Elements of Mathematics for Economics and Finance
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1 Elements of Mathematics for Economics and Finance
2 Vassilis C. Mavron and Timothy N. Phillips Elements of Mathematics for Economics and Finance With 77 Figures
3 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: ISBN-10: e-isbn Printed on acid-free paper ISBN-13: 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) Springer Science + Business Media, LLC springer.com
4 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
5 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
6 Contents 1. Essential Skills Introduction Numbers Addition and Subtraction Multiplication and Division Evaluation of Arithmetical Expressions Fractions Multiplication and Division Decimal Representation of Numbers Standard Form Percentages PowersandIndices Simplifying Algebraic Expressions Multiplying Brackets Factorization Linear Equations Introduction Solution of Linear Equations Solution of Simultaneous Linear Equations GraphsofLinearEquations Slope of a Straight Line BudgetLines Supply and Demand Analysis Multicommodity Markets vii
7 viii Contents 3. Quadratic Equations Introduction GraphsofQuadraticFunctions Quadratic Equations Applications to Economics Functions of a Single Variable Introduction Limits Polynomial Functions ReciprocalFunctions Inverse Functions The Exponential and Logarithmic Functions Introduction ExponentialFunctions Logarithmic Functions ReturnstoScaleofProductionFunctions Cobb-DouglasProductionFunctions Compounding of Interest Applications of the Exponential Function in Economic Modelling Differentiation Introduction RulesofDifferentiation ConstantFunctions LinearFunctions PowerFunctions SumsandDifferencesofFunctions ProductofFunctions QuotientofFunctions TheChainRule Exponential and Logarithmic Functions MarginalFunctionsinEconomics Marginal Revenue and Marginal Cost Marginal Propensities ApproximationtoMarginalFunctions HigherOrderDerivatives ProductionFunctions...129
8 Contents ix 7. Maxima and Minima Introduction Local Properties of Functions Increasing and Decreasing Functions ConcaveandConvexFunctions Local or Relative Extrema GlobalorAbsoluteExtrema PointsofInflection Optimization of Production Functions Optimization of Profit Functions OtherExamples Partial Differentiation Introduction FunctionsofTwoorMoreVariables Partial Derivatives Higher Order Partial Derivatives Partial Rate of Change TheChainRuleandTotalDerivatives Some Applications of Partial Derivatives ImplicitDifferentiation ElasticityofDemand Utility Production Graphical Representations Optimization Introduction UnconstrainedOptimization Constrained Optimization Substitution Method Lagrange Multipliers The Lagrange Multiplier λ:aninterpretation IsoCurves Matrices and Determinants Introduction Matrix Operations Scalar Multiplication Matrix Addition Matrix Multiplication Solutions of Linear Systems of Equations
9 x Contents 10.4 Cramer srule MoreDeterminants SpecialCases Integration Introduction RulesofIntegration DefiniteIntegrals DefiniteIntegration:AreaandSummation Producer s Surplus Consumer ssurplus Linear Difference Equations Introduction DifferenceEquations First Order Linear Difference Equations Stability TheCobwebModel Second Order Linear Difference Equations Complementary Solutions Particular Solutions Stability Differential Equations Introduction First Order Linear Differential Equations Stability Nonlinear First Order Differential Equations SeparationofVariables SecondOrderLinearDifferentialEquations TheHomogeneousCase TheGeneralCase Stability A. Differentials Index...309
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