Über die Autorin bzw. den Autor

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.

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"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

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"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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"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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AN INTRODUCTION TO MATHEMATICAL ANALYSIS FOR ECONOMIC THEORY AND ECONOMETRICS

PRINCETON UNIVERSITY PRESS

Contents

Preface...............................................................................................................xiUser's Guide..........................................................................................................xiiiNotation..............................................................................................................xixChapter 1 * Logic.....................................................................................................1Chapter 2 * Set Theory................................................................................................15Chapter 3 * The Space of Real Numbers.................................................................................72Chapter 4 * The Finite-Dimensional Metric Space of Real Vectors.......................................................106Chapter 5 * Finite-Dimensional Convex Analysis........................................................................172Chapter 6 * Metric Spaces.............................................................................................259Chapter 7 * Measure Spaces and Probability............................................................................355Chapter 8 * The Lp(Ω, F, P) and lp Spaces, p [member of] [1, ∞].....................452Chapter 9 * Probabilities on Metric Spaces............................................................................551Chapter 10 * Infinite-Dimensional Convex Analysis.....................................................................595Chapter 11 * Expanded Spaces..........................................................................................627Index.................................................................................................................655

Chapter One

The building blocks of modern economics are based on logical reasoning to prove the validity of a conclusion, B, from well-defined premises, A. In general, statements such as A and/or B can be represented using sets, and a "proof" is constructed by applying, sometimes ingeniously, a fixed set of rules to establish that the statement B is true whenever A is true. We begin with examples of how we represent statements as sets, then turn to the rules that allow us to form more and more complex statements, and then give a taxonomy of the major types of proofs that we use in this book.

1.1 * Statements, Sets, Subsets, and Implication

The idea of a set (of things), or group, or collection is a "primitive," one that we use without being able to clearly define it. The idea of belonging to a set (group, collection) is primitive in exactly the same sense. Our first step is to give the allowable rules by which we evaluate whether statements about sets are true.

We begin by fixing a set X of things that we might have an interest in. When talking about demand behavior, the set X has to include, at the very least, prices, incomes, affordable consumption sets, preference relations, and preference-optimal sets. The set X varies with the context, and is often not mentioned at all.

We express the primitive notion of membership by "[member of]," so that "x [member of] A" means that "x is an element of the set A" and "y [not member of] A" means that "y is not an element of A."

Notation Alert 1.1.A Capitalized letters are usually reserved for sets and smaller letters for points/things in the set that we are studying. Sometimes, several levels of analysis are present simultaneously, and we cannot do this. Consider the study of utility functions, u, on a set of options, X. A function u is a set of pairs of the form (x, u(x)), with x an option and u(x) the number representing its utility. However, in our study of demand behavior, we want to see what happens as u varies. From this perspective, u is a point in the set of possible utility functions.

Membership allows us to define subsets. We say "A is a subset of B," written "A [subset] B," if every x [member of] A satisfies x [member of] B. Thus, subsets are defined in terms of the primitive relation "[member of]." We write A = B if A [subset] B and B [subset] A, and A [not equal to] B otherwise.

We usually specify the elements of a set explicitly by saying "The set A is the set of all elements x in X such that each x has the property A, that is, that A(x) is true," and write "A = {x [member of] X : A(x)}" as a shorter version of this. For example, with X = R^l, the statement "x ≥ 0" is identified with the set R^l₊ = {x [member of] X : x ≥ 0}. In this way, we identify a statement with the set of elements of X for which the statement is true. There are deep issues in logic and the foundations of mathematics relating to the question of whether or not all sets can be identified by "properties." Fortunately, these issues rarely impinge on the mathematics that economists need. Chapter 2 is more explicit about these issues.

We are very often interested in establishing the truth of statements of the form "If A, then B." There are many equivalent ways of writing such a statement: "A -> B," "A implies B," "A only if B," "A is sufficient for B," or "B is necessary for A." To remember the sufficiency and necessity, it may help to subvocalize them as "A is sufficiently strong to guarantee B" and "B is necessarily true if A is true."

The logical relation of implication is a subset relation. If A = {x [member of] X : A(x)} and B = {x [member of] X :B(x)}, then "A -> B" is the same as "A [subset] B."

Example 1.1.1 Let X be the set of numbers, A(x) the statement "x² < 1," and B(x) the statement "|x| ≤ 1." Now, A -> B. In terms of sets, A = {x [member of] X :A(x)} is the set of numbers strictly between -1 and +1, B = {x [member of] X :B(x)} is the set of numbers greater than or equal to -1 and less than or equal to +1, and A [subset] B.

