Basic Mathematics for Economists

Basic Mathematics for Economists

Economics students will welcome the new edition of this excellent textbook. Given

that many students come into economics courses without having studied mathematics

for a number of years, this clearly written book will help to develop quantitative skills

in even the least numerate student up to the required level for a general Economics

or Business Studies course. All explanations of mathematical concepts are set out in

the context of applications in economics.

This new edition incorporates several new features, including new sections on:

• financial mathematics

• continuous growth

• matrix algebra

Improved pedagogical features, such as learning objectives and end of chapter ques￾tions, along with an overall example-led format and the use of Microsoft Excel for

relevant applications mean that this textbook will continue to be a popular choice for

both students and their lecturers.

Mike Rosser is Principal Lecturer in Economics in the Business School at Coventry

University.

© 1993, 2003 Mike Rosser

Basic Mathematics for

Economists

Second Edition

Mike Rosser

© 1993, 2003 Mike Rosser

First edition published 1993

by Routledge

This edition published 2003

by Routledge

11 New Fetter Lane, London EC4P4EE

Simultaneously published in the USA and Canada

by Routledge

29 West 35th Street, New York, NY 10001

Routledge is an imprint of the Taylor & Francis Group

© 1993, 2003 Mike Rosser

All rights reserved. No part of this book may be reprinted or reproduced or

utilised in any form or by any electronic, mechanical, or other means, now

known or hereafter invented, including photocopying and recording, or in any

information storage or retrieval system, without permission in writing from

the publishers.

British Library Cataloguing in Publication Data

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

Library of Congress Cataloging in Publication Data

A catalog record for this book has been requested

ISBN 0–415–26783–8 (hbk)

ISBN 0–415–26784–6 (pbk)

This edition published in the Taylor & Francis e-Library, 2003.

ISBN 0-203-42263-5 Master e-book ISBN

ISBN 0-203-42439-5 (Adobe eReader Format)

