Basic MatheMatics for econoMists Basic Mathematics for Economists, now in its third edition, is a classic of its genre and this new edition builds on the success of previous editions. Suitable for students who may only have a basic mathematics background, as well as students who may have followed more advanced mathematics courses but who still want a clear explanation of fundamental concepts, this book covers all the basic tenets required for an under- standing of mathematics and how it is applied in economics, finance and business. Starting with revisions of the essentials of arithmetic and algebra, students are then taken through to more advanced topics in calculus, comparative statics, dynamic analysis and matrix algebra, with all topics explained in the context of relevant applications. New features in this third edition reflect the increased emphasis on finance in many economics and related degree courses, with fuller analysis of topics such as: DD savings and pension schemes, including drawdown pensions DD asset valuation techniques for bond and share prices DD the application of integration to concepts in economics and finance DD input-output analysis, using spreadsheets to do matrix algebra calculations. In developing new topics the book never loses sight of their applied context and examples are always used to help explain analysis. This book is the most logical, user-friendly book on the market and is usable for mathematics of economics, finance and business courses in all countries. Mike Rosser is a former Principal Lecturer in Economics at Coventry University, UK. Piotr Lis is a Senior Lecturer in Economics at Coventry University, UK. This page intentionally left blank Basic MatheMatics for econoMists third edition Mike Rosser and Piotr Lis Third edition published 2016 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2016 Mike Rosser and Piotr Lis The right of Mike Rosser and Piotr Lis to be identified as authors of this work has been asserted in accordance with the Copyright, Designs and Patent Act 1988. 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. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. First edition published by HarperCollins 1993 Second edition published by Routledge 2003 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 Rosser, M. J., 1949- author. Basic mathematics for economists / Mike Rosser and Piotr Lis. – Third edition. 1. Economics, Mathematical. 2. Business mathematics. I. Title. HB135.R665 2016 510–dc23 2015032713 ISBN: 978-0-415-48591-3 (hbk) ISBN: 978-0-415-48592-0 (pbk) ISBN: 978-1-315-64171-3 (ebk) Typeset in 10.5/13pt Times New Roman by Graphicraft Limited, Hong Kong contents Preface x Preface to second edition xi Preface to third edition xii Acknowledgements xii 1 Introduction 1 1.1 Why study mathematics? 1 1.2 Calculators and computers 3 1.3 Using this book 5 2 Arithmetic 8 2.1 Revision of basic concepts 8 2.2 Multiple operations 9 2.3 Brackets 11 2.4 Fractions 12 2.5 Elasticity of demand 15 2.6 Decimals 18 2.7 Negative numbers 21 2.8 Powers 23 2.9 Roots and fractional powers 25 2.10 Logarithms 28 3 Introduction to algebra 32 3.1 Representation 32 3.2 Evaluation 35 3.3 Simplification: addition and subtraction 37 contents 3.4 Simplification: multiplication 39 3.5 Simplification: factorizing 43 3.6 Simplification: division 47 3.7 Solving simple equations 49 3.8 The summation sign ∑ and price indexes 54 3.9 Inequality signs 59 4 Graphs and functions 63 4.1 Functions 63 4.2 Inverse functions 66 4.3 Graphs of linear functions 68 4.4 Fitting linear functions 73 4.5 Slope 76 4.6 Budget constraints 81 4.7 Non-linear functions 86 4.8 Composite functions 88 4.9 Using a spreadsheet to plot functions 93 4.10 Functions with two independent variables 97 4.11 Summing functions horizontally 102 5 Simultaneous linear equations 107 5.1 Systems of simultaneous linear equations 107 5.2 Solving simultaneous linear equations 108 5.3 Graphical solution 108 5.4 Equating to same variable 110 5.5 Substitution 112 5.6 Row operations 114 5.7 More than two unknowns 116 5.8 Which method? 