1、AN INTRODUCTION TO MATHEMATICS FOR ECONOMICSAKIHITO ASANOCONTENTS1 Demand and supply in competitive markets2 Basic mathematics3 Financial mathematics4 Differential calculus 15 Differential calculus 26 Multivariate calculus7 Integral calculusAppendix A Matrix algebraAppendix B An introduction to diff
2、erence and differential equationsASANOAN INTRODUCTION TOMATHEMATICS FOR ECONOMICSA concise, accessible introduction to quantitative methods for economics and finance students, this textbook con-tains lots of practical applications to show why maths is necessary and relevant to economics, as well as
3、worked examples and exercises to help students learn and prepare for exams. Introduces mathematical techniques in the context of introductory economics, bridging the gap between the two subjects Written in a friendly conversational style, but with precise presentation of mathematics Explains applica
4、tions in detail, enabling students to learn how to put mathematics into practice Encourages students to develop confidence in their numeracy skills by solving arithmetical problems without a calculator Extensive provision of worked examples and exercises to underpin the readers knowledge and learnin
5、g This outstanding textbook is the by-product of lecture notes writ-ten by a dedicated teacher. Mathematics is carefully exposited for first-year students, using familiar applications drawn from economics and finance. By working through the problems provided, students can overcome any fear they migh
6、t have of mathematics to make it an enjoyable companion.Chris Jones is Associate Professor of Economics at Australian National UniversityIn this well-written text, mathematical techniques are introduced together with basic economic ideas, underlining the fact that mathematics should not be treated s
7、eparately, but is an integral and essential part of economics. The style is friendly and conversational, and the mathematical techniques are treated rigorously, with many clearly presented examples. Dr Asano is adept in pinpointing those areas that students find difficult, making this a very useful
8、and comprehensive text for anyone undertaking the study of economics.Valerie Haggan-Ozaki, Faculty of Liberal Arts, Sophia UniversityDr Asano is a renowned teacher, who transformed the course he has taught on this subject into a popular, albeit challenging, course that laid an excellent foundation f
9、or Economics majors. He has brought this style to this textbook and students will find it very thorough in its treatment of each topic, and find that they will learn by doing as much as by reading. Students new to economics will find the style easy to follow and this textbook will facilitate the tea
10、ching of this material for lecturers too. Martin Richardson, Professor of Economics, Australian National UniversityCOVER DESIGN:SUE WATSONAn Introduction toMathematics for EconomicsAnIntroductiontoMathematicsforEconomics introduces quantitative methods to studentsof economics and finance in a succin
11、ct and accessible style. The introductory nature ofthis textbook means a background in economics is not essential, as it aims to help studentsappreciate that learning mathematics is relevant to their overall understanding of thesubject. Economic and financial applications are explained in detail bef
12、ore students learnhow mathematics can be used, enabling students to learn how to put mathematics intopractice. Starting with a revision of basic mathematical principles the second half of thebook introduces calculus, emphasising economic applications throughout. Appendiceson matrix algebra and diffe
13、rence/differential equations are included for the benefit ofmore advanced students. Other features, including worked examples and exercises, helpto underpin the readers knowledge and learning. Akihito Asano has drawn upon his ownextensive teaching experience to create an unintimidating yet rigorous
14、textbook.Akihito Asano is Associate Professor of Economics at the Faculty of Liberal Arts, SophiaUniversity, Tokyo. He has previously held positions at the University of Melbourne andthe Australian National University (ANU). In 2008 he received the Award for TeachingExcellence from the College of Bu
15、siness and Economics at the ANU. He currently teachesintroductory and intermediate microeconomics, international trade and introduction togame theory to undergraduate students, and mathematical techniques in economics tograduate students.An Introduction toMathematics forEconomicsAKIHITO ASANOcambrid
16、ge university pressCambridge, New York, Melbourne, Madrid, Cape Town,Singapore, Sao Paulo, Delhi, Mexico CityCambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UKPublished in the United States of America by Cambridge University Press, New Yorkwww.cambridge.orgInformation on this ti
17、tle: www.cambridge.org/9780521189460CAkihito Asano 2013This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.First publis
