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
Trang 3Basic 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.
Trang 4Basic Mathematics for Economists
Second Edition
Mike Rosser
Trang 5First 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)
Trang 6Preface
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
Trang 74 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
Trang 87.8 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
Trang 912.5 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
Trang 10Over 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
Trang 11Although 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
Trang 12Preface 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 ofChapter 14, which now includes sections 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 inChapter 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 5using linear algebra, and in
Chapter 9using 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
Trang 13Microsoft 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