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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
© Springer-Verlag London Limited 2007
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
Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to 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
v
vi
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
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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