18、hed 2013Printed and Bound in Great Britain by the MPG Books GroupA catalogue record for this publication is available from the British LibraryISBN 978-1-107-00760-4 HardbackISBN 978-0-521-18946-0 PaperbackCambridge University Press has no responsibility for the persistence oraccuracy of URLs for ext
19、ernal or third-party internet websites referred toin this publication, and does not guarantee that any content on suchwebsites is, or will remain, accurate or appropriate.ContentsList of illustrations page viiiList of tables xiPreface xiiiAcknowledgements xvi1 Demand and supply in competitive market
20、s 11.1 Markets 11.2 Demand and supply schedules 31.3 Market equilibrium 51.4 Rest of this book 72 Basic mathematics 82.1 Numbers 92.2 Fractions, decimal numbers and the use of a calculator 102.3 Some algebraic properties of real numbers 112.4 Equalities, inequalities and intervals 122.5 Powers 132.6
21、 An imaginary number and complex numbers 162.7 Factorisation: reducing polynomial expressions 162.8 Equations 192.9 Functions 222.10 Simultaneous equations: the demand and supply analysis 342.11 Logic 432.12 Proofs 472.13 Additional exercises 533 Financial mathematics 573.1 Limits 573.2 Summation 61
22、3.3 A geometric series 643.4 Compound interest 653.5 The exponential function: how can we calculate the compound amount ofthe principal if interest is compounded continuously? 703.6 Logarithms: how many years will it take for my money to double? 753.7 Present values 803.8 Annuities: what is the valu
23、e of your home loan? 82vi Contents3.9 Perpetuity 863.10 Additional exercises 874 Differential calculus 1 904.1 Cost function 904.2 The marginal cost and the average costs 924.3 Production function 954.4 Firms supply curve 984.5 From a one-unit change to an infinitesimally small change 1034.6 The rel
24、ative positions of MC, AC and AV C revisited 1104.7 Profit maximisation 1114.8 Additional exercises 1215 Differential calculus 2 1245.1 Curve sketching 1245.2 The differential 1345.3 Elasticity 1365.4 Additional exercises 1446 Multivariate calculus 1476.1 The utility function 1486.2 Indifference cur
25、ves 1516.3 The marginal utility for the two-good case 1516.4 The marginal rate of substitution 1576.5 Total differentiation and implicit differentiation 1596.6 Maxima and minima revisited 1646.7 The utility maximisation problem: constrainedoptimisation 1696.8 The substitution method 1746.9 The Lagra
26、nge multiplier method 1766.10 The individual demand function 1796.11 Additional exercises 1827 Integral calculus 1847.1 An anti-derivative and the indefinite integral 1847.2 The fundamental theorem of integral calculus 1877.3 Application of integration to finance: the present value of a continuousan
27、nuity 1907.4 Demand and supply analysis revisited 1957.5 The deadweight loss of taxation 2067.6 Additional exercises 214vii ContentsAppendix A Matrix algebra 218A.1 Matrices and vectors 219A.2 An inverse of a matrix and the determinant: solving a system of equations 228A.3 An unconstrained optimisat
28、ion problem 234Appendix B An introduction to difference and differentialequations 243B.1 The cobweb model of price adjustment 243B.2 The linear first-order autonomous difference equation 248B.3 The linear first-order autonomous differential equation 255Index 262Illustrations1.1 The market demand for
29、 sausages page 31.2 The market supply for sausages 51.3 The market equilibrium 62.1 The real line 112.2 The coordinate plane 242.3 C(q) = 10q + 30 262.4 p(q) =12q + 80 262.5 f (x) = x 272.6 f (x) =4 and x = 3272.7 Diagrammatic representation of the solutions to a quadratic equation 312.8 The graph o
30、f p(q) =4q322.9 Case-defined function 332.10 Inverse demand and inverse supply functions 352.11 Infinitely many solutions 372.12 No solutions 372.13 A unique solution 382.14 A system of non-linear equations 402.15 A shift of the demand curve (schedule) 412.16 Comparative statics 423.1 Continuous fun
31、ction 583.2 A function not continuous 593.3 Another function not continuous 603.4 A graph of the natural exponential function 743.5 The compound amount increases at an increasing rate 743.6 A graph of the natural logarithmic function 793.7 A time line 1 833.8 A time line 2 854.1 A rough sketch of th
32、e total cost curve 924.2 A rough sketch of the marginal cost curve 934.3 A rough sketch of AC and AV C curves 944.4 A rough sketch of MC, AC and AV C curves 944.5 A rough sketch of the production function 964.6 A rough sketch of MC, AC and AV C curves 994.7 A firm makes positive profits: ppB1004.8 A
33、 firm is break-even: p = pB1014.9 A firm makes losses but produces: pS0: slope is increasing 1144.18 primeprime(q) 0: slope is increasing; strictly convex function (in this domain) 1275.4 Curve sketching: Steps (1), (2) and (3) 1285.5 Curve sketching: f (x) =12x2 8x + 40 1305.6 Curve sketching: f (x