© 1993, 2003 Mike Rosser

Contents

Preface

Preface to Second Edition

Acknowledgements

1 Introduction

1.1 Why study mathematics?

1.2 Calculators and computers

1.3 Using the book

2 Arithmetic

2.1 Revision of basic concepts

2.2 Multiple operations

2.3 Brackets

2.4 Fractions

2.5 Elasticity of demand

2.6 Decimals

2.7 Negative numbers

2.8 Powers

2.9 Roots and fractional powers

2.10 Logarithms

3 Introduction to algebra

3.1 Representation

3.2 Evaluation

3.3 Simplification: addition and subtraction

3.4 Simplification: multiplication

3.5 Simplification: factorizing

3.6 Simplification: division

3.7 Solving simple equations

3.8 The summation sign

3.9 Inequality signs

© 1993, 2003 Mike Rosser

4 Graphs and functions

4.1 Functions

4.2 Inverse functions

4.3 Graphs of linear functions

4.4 Fitting linear functions

4.5 Slope

4.6 Budget constraints

4.7 Non-linear functions

4.8 Composite functions

4.9 Using Excel to plot functions

4.10 Functions with two independent variables

4.11 Summing functions horizontally

5Linear equations

5.1 Simultaneous linear equation systems

5.2 Solving simultaneous linear equations

5.3 Graphical solution

5.4 Equating to same variable

5.5 Substitution

5.6 Row operations

5.7 More than two unknowns

5.8 Which method?

5.9 Comparative statics and the reduced form of

an economic model

5.10 Price discrimination

5.11 Multiplant monopoly

Appendix: linear programming

6 Quadratic equations

6.1 Solving quadratic equations

6.2 Graphical solution

6.3 Factorization

6.4 The quadratic formula

6.5 Quadratic simultaneous equations

6.6 Polynomials

7 Financial mathematics: series, time and investment

7.1 Discrete and continuous growth

7.2 Interest

7.3 Part year investment and the annual equivalent rate

7.4 Time periods, initial amounts and interest rates

7.5 Investment appraisal: net present value

7.6 The internal rate of return

7.7 Geometric series and annuities

© 1993, 2003 Mike Rosser

7.8 Perpetual annuities

7.9 Loan repayments

7.10 Other applications of growth and decline

8 Introduction to calculus

8.1 The differential calculus

8.2 Rules for differentiation

8.3 Marginal revenue and total revenue

8.4 Marginal cost and total cost

8.5 Profit maximization

8.6 Respecifying functions

8.7 Point elasticity of demand

8.8 Tax yield

8.9 The Keynesian multiplier

9 Unconstrained optimization

9.1 First-order conditions for a maximum

9.2 Second-order condition for a maximum

9.3 Second-order condition for a minimum

9.4 Summary of second-order conditions

9.5 Profit maximization

9.6 Inventory control

9.7 Comparative static effects of taxes

10 Partial differentiation

10.1 Partial differentiation and the marginal product

10.2 Further applications of partial differentiation

10.3 Second-order partial derivatives

10.4 Unconstrained optimization: functions with two variables

10.5 Total differentials and total derivatives

11 Constrained optimization

11.1 Constrained optimization and resource allocation

11.2 Constrained optimization by substitution

11.3 The Lagrange multiplier: constrained maximization

with two variables

11.4 The Lagrange multiplier: second-order conditions

11.5 Constrained minimization using the Lagrange multiplier

11.6 Constrained optimization with more than two variables

12 Further topics in calculus

12.1 Overview

12.2 The chain rule

12.3 The product rule

12.4 The quotient rule

© 1993, 2003 Mike Rosser

12.5 Individual labour supply

12.6 Integration

12.7 Definite integrals

13 Dynamics and difference equations

13.1 Dynamic economic analysis

13.2 The cobweb: iterative solutions

13.3 The cobweb: difference equation solutions

13.4 The lagged Keynesian macroeconomic model

13.5 Duopoly price adjustment

14 Exponential functions, continuous growth and

differential equations

14.1 Continuous growth and the exponential function

14.2 Accumulated final values after continuous growth

14.3 Continuous growth rates and initial amounts

14.4 Natural logarithms

14.5 Differentiation of logarithmic functions

14.6 Continuous time and differential equations

14.7 Solution of homogeneous differential equations

14.8 Solution of non-homogeneous differential equations

14.9 Continuous adjustment of market price

14.10 Continuous adjustment in a Keynesian macroeconomic model

15Matrix algebra

15.1 Introduction to matrices and vectors

15.2 Basic principles of matrix multiplication

15.3 Matrix multiplication – the general case

15.4 The matrix inverse and the solution of

simultaneous equations

15.5 Determinants

15.6 Minors, cofactors and the Laplace expansion

15.7 The transpose matrix, the cofactor matrix, the adjoint

and the matrix inverse formula

15.8 Application of the matrix inverse to the solution of

linear simultaneous equations

15.9 Cramer’s rule

15.10 Second-order conditions and the Hessian matrix

15.11 Constrained optimization and the bordered Hessian

Answers

Symbols and terminology

© 1993, 2003 Mike Rosser

Preface

Over half of the students who enrol on economics degree courses have not studied mathe￾matics beyond GCSE or an equivalent level. These include many mature students whose last

encounter with algebra, or any other mathematics beyond basic arithmetic, is now a dim and

distant memory. It is mainly for these students that this book is intended. It aims to develop

their mathematical ability up to the level required for a general economics degree course (i.e.

one not specializing in mathematical economics) or for a modular degree course in economics

and related subjects, such as business studies. To achieve this aim it has several objectives.

First, it provides a revision of arithmetical and algebraic methods that students probably

studied at school but have now largely forgotten. It is a misconception to assume that, just

because a GCSE mathematics syllabus includes certain topics, students who passed exami￾nations on that syllabus two or more years ago are all still familiar with the material. They

usually require some revision exercises to jog their memories and to get into the habit of

using the different mathematical techniques again. The first few chapters are mainly devoted

to this revision, set out where possible in the context of applications in economics.

Second, this book introduces mathematical techniques that will be new to most students

through examples of their application to economic concepts. It also tries to get students

tackling problems in economics using these techniques as soon as possible so that they can

see how useful they are. Students are not required to work through unnecessary proofs, or

wrestle with complicated special cases that they are unlikely ever to encounter again. For

example, when covering the topic of calculus, some other textbooks require students to

plough through abstract theoretical applications of the technique of differentiation to every

conceivable type of function and special case before any mention of its uses in economics

is made. In this book, however, we introduce the basic concept of differentiation followed

by examples of economic applications in Chapter 8. Further developments of the topic,

such as the second-order conditions for optimization, partial differentiation, and the rules

for differentiation of composite functions, are then gradually brought in over the next few

chapters, again in the context of economics application.

Third, this book tries to cover those mathematical techniques that will be relevant to stu￾dents’ economics degree programmes. Most applications are in the field of microeconomics,

rather than macroeconomics, given the increased emphasis on business economics within

many degree courses. In particular, Chapter 7 concentrates on a number of mathematical

techniques that are relevant to finance and investment decision-making.

Given that most students now have access to computing facilities, ways of using a spread￾sheet package to solve certain problems that are extremely difficult or time-consuming to

solve manually are also explained.

