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 questions, 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 EC4P 4EE 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 This edition published in the Taylor & Francis e-Library, 2003 © 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 0-203-42263-5 Master e-book ISBN ISBN 0-203-42439-5 (Adobe eReader Format) ISBN 0–415–26783–8 (hbk) ISBN 0–415–26784–6 (pbk) © 1993, 2003 Mike Rosser Contents Preface Preface to Second Edition Acknowledgements Introduction 1.1 Why study mathematics? 1.2 Calculators and computers 1.3 Using the book 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 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 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 Linear 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 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 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 7.9 7.10 Perpetual annuities Loan repayments Other applications of growth and decline 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 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 12.6 12.7 Individual labour supply Integration 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 15 Matrix 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 mathematics 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 examinations 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 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 students’ 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 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 spreadsheet package to solve certain problems that are extremely difficult or time-consuming to solve manually are also explained © 1993, 2003 Mike Rosser ... MR = 33.33 − 0.00667Q for Q ≥ 500 MR = 76 − 0.222Q for Q ≥ 22.5 MR = 80 − 0.555Q for Q ≥ 562.5 MC = 30 + 0.0714Q for Q ≥ 56 MC = 56 + 0.1333Q for Q ≥ 30 MC = + 0.0714Q for Q ≥ 59 Chapter 5.1... 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. .. depends on which order you perform the calculations and the type of calculator you use There are set rules for the order in which basic arithmetic operations should be performed, which are explained