34、) =1x1315.7 Curve sketching: f (x) = x +1x1325.8 Curve sketching: f (x) = ex1335.9 Curve sketching: f (x) = ln x 1345.10 Visualising the differential 1355.11 Price elasticity of demand 1395.12 Price elasticity and total expenditure 1405.13 Elasticity increases over time: the second law of the demand
35、 1446.1 Utility function (for one good) 1486.2 Consumption bundles 1506.3 Indifference curves 1516.4 Marginal utility of Good 1 and Good 2 1526.5 The utility function and indifference curves 1556.6 The utility function and the marginal utility 1556.7 The marginal rate of substitution 1586.8 The tota
36、l derivative 1616.9 Obtaining MRS by using implicit differentiation 1636.10 Local maximum 1646.11 Local minimum 1656.12 A saddle point 1666.13 The budget constraint 1706.14 Utility maximisation occurs at Point A 1716.15 Utility maximisation does NOT occur at Point A 1726.16 The individual demand sch
37、edule 1807.1 The derivative and anti-derivatives 185x List of illustrations7.2 The definite integral 1877.3 The present value of an ordinary annuity 1907.4 The present value of continuous payments 1917.5 The present value of a continuous annuity 1917.6 Adding supply schedules (horizontally) 1967.7 P
38、roducer surplus 1987.8 Producer surplus when the market supply schedule is smooth 1997.9 The change in producer surplus 2017.10 Individual and market demand schedules 2017.11 Consumer surplus 2037.12 Total surplus 2057.13 There will be an excess demand if suppliers bear the entire tax burden 2077.14
39、 The economic incidence of a production tax 2077.15 The deadweight loss (DWL) of taxation 2087.16 There will be an excess demand if producers bear the tax burden 2127.17 The market clears if consumers bear the entire tax burden 212B.1 The cobweb model of price adjustment (stable case; p0= 2) 245B.2
40、The cobweb model of price adjustment (stable case; p0= 1) 246B.3 The cobweb model of price adjustment (unstable case) 247B.4 The demand and supply model (unstable case) 260Tables3.1 Values ofparenleftbigg1 +1kparenrightbiggkpage 723.2 Cash flow schedule 823.3 Net cash flows 884.1 Costs 914.2 The mar
41、ginal cost 924.3 The average cost and the average variable cost 934.4 Costs and labour input 964.5 The marginal revenue and marginal cost 984.6 Firms profit maximisation when p = 400 1004.7 Firms profit maximisation when p = 200 1014.8 Linear cost function 1215.1 Information for curve sketching: Ste
42、ps (1) and (2) 1255.2 Information for curve sketching: Steps (1), (2) and (3) 1275.3 Information for curve sketching: f (x) =12x2 8x + 40 1295.4 Information for curve sketching: f (x) =1x1315.5 Information for curve sketching: f (x) = x +1x1325.6 Information for curve sketching: f (x) = ex1335.7 Inf
43、ormation for curve sketching: f (x) = ln x 133A.1 The price information on three goods 225B.1 The evolutions of ptand qt(stable case; p0= 2) 244B.2 The evolutions of ptand qt(stable case; p0= 1) 246B.3 The evolutions of ptand qt(unstable case) 246PrefaceThis book is based on lecture notes I wrote fo
44、r a first-year compulsory quantitative meth-ods course in the Australian National University (ANU) over a period of seven years.Before I started teaching the course in 2002, an encyclopaedic textbook on introductoryquantitative methods that had limited focus on economics was used. However, teachingm
45、athematics out of such a textbook to my students seemed ineffective because many ofthem disliked studying mathematics unless they saw practical applications. Accordingly,as an economist, I looked for other textbooks in mathematical economics. Many goodtextbooks were available, but they were too adva
46、nced for an introductory course, in termsof both mathematics and economics. Although they contained many applications to eco-nomics, they usually assumed that students have learnt some introductory economics.These applications are not straightforward for first-year students with little, if any, back
47、-ground in economics. I decided to write my own lecture notes given this unsatisfactorysituation.ScopeThe material in the main text ranges from a revision of high-school mathematics to appli-cations of calculus (single-variate, multivariate and integral) to economics and finance.For example: linear
48、and quadratic functions are introduced in the context of demand andsupply analysis; geometric sequences, exponential and logarithmic functions are intro-duced in the context of finance; single-variate calculus is explained in the course ofsolving a firms profit maximisation problem; a consumers util
49、ity maximisation is usedto motivate introducing multivariate calculus; and integral calculus is explained in thecontext of calculating the deadweight loss of taxation. The material can be taught in 1315 weeks (3945 hours). To give some flexibility, matrix algebra and an introduction todifference/differential equations are covered in appendices.Features, approach and styleOne of the distinctive features of this book is that, where possible, mathematical tech-niques are introduced in the context of introductory economics. Many students tend todisli
