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Vassilis C. Mavron and Timothy N. Phillips

Elements of

Mathematics for

Economics and Finance

With 77 Figures

Vassilis C. Mavron, MA, MSc, PhD

Institute of Mathematical and Physical

Sciences

University of Wales Aberystwyth

Aberystwyth SY23 3BZ

Wales, UK

Timothy N. Phillips, MA, MSc, DPhil,

DSc

Cardiff School of Mathematics

Cardiff University

Senghennydd Road

Cardiff CF24 4AG

Wales, UK

Mathematics Subject Classification (2000): 91-01; 91B02

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2006928729

ISBN-10: 1-84628-560-7

e-ISBN 1-84628-561-5

Printed on acid-free paper

ISBN-13: 978-1-84628-560-8

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as

permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,

stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers,

or in the case of reprographic reproduction in accordance with the terms of licences issued by the

publishers.

The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a

specific statement, that such names are exempt from the relevant laws and regulations and therefore free

for general use.

The publisher makes no representation, express or implied, with regard to the accuracy of the information

contained in this book and cannot accept any legal responsibility or liability for any errors or omissions

that may be made.

Printed in the United States of America

(HAM)

987654321

Springer Science + Business Media, LLC

springer.com

Preface

The mathematics contained in this book for students of economics and nance

has, for many years, been given by the authors in two single-semester courses

at the University of Wales Aberystwyth. These were mathematics courses in

an economics setting, given by mathematicians based in the Department of

Mathematics for students in the Faculty of Social Sciences or School of Man-

agement. The choice of subject matter and arrangement of material reect this

collaboration and are a result of the experience thus obtained.

The majority of students to whom these courses were given were study-

ing for degrees in economics or business administration and had not acquired

any mathematical knowledge beyond pre-calculus mathematics, i.e., elementary

algebra. Therefore, the rst-semester course assumed little more than basic pre-

calculus mathematics and was based on Chapters 1–7. This course led on to

the more advanced second-semester course, which was also suitable for students

who had already covered basic calculus. The second course contained at most

one of the three Chapters 10, 12, and 13. In any particular year, their inclusion

or exclusion would depend on the requirements of the economics or business

studies degree syllabuses. An appendix on dierentials has been included as an

optional addition to an advanced course.

The students taking these courses were chiey interested in learning the

mathematics that had applications to economics and were not primarily in-

terested in theoretical aspects of the subject per se. The authors have not at-

tempted to write an undergraduate text in economics but instead have written

a text in mathematics to complement those in economics.

The simplicity of a mathematical theory is sometimes lost or obfuscated

by a dense covering of applications at too early a stage. For this reason, the

aim of the authors has been to present the mathematics in its simplest form,

highlighting threads of common mathematical theory in the various topics of

Elements of Mathematics for Economics and Finance

economics.

Some knowledge of theory is necessary if correct use is to be made of the

techniques; therefore, the authors have endeavoured to introduce some basic

theory in the expectation and hope that this will improve understanding and

incite a desire for a more thorough knowledge.

Students who master the simpler cases of a theory will nd it easier to go on

to the more di cult cases when required. They will also be in a better position

to understand and be in control of calculations done by hand or calculator

and also to be able to visualise problems graphically or geometrically. It is

still true that the best way to understand a technique thoroughly is through

practice. Mathematical techniques are no exception, and for this reason the

book illustrates theory through many examples and exercises.

We are grateful to Noreen Davies and Joe Hill for invaluable help in prepar-

ing the manuscript of this book for publication.

Above all, we are grateful to our wives, Nesta and Gill, and to our chil-

dren, Nicholas and Christiana, and Rebecca, Christopher, and Emily, for their

patience, support, and understanding: this book is dedicated to them.

Vassilis C. Mavron

Timothy N. Phillips

Aberystwyth

Cardi

United Kingdom

March 2006

Contents

1. Essential Skills .............................................. 1

1.1 Introduction ............................................. 1

1.2 Numbers ................................................ 2

1.2.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.3 Evaluation of Arithmetical Expressions . . . . . . . . . . . . . . . . 4

1.3 Fractions ................................................ 5

1.3.1 Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Decimal Representation of Numbers . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Percentages.............................................. 10

1.6 Powers and Indices ....................................... 12

1.7 Simplifying Algebraic Expressions .......................... 16

1.7.1 Multiplying Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.7.2 Factorization ...................................... 18

2. Linear Equations ........................................... 23

2.1 Introduction ............................................. 23

2.2 Solution of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Solution of Simultaneous Linear Equations . . . . . . . . . . . . . . . . . . . 27

2.4 Graphs of Linear Equations................................ 30

2.4.1 Slope of a Straight Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Budget Lines ............................................ 37

2.6 Supply and Demand Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.6.1 Multicommodity Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

vii

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