Mathematical Methods in Economics

Norman Schofield

• Publisher: Routledge
• Year: 2018
• eBook ISBN: 9781351140423
• Topic: Business
• Subtopic: Business General
• Index: Business

Originally published in 1984. Since the logic underlying economic theory can only be grasped fully by a thorough understanding of the mathematics, this book will be invaluable to economists wishing to understand vast areas of important research. It provides a basic introduction to the fundamental mathematical ideas of topology and calculus, and uses these to present modern singularity theory and recent results on the generic existence of isolated price equilibria in exchange economies.

Chapter 1.
SETS, RELATIONS AND PREFERENCES

In this chapter we introduce the elementary set theory and notation to be used throughout the book. We also define the notions of binary relation, function, as well as the axioms of a group and field. Finally we discuss the idea of an individual and social preference relation, and mention some of the concepts of social choice and welfare economics.

1.1. Elements of Set Theory

Let U be a collection of objects, which we shall call the domain of discourse, universal set or universe. A set B in this universe (or subset of U) is a subcollection of objects from U. B may be defined either explicitly by enumerating the objects, for example by writing

B={Tom, Dick, Harry}or B={x1,x2,x3,…}.

Alternatively B may be defined implicitly be reference to some property P(B), which characterises the elements of B, thus

For example:

B = {x: x is an integer satisfying 1 ≤ x ≤ 5} is a satisfactory definition of the set B, where the universal set could be the collection of all integers. If B is a set write x ɛ B to mean that the element x is a member of B. Write {x} for the set which contains only one element, x.

If A, B are two sets write A ∩ B for the set which contains only those elements which are both in A and B, and A ∪ B for the set whose elements are either in A or B. The null set Φ, is that subset of U which contains no elements in U.

Finally if A is a subset of U, define the negation of A, or complement of A in U to be the set

A¯={x: x is in U but not in A}.

1.1.1. A Set Theory

Now let T be a family of subsets of U, where T includes both U and Φ, i.e.

T={U,Φ,A,B,…}.

If A is a member of T, then write A ɛ T. Note here that T is a set of sets.

Suppose that T satisfies the following properties:

1. for any A ɛ T, A¯∈T
2. for any A, B in in T, A ∪...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright page
• Contents
• Foreword
• 1. Sets, Relations and Preferences
• 2. Linear Systems
• 3. Topology and Convex Optimisation
• 4. Differential Calculus and Smooth Optimisation
• 5. Singularity Theory and General Equilibrium
• Further Reading
• Review Exercises
• Index