119 5.9 Comparative statics and the reduced form of an economic model 124 5.10 Price discrimination 133 5.11 Multiplant monopoly 140 5A Appendix: linear programming 148 5A.1 Constrained maximization 148 5A.2 Constrained minimization 158 5A.3 Mixed constraints 165 5A.4 More than two variables 167 6 Quadratic equations 168 6.1 Solving quadratic equations 168 6.2 Graphical solution 170 6.3 Factorization 174 6.4 The quadratic formula 176 6.5 Quadratic simultaneous equations 177 6.6 Polynomials 182 vi contents 7 Financial mathematics – series, time and investment 189 7.1 Discrete and continuous growth 189 7.2 Interest 191 7.3 Part year investment and the annual equivalent rate 196 7.4 Time periods, initial amounts and interest rates 202 7.5 Investment appraisal: net present value 207 7.6 The internal rate of return 217 7.7 Geometric series and annuities 224 7.8 Perpetual annuities 230 7.9 Pension pots, annuity income and drawdown pensions 234 7.10 Drawdown pension income 242 7.11 Loan repayments and mortgages 244 7.12 Savings schemes 252 7.13 Sinking fund savings schemes 257 7.14 Other applications of growth and decline 260 7A Appendix: asset valuation 267 7A.1 Valuation of bonds 267 7A.2 Valuation of shares 273 8 Introduction to calculus 280 8.1 Differential calculus 280 8.2 Rules for differentiation 282 8.3 Marginal revenue and total revenue 286 8.4 Marginal cost and total cost 291 8.5 Profit maximization 293 8.6 Re-specifying functions 295 8.7 Point elasticity of demand 297 8.8 Tax yield 300 8.9 The Keynesian multiplier 303 9 Unconstrained optimization 305 9.1 First-order conditions for a maximum 305 9.2 Second-order conditions for a maximum 306 9.3 Second-order conditions for a minimum 309 9.4 Summary of second-order conditions 310 9.5 Profit maximization 313 9.6 Inventory control 316 9.7 Comparative static effects of taxes 320 10 Partial differentiation 326 10.1 Partial differentiation and the marginal product 326 10.2 Further applications of partial differentiation 332 10.3 Second-order partial derivatives 344 10.4 Unconstrained optimization: functions with two variables 349 10.5 Total differentials and total derivatives 364 vii contents 11 Constrained optimization 374 11.1 Constrained optimization and resource allocation 374 11.2 Constrained optimization by substitution 375 11.3 The Lagrange multiplier: constrained maximization with two variables 383 11.4 The Lagrange multiplier: second-order conditions 389 11.5 Constrained minimization using the Lagrange multiplier 392 11.6 Constrained optimization with more than two variables 398 12 Further topics in differentiation and integration 407 12.1 Overview 407 12.2 The chain rule 407 12.3 The product rule 416 12.4 The quotient rule 422 12.5 Integration 429 12.6 Definite integrals 435 12.7 Integration by substitution and integration by parts 442 13 Dynamics and difference equations 449 13.1 Dynamic economic analysis 449 13.2 The cobweb: iterative solutions 450 13.3 The cobweb: difference equation solutions 460 13.4 The lagged Keynesian macroeconomic model 470 13.5 Duopoly price adjustment 482 14 Exponential functions, continuous growth and differential equations 488 14.1 Continuous growth and the exponential function 488 14.2 Accumulated final values after continuous growth 491 14.3 Continuous growth rates and initial amounts 494 14.4 Natural logarithms 499 14.5 Differentiation of logarithmic functions 504 14.6 Continuous time and differential equations 506 14.7 Solution of homogeneous differential equations 507 14.8 Solution of non-homogeneous differential equations 511 14.9 Continuous adjustment of market price 516 14.10 Continuous adjustment in a Keynesian macroeconomic model 521 15 Matrix algebra 526 15.1 Introduction to matrices and vectors 526 15.2 Basic principles of matrix multiplication 531 15.3 Matrix multiplication – the general case 534 15.4 The matrix inverse and the solution of simultaneous equations 540 15.5 Determinants 544 15.6 Minors, cofactors and the Laplace expansion 547 viii contents 15.7 The transpose matrix, the cofactor matrix, the adjoint and the matrix inverse formula 551 15.8 Application of the matrix inverse to the solution of linear simultaneous equations 556 15.9 Cramer’s rule 562 15.10 Second-order conditions and the Hessian matrix 564 15.11 Constrained optimization and the bordered Hessian 571 15.12 Input-output analysis 575 15.13 Multiple industry input-output models 581 Answers 589 Index 600 ix