© 1993, 2003 Mike Rosser

Although it starts at a gentle pace through fairly elementary material, so that the students

who gave up mathematics some years ago because they thought that they could not cope with

A-level maths are able to build up their confidence, this is not a watered-down ‘mathematics

without tears or effort’ type of textbook. As the book progresses the pace is increased and

students are expected to put in a serious amount of time and effort to master the material.

However, given the way in which this material is developed, it is hoped that students will be

motivated to do so. Not everyone finds mathematics easy, but at least it helps if you can see

the reason for having to study it.

© 1993, 2003 Mike Rosser

Preface to Second Edition

The approach and style of the first edition have proved popular with students and I have tried

to maintain both in the new material introduced in this second edition. The emphasis is on the

introduction of mathematical concepts in the context of economics applications, with each

step of the workings clearly explained in all the worked examples. Although the first edition

was originally aimed at less mathematically able students, many others have also found it

useful, some as a foundation for further study in mathematical economics and others as a

helpful reference for specific topics that they have had difficulty understanding.

The main changes introduced in this second edition are a new chapter on matrix algebra

(Chapter 15) and a rewrite of most of Chapter 14, which now includessections on differential

equations and has been retitled ‘Exponential functions, continuous growth and differential

equations’. A new section on part-year investment has been added and the section on interest

rates rewritten in Chapter 7, which is now called ‘Financial mathematics – series, time and

investment’. There are also new sections on the reduced form of an economic model and

the derivation of comparative static predictions, in Chapter 5 using linear algebra, and in

Chapter 9 using calculus. All spreadsheet applications are now based on Excel, as this is now

the most commonly used spreadsheet program. Other minor changes and corrections have

been made throughout the rest of the book.

The Learning Objectives are now set out at the start of each chapter. It is hoped that students

will find these useful as a guide to what they should expect to achieve, and their lecturers

will find them useful when drawing up course guides. The layout of the pages in this second

edition is also an improvement on the rather cramped style of the first edition.

I hope that both students and their lecturers will find these changes helpful.

Mike Rosser

Coventry

© 1993, 2003 Mike Rosser

Acknowledgements

Microsoft
Windows and Microsoft
Excel
are registered trademarks of the Microsoft

Corporation. Screen shot(s) reprinted by permission from Microsoft Corporation.

I am still grateful to those who helped in the production of the first edition of this book,

including Joy Warren for her efficiency in typing the final manuscript, Mrs M. Fyvie and

Chandrika Chauhan for their help in typing earlier drafts, and Mick Hayes for his help in

checking the proofs.

The comments I have received from those people who have used the first edition have been

very helpful for the revisions and corrections made in this second edition. I would particularly

like to thank Alison Johnson at the Centre for International Studies in Economics, SOAS,

London, and Ray Lewis at the University of Adelaide, Australia, for their help in checking

the answers to the questions. I am also indebted to my colleague at Coventry, Keith Redhead,

for his advice on the revised chapter on financial mathematics, to Gurpreet Dosanjh for his

help in checking the second edition proofs, and to the two anonymous publisher’s referees

whose comments helped me to formulate this revised second edition.

Last, but certainly not least, I wish to acknowledge the help of my students in shaping

the way that this book was originally developed and has since been revised. I, of course, am

responsible for any remaining errors or omissions.

© 1993, 2003 Mike Rosser

1 Introduction

Learning objective

After completing this chapter students should be able to:

• Understand why mathematics is useful to economists.

1.1 Why study mathematics?

Economics is a social science. It does not just describe what goes on in the economy. It

attempts to explain how the economy operates and to make predictions about what may

happen to specified economic variables if certain changes take place, e.g. what effect a crop

failure will have on crop prices, what effect a given increase in sales tax will have on the

price of finished goods, what will happen to unemployment if government expenditure is

increased. It also suggests some guidelines that firms, governments or other economic agents

might follow if they wished to allocate resources efficiently. Mathematics is fundamental to

any serious application of economics to these areas.

Quantification

In introductory economic analysis predictions are often explained with the aid of sketch

diagrams. For example, supply and demand analysis predicts that in a competitive market if

supply is restricted then the price of a good will rise. However, this is really only common

sense, as any market trader will tell you. An economist also needs to be able to say by how

much price is expected to rise if supply contracts by a specified amount. This quantification

of economic predictions requires the use of mathematics.

Although non-mathematical economic analysis may sometimes be useful for making qual￾itative predictions (i.e. predicting the direction of any expected changes), it cannot by itself

provide the quantification that users of economic predictions require. A firm needs to know

how much quantity sold is expected to change in response to a price increase. The government

wants to know how much consumer demand will change if it increases a sales tax.

Simplification

Sometimes students believe that mathematics makes economics more complicated. Algebraic

notation, which is essentially a form of shorthand, can, however, make certain concepts much

© 1993, 2003 Mike Rosser