . . . . 158
5.1.4 -measurable mappings . . . . . . . . . . . . . . . . . 159
L
5.2 Lebesgue Integration . . . . . . . . . . . . . . . . . . . . . . . 170
5.2.1 Riemann integrals . . . . . . . . . . . . . . . . . . . . . 170
5.2.2 Lebesgue integrals . . . . . . . . . . . . . . . . . . . . 172
CONTENTS 5
5.3 General Measure . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.3.1 Signed Measures . . . . . . . . . . . . . . . . . . . . . 185
5.4 Examples Using Measure Theory . . . . . . . . . . . . . . . . 194
5.4.1 Probability Spaces . . . . . . . . . . . . . . . . . . . . 194
5.4.2 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
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5.5 Appendix - Proofs in Chapter 5 . . . . . . . . . . . . . . . . . 200
5.6 Bibilography for Chapter 5 . . . . . . . . . . . . . . . . . . . . 211
6 Function Spaces 213
6.1 The set of bounded continuous functions . . . . . . . . . . . . 216
6.1.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . 216
6.1.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . 218
6.1.3 Approximation . . . . . . . . . . . . . . . . . . . . . . 221
6.1.4 Separability of (X) . . . . . . . . . . . . . . . . . . . 227
C
6.1.5 Fixed point theorems . . . . . . . . . . . . . . . . . . . 227
6.2 Classical Banach spaces: L . . . . . . . . . . . . . . . . . . . 229
p
6.2.1 Additional Topics in L (X) . . . . . . . . . . . . . . . 235
p
6.2.2 Hilbert Spaces (L (X)) . . . . . . . . . . . . . . . . . . 237
2
6.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . 241
6.4 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . 245
6.4.1 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . 248
6.4.2 Second Dual Space . . . . . . . . . . . . . . . . . . . . 252
6.5 Separation Results . . . . . . . . . . . . . . . . . . . . . . . . 254
6.5.1 Existence of equilibrium . . . . . . . . . . . . . . . . . 260
6.6 Optimization of Nonlinear Operators . . . . . . . . . . . . . . 262
6.6.1 Variationalmethodsonin(cid:222)nitedimensionalvectorspaces262
6.6.2 Dynamic Programming . . . . . . . . . . . . . . . . . . 274
6.7 Appendix - Proofs for Chapter 6 . . . . . . . . . . . . . . . . . 284
6.8 Bibilography for Chapter 6 . . . . . . . . . . . . . . . . . . . . 297
7 Topological Spaces 299
7.1 Continuous Functions and Homeomorphisms . . . . . . . . . . 302
7.2 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . 303
7.3 Convergence and Completeness . . . . . . . . . . . . . . . . . 305
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Acknowledgements
To my family: those who put up with me in the past - Jo and Phil - and
especially those who put up with me in the present - Margaret, Bethany,
Paul, and Elena. D.C.
To my family. J.Z.
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8 CONTENTS
Preface
The objective of this book is to provide a simple introduction to mathemat-
ical analysis with applications in economics. There is increasing use of real
and functional analysis in economics, but few books cover that material at
an elementary level. Our rationale for writing this book is to bridge the gap
between basic mathematical economics books (which deal with introductory
calculus and linear algebra) and advanced economics books such as Stokey
and Lucas(cid:146) Recursive Methods in Economic Dynamics that presume a work-
ing knowledge of functional analysis. The major innovations in this book
relative to classic mathematics books in this area (such as Royden(cid:146)s Real
Analysis or Munkres(cid:146) Topology) are that we provide: (i) extensive simple
examples (we believe strongly that examples provide the intuition necessary
to grasp difficult ideas); (ii) sketches of complicated proofs (followed by the
complete proof at the end of the book); and (iii) only material that is rel-
evant to economists (which means we drop some material and add other
topics (e.g. we focus extensively on set valued mappings instead of just point
valued ones)). It is important to emphasize that while we aim to make this
material as accessible as possible, we have not excluded demanding mathe-
matical concepts used by economists and that the book is self-contained (i.e.
virtually any theorem used in proving a given result is itself proven in our
book).
Road Map
Chapter 1 is a brief introduction to logical reasoning and how to construct
direct versus indirect proofs. Proving the truth of the compound statement
(cid:147)If A, then B(cid:148) captures the essence of mathematical reasoning; we take the
truth of statement (cid:147)A(cid:148) as given and then establish logically the truth of
statement (cid:148)B(cid:148) follows. We do so by introducing logical connectives and the
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idea of a truth table.
We introduce set operations, relations, functions and correspondences
in Chapter 2 . Then we study the (cid:147)size(cid:148) of sets and show the differences
between countable and uncountable in(cid:222)nite sets. Finally, we introduce the
notion of an algebra (just a collection of sets that satisfy certain properties)
and (cid:147)generate(cid:148) (i.e. establish that there always exists) a smallest collection
of subsets of a given set where all results of set operations (like complements,
union, and intersection) remain in the collection.
Chapter 3 focuses on the set of real numbers (denoted R), which is one
of the simplest but most economic (both literally and (cid:222)guratively) sets to
introduce students to the ideas of algebraic, order, and completeness prop-
erties. Here we expose students to the most elementary notions of distance,
openandclosedness, boundedness, andsimplefactslikebetweenanytworeal
numbers is another real number. One critical result we prove is the Bolzano-
Weierstrass Theorem which says that every bounded in(cid:222)nite subset of R has
a point with sufficiently many points in any subset around it. This result
has important implications for issues like convergence of a sequence of points
which is introduced in more general metric spaces. We end by generating the
smallest collection of all open sets in R known as the Borel (σ-)algebra.
InChapter4weintroducesequencesandthenotionsofconvergence, com-
pleteness, compactness, and connectedness in general metric spaces, where
weaugmentanarbitrarysetwithanabstractnotionofa(cid:147)distance(cid:148)function.
Understanding these (cid:147)C(cid:148) properties are absolutely essential for economists.
For instance, the completeness of a metric space is a very important property
for problem solving. In particular, one can construct a sequence of approxi-
mate solutions that get closer and closer together and provided the space is
complete, thenthelimitofthissequenceexists andisthesolutionoftheorig-
inal problem. We also present properties of normed vector spaces and study
twoimportantexamples, bothofwhicharetheusedextensivelyineconomics:
(cid:222)nite dimensional Euclidean space (denoted Rn) and the space of (in(cid:222)nite
dimensional) sequences (denoted ! ). Then we study continuity of functions
p
and hemicontinuity of correspondences. Particular attention is paid to the
properties of a continuous function on a connected domain (a generalization
of the Intermediate Value Theorem) as well as a continuous function on a
compact domain (a generalization of the Extreme Value Theorem). We end
by providing (cid:222)xed point theorems for functions and correspondences that
are useful in proving, for instance, the existence of general equilibrium with
competitive markets or a Nash Equilibrium of a noncooperative game.