Elements of Mathematics for Economics and Finance

Elements of Mathematics for Economics and Finance

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) 9 8 7 6 5 4 3 2 1 Springer Science + Business Media, LLC springer.com

Preface The mathematics contained in this book for students of economics and finance 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 Management. The choice of subject matter and arrangement of material reflect this collaboration and are a result of the experience thus obtained. The majority of students to whom these courses were given were studying for degrees in economics or business administration and had not acquired any mathematical knowledge beyond pre-calculus mathematics, i.e., elementary algebra. Therefore, the first-semester course assumed little more than basic precalculus 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 differentials has been included as an optional addition to an advanced course. The students taking these courses were chiefly interested in learning the mathematics that had applications to economics and were not primarily interested in theoretical aspects of the subject per se. The authors have not attempted 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 find it easier to go on to the more difficult 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 preparing the manuscript of this book for publication. Above all, we are grateful to our wives, Nesta and Gill, and to our children, Nicholas and Christiana, and Rebecca, Christopher, and Emily, for their patience, support, and understanding: this book is dedicated to them. Vassilis C. Mavron Aberystwyth United Kingdom Timothy N. Phillips Cardiff 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 PowersandIndices... 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 GraphsofLinearEquations... 30 2.4.1 Slope of a Straight Line............................. 34 2.5 BudgetLines... 37 2.6 Supply and Demand Analysis.............................. 40 2.6.1 Multicommodity Markets............................ 44 vii

viii Contents 3. Quadratic Equations... 49 3.1 Introduction... 49 3.2 GraphsofQuadraticFunctions... 50 3.3 Quadratic Equations...................................... 56 3.4 Applications to Economics................................. 61 4. Functions of a Single Variable... 69 4.1 Introduction... 69 4.2 Limits... 72 4.3 Polynomial Functions..................................... 72 4.4 ReciprocalFunctions... 75 4.5 Inverse Functions......................................... 81 5. The Exponential and Logarithmic Functions... 87 5.1 Introduction... 87 5.2 ExponentialFunctions... 88 5.3 Logarithmic Functions.................................... 90 5.4 ReturnstoScaleofProductionFunctions... 95 5.4.1 Cobb-DouglasProductionFunctions... 97 5.5 Compounding of Interest.................................. 98 5.6 Applications of the Exponential Function in Economic Modelling...................................... 102 6. Differentiation...109 6.1 Introduction...109 6.2 RulesofDifferentiation...113 6.2.1 ConstantFunctions...113 6.2.2 LinearFunctions...114 6.2.3 PowerFunctions...114 6.2.4 SumsandDifferencesofFunctions...114 6.2.5 ProductofFunctions...116 6.2.6 QuotientofFunctions...117 6.2.7 TheChainRule...117 6.3 Exponential and Logarithmic Functions..................... 119 6.4 MarginalFunctionsinEconomics...121 6.4.1 Marginal Revenue and Marginal Cost................. 121 6.4.2 Marginal Propensities............................... 123 6.5 ApproximationtoMarginalFunctions...125 6.6 HigherOrderDerivatives...127 6.7 ProductionFunctions...129

Contents ix 7. Maxima and Minima...137 7.1 Introduction...137 7.2 Local Properties of Functions.............................. 138 7.2.1 Increasing and Decreasing Functions.................. 138 7.2.2 ConcaveandConvexFunctions...138 7.3 Local or Relative Extrema................................. 139 7.4 GlobalorAbsoluteExtrema...144 7.5 PointsofInflection...145 7.6 Optimization of Production Functions....................... 146 7.7 Optimization of Profit Functions........................... 151 7.8 OtherExamples...154 8. Partial Differentiation...159 8.1 Introduction...159 8.2 FunctionsofTwoorMoreVariables...160 8.3 Partial Derivatives........................................ 160 8.4 Higher Order Partial Derivatives........................... 163 8.5 Partial Rate of Change.................................... 165 8.6 TheChainRuleandTotalDerivatives...168 8.7 Some Applications of Partial Derivatives.................... 171 8.7.1 ImplicitDifferentiation...171 8.7.2 ElasticityofDemand...173 8.7.3 Utility............................................ 176 8.7.4 Production...179 8.7.5 Graphical Representations........................... 181 9. Optimization...185 9.1 Introduction...185 9.2 UnconstrainedOptimization...186 9.3 Constrained Optimization................................. 193 9.3.1 Substitution Method................................ 193 9.3.2 Lagrange Multipliers................................ 197 9.3.3 The Lagrange Multiplier λ:aninterpretation...201 9.4 IsoCurves...204 10. Matrices and Determinants...209 10.1 Introduction...209 10.2 Matrix Operations........................................ 209 10.2.1 Scalar Multiplication................................ 211 10.2.2 Matrix Addition.................................... 212 10.2.3 Matrix Multiplication............................... 212 10.3 Solutions of Linear Systems of Equations.................... 220

x Contents 10.4 Cramer srule...222 10.5 MoreDeterminants...223 10.6 SpecialCases...230 11. Integration...233 11.1 Introduction...233 11.2 RulesofIntegration...236 11.3 DefiniteIntegrals...241 11.4 DefiniteIntegration:AreaandSummation...243 11.5 Producer s Surplus....................................... 250 11.6 Consumer ssurplus...251 12. Linear Difference Equations...261 12.1 Introduction...261 12.2 DifferenceEquations...261 12.3 First Order Linear Difference Equations..................... 264 12.4 Stability................................................. 267 12.5 TheCobwebModel...270 12.6 Second Order Linear Difference Equations................... 273 12.6.1 Complementary Solutions........................... 274 12.6.2 Particular Solutions................................ 277 12.6.3 Stability.......................................... 282 13. Differential Equations...287 13.1 Introduction...287 13.2 First Order Linear Differential Equations.................... 288 13.2.1 Stability.......................................... 292 13.3 Nonlinear First Order Differential Equations................. 292 13.3.1 SeparationofVariables...294 13.4 SecondOrderLinearDifferentialEquations...296 13.4.1 TheHomogeneousCase...297 13.4.2 TheGeneralCase...300 13.4.3 Stability.......................................... 302 A. Differentials...305 Index...309