The statements of interest can be quite complex to write out in their entirety. If X is the set of allocations in a model E of an economy and A(x) is the statement "x is a Walrasian equilibrium allocated for the economy ε," then a complete specification of the statement takes a great deal of work. Presuming some familiarity with general equilibrium models, we offer the following.

Example 1.1.2 Let X be the set of allocations in a model ε of an economy; let A(x) be the statement "x is a Walrasian equilibrium allocation"; and B(x) be the statement "x is Pareto efficient for ε." The first fundamental theorem of welfare economics is A -> B. In terms of the definition of subsets, this is expressed as, "Every Walrasian equilibrium allocation is Pareto efficient."

In other cases, we are interested in the truth of statements of the form "A if and only if B," often written "A iff B." Equivalently, such a statement can be written: "A -> B and B -> A," which is often shortened to "A <-> B." Other frequently used formulations are: "A implies B and B implies A," "A is necessary and sufficient for B," or "A is equivalent to B." In terms of the corresponding sets A and B, these are all different ways of writing "A = B."

Example 1.1.3 Let X be the set of numbers, A(x) the statement "0 ≤ x ≤ 1," and B(x) the statement "x² ≤ x." From high school algebra, A <-> B. In terms of sets, A = {x [member of] X :A(x)} and B = {x [member of] X : B(x)} are both the sets of numbers greater than or equal to 0 and less than or equal to 1.

1.2 * Statements and Their Truth Values

Note that a statement of the form "A -> B" is simply a construct of two simple statements connected by "->." This is one of seven ways of constructing new statements that we use. In this section, we cover the first five of them: ands, ors, nots, implies, and equivalence. Repeated applications of these seven ways of constructing statements yield more and more elaboration and complication.

We begin with the simplest three methods, which construct new sets directly from a set or pair of sets that we start with. We then turn to the statements that are about relations between sets and introduce another formulation in terms of indicator functions. Later we give the other two methods, which involve the logical quantifiers "for all" and "there exists." Throughout, interest focuses on methods of establishing the truth or falsity of statements, that is, on methods of proof.

1.2.a Ands/Ors/Nots as Intersections/Unions/Complements

The simplest three ways of constructing new statements from other ones are using the connectives "and" or "or," or by "not," which is negation. Notationally: "A [conjunction] B" means "A and B," "A [disjunction] B" means "A or B," and "[logical not] A" means "not A."

In terms of the corresponding sets: "A [conjunction] B" is A [intersection] B, the intersection of A and B, that is, the set of all points that belong to both A and B; "A [disjunction] B" is A [union] B, the union of A and B, that is, the set of all points that belong to A or belong to B; and "[logical not] A" is A^c = {x [member of] X : x [not member of] A}, the complement of A, is the set of all elements of X that do not belong to A.

The meanings of these new statements, [logical not] A, A [conjunction] B, and A [disjunction] B, are given by a truth table, Table 1.a. The corresponding Table 1.b gives the corresponding set versions of the new statements.

The first two columns of Table 1.a give possible truth values for the statements A and B. The last three columns give the truth values for [logical not] A, A [conjunction] B, and A [disjunction] B as a function of the truth values of A and B. The first two columns of Table 1.b give the corresponding membership properties of an element x, and the last three columns give the corresponding membership properties of x in the sets A^c, A [intersection] B, and A [union] B.

Consider the second rows of both tables, the row where A is true and B is false. This corresponds to discussing an x with the properties that it belongs to A and does not belong to B. The statement "not A," that is, [logical not] A, is false, which corresponds to x not belonging to A^c, x [not member of] A^c. The statement "A and B," that is, "A [conjunction] B," is also false. This is sensible: since B is false, it is not the case that both A and B are true. This corresponds to x not being in the intersection of A and B, that is, x [not member of] A [intersection] B. The statement "A or B," that is, "A [disjunction] B," is true. This is sensible: since A is true, it is the case that at least one of A and B is true, corresponding to x being in the union of A and B.

It is important to note that we use the word "or" in its nonexclusive sense. When we describe someone as "tall or red-headed," we mean to allow tall red-headed people. We do not mean "or" in the exclusive sense that the person is either tall or red-headed but not both. One sees this by considering the last columns in the two tables, the ones with the patterns TTTF and [member of][member of][member of][not member of]. "A or B" is true as long as at least one of A and B is true, and we do not exclude the possibility that both are true. The exclusive "or" is defined by (A [disjunction] B) [conjunction] ([logical not] (A [conjunction] B)^c, which has the truth pattern FTTF. In terms of sets, the exclusive "or" is (A [union] B) [intersection] (A [intersection] B)^c, which has the corresponding membership pattern [not member of][member of][member of][not member of].

1.2.b Implies/Equivalence as Subset/Equality

Two of the remaining four ways of constructing new statements are: "A -> B," which means "A implies B" and "A <-> B," which means "A is equivalent to B." In terms of sets, these are "A [subset] B" and "A = B." These are statements about relations between subsets of X.

Indicator functions are a very useful way to talk about the relations between subsets. For each x [member of] X and A [subset] X, define the indicator of the set A by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

Remember, a proposition, A, is a statement about elements x [member of] X that can be either true or false. When it is true, we write A(x). The corresponding set A is {x [member of] X : A(x)}. The indicator of A takes on the value 1 for exactly those x for which A is true and takes on the value 0 for those x for which A is false.

Indicator functions are ordered pointwise; that is, 1_A ≤ 1_B when 1_A(x) ≤ 1_B(x) for every point x in the set X. Saying "1_A ≤ 1_B" is the same as saying that "A [subset] B." It is easy to give sets A and B that satisfy neither A [subset] B nor B [subset] A. Therefore, unlike pairs of numbers r and s, for which it is always true that either r ≤ s or s ≤ r, pairs of indicator functions may not be ranked by "≤."

Example 1.2.1 If X is the three-point set {a, b, c}, A = {a, b}, B = {b, c}, and C = {c}, then 1_A ≤ 1_X, 1_B = 1_X, 1C = 1B, ?(1A = 1B), and ?(1B = 1A).

Proving statements of the form A -> B and A <-> B is the essential part of mathematical reasoning. For the first, we take the truth of A as given and then establish logically that the truth of B follows. For the second, we take the additional step of taking the truth of B as given and then establish logically that the truth of A follows. In terms of sets, for proving the first, we take a point, x, assume only that x [member of] A, and establish that this implies that x [member of] B, thus proving that A [subset] B. For proving the second, we take the additional step of taking a point, x, assume only that x [member of] B, and establish that this implies that x [member of] A. Here is the truth table for? and?, both for statements and for indicator functions.

1.2.c The Empty Set and Vacuously True Statements

We now come to the idea of something that is vacuously true, and a substantial proportion of people find this idea tricky or annoying, or both. The idea that we are after is that starting from false premises, one can establish anything. In Table 1.c, if A is false, then the statement A -> B is true, whether B is true or false.

A statement that is false for all x [member of] X corresponds to having an indicator function with the property that for all x [member of] X, 1_A(x) = 0. In terms of sets, the notation for this is A = [empty set], where we read "[empty set]" as the empty set, that is, the vacuous set, the one that contains no elements. No matter what the set B is, if A = [empty set], then 1_A(x) ≤ 1_B(x) for all x [member of] X.

Definition 1.2.2 The statement A -> B is vacuously true if A = [empty set]. This definition follows the convention that we use throughout: we show the term or terms being defined in boldface type.

In terms of sets, this is the observation that for all B, [empty set] B, that is, that every element of [empty set] belongs to B. What many people find distasteful is that "every element of [empty set] belongs to B" suggests that there is an element of [empty set], and since there is no such element, the statement feels wrong to them. There is nothing to be done except to get over the feeling.

1.2.d Indicators and Ands/Ors/Nots

Indicator functions can also be used to capture ands, ors, and nots. Often this makes proofs simpler.

The pointwise minimum of a pair of indicator functions, 1_A and 1_B, is written as "1_A [conjunction] 1_B," and is defined by (1_A [conjunction] 1_B)(x) = min{1_A(x), 1_B(x)}. Now, 1_A(x) and 1_B(x) are equal either to 0 or to 1. Since the minimum of 1 and 1 is 1, the minimum of 0 and 1 is 0, and the minimum of 0 and 0 is 0, 1_{A [intersection] B} = 1_A [conjunction] 1_B. This means that the indicator associated with the statement "A ? B" is 1A ? 1B. By checking cases, we note that for all x [member of] X, (1_A [conjunction] 1_B(x) = 1_A(x) x 1_B(x). As a result, 1_A [conjunction] 1_B is often written as 1_A x 1_B.

(Continues...)

Excerpted from AN INTRODUCTION TO MATHEMATICAL ANALYSIS FOR ECONOMIC THEORY AND ECONOMETRICSby Dean Corbae Maxwell B. Stinchcombe Juraj Zeman Copyright © 2009 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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