Essential Mathematics for Economics and Business Fourth Edition
Essential Mathematics for Economics and Business
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Typeset in 10/12pt Goudy and Helvetica by Aptara Inc., New Delhi, India Printed in Great Britain by Bell & Bain Ltd, Glasgow
Senan and Ferdia
Mathematical Preliminaries 1.1 Some Mathematical Preliminaries 1.2 Arithmetic Operations 1.3 Fractions 1.4 Solving Equations 1.5 Currency Conversions 1.6 Simple Inequalities 1.7 Calculating Percentages 1.8 The Calculator. Evaluation and Transposition of Formulae 1.9 Introducing Excel
1 2 3 6 11 14 18 21 24 28
CHAPTER 2 The Straight Line and Applications 2.1 The Straight Line 2.2 Mathematical Modelling 2.3 Applications: Demand, Supply, Cost, Revenue 2.4 More Mathematics on the Straight Line 2.5 Translations of Linear Functions 2.6 Elasticity of Demand, Supply and Income 2.7 Budget and Cost Constraints 2.8 Excel for Linear Functions 2.9 Summary
37 38 54 59 76 82 83 91 92 97
CHAPTER 3 Simultaneous Equations 3.1 Solving Simultaneous Linear Equations 3.2 Equilibrium and Break-even 3.3 Consumer and Producer Surplus 3.4 The National Income Model and the IS-LM Model 3.5 Excel for Simultaneous Linear Equations 3.6 Summary Appendix
101 102 111 128 133 137 142 143
[x] CONTENTS CHAPTER 4 Non-linear Functions and Applications 4.1 Quadratic, Cubic and Other Polynomial Functions 4.2 Exponential Functions 4.3 Logarithmic Functions 4.4 Hyperbolic (Rational) Functions of the Form a /(b x + c) 4.5 Excel for Non-linear Functions 4.6 Summary
147 148 170 184 197 202 205
CHAPTER 5 Financial Mathematics 5.1 Arithmetic and Geometric Sequences and Series 5.2 Simple Interest, Compound Interest and Annual Percentage Rates 5.3 Depreciation 5.4 Net Present Value and Internal Rate of Return 5.5 Annuities, Debt Repayments, Sinking Funds 5.6 The Relationship between Interest Rates and the Price of Bonds 5.7 Excel for Financial Mathematics 5.8 Summary Appendix
209 210 218 228 230 236 248 251 254 256
CHAPTER 6 Differentiation and Applications 6.1 Slope of a Curve and Differentiation 6.2 Applications of Differentiation, Marginal Functions, Average Functions 6.3 Optimisation for Functions of One Variable 6.4 Economic Applications of Maximum and Minimum Points 6.5 Curvature and Other Applications 6.6 Further Differentiation and Applications 6.7 Elasticity and the Derivative 6.8 Summary
259 260 270 286 304 320 334 347 357
CHAPTER 7 Functions of Several Variables 7.1 Partial Differentiation 7.2 Applications of Partial Differentiation 7.3 Unconstrained Optimisation 7.4 Constrained Optimisation and Lagrange Multipliers 7.5 Summary
361 362 380 400 410 422
CHAPTER 8 Integration and Applications 8.1 Integration as the Reverse of Differentiation 8.2 The Power Rule for Integration 8.3 Integration of the Natural Exponential Function 8.4 Integration by Algebraic Substitution 8.5 The Definite Integral and the Area under a Curve
427 428 429 435 436 441
[ xi ] CONTENTS 8.6 8.7 8.8 8.9 8.10
Consumer and Producer Surplus First-order Differential Equations and Applications Differential Equations for Limited and Unlimited Growth Integration by Substitution and Integration by Parts Summary
448 456 468 website only 474
CHAPTER 9 Linear Algebra and Applications 9.1 Linear Programming 9.2 Matrices 9.3 Solution of Equations: Elimination Methods 9.4 Determinants 9.5 The Inverse Matrix and Input/Output Analysis 9.6 Excel for Linear Algebra 9.7 Summary
477 478 488 498 504 518 531 534
CHAPTER 10 Difference Equations 10.1 Introduction to Difference Equations 10.2 Solution of Difference Equations (First-order) 10.3 Applications of Difference Equations (First-order) 10.4 Summary
539 540 542 554 564
Solutions to Progress Exercises
Many students who embark on the study of economics and/or business are surprised and apprehensive to find that mathematics is a core subject on their course. Yet, to progress beyond a descriptive level in most subjects, an understanding and a certain fluency in basic mathematics is essential. In this text a minimal background in mathematics is assumed: the text starts in Chapter 1 with a review of basic mathematical operations such as multiplying brackets, manipulating fractions, percentages, use of the calculator, evaluating and transposing formulae, the concept of an equation and the solution of simple equations. Throughout the text worked examples demonstrate concepts and mathematical methods with a simple numerical example followed by further worked examples applied to real-world situations. The worked examples are also useful for practice. Start by reading the worked example to make sure you understand the method; then test yourself by attempting the example with a blank sheet of paper! You can always refer back to the detailed worked example if you get stuck. You should then be in a position to attempt the progress exercises. In this new edition, the worked examples in the text are complemented by an extensive question bank in WileyPLUS and MapleTA.
An Approach to Learning The presentation of content is designed to encourage a metacognitive approach to manage your own learning. This requires a clear understanding of the goals or content of each topic; a plan of action to understand and become competent in the material covered; and to test that this has been achieved.
1. Goals The learning goals must be clear. Each chapter is introduced with an overview and chapter objectives. Topics within chapters are divided into sections and subsections to maintain an overview of the chapter’s logical development. Key concepts and formulae are highlighted throughout. A summary, to review and consolidate the main ideas, is given within chapters where appropriate, with a final overview and summary at the end.
2. Plan of Action to Understand and Become Competent in the Material Covered r Understanding the rationale underlying methods is essential. To this end, concepts and methods are reinforced by verbal explanations and then demonstrated in worked examples also interjected with explanations, comments and reminders on basic algebra. More formal mathematical terminologies are introduced, not only for conciseness and to further enhance understanding, but to enable the readers to transfer their mathematical skills to related subjects in economics and business.
[ xiv ] INTRODUCTION
r Mathematics is an analytical tool in economics and business. Each mathematical method is followed immediately by one or more applications. For example, demand, supply, cost and revenue functions immediately follow the introduction of the straight line in Chapter 2. Simultaneous linear equations in Chapters 3 and 9 are followed by equilibrium, taxes and subsides (see Worked Example 3.12 below), break-even analysis, and consumer and producer surplus and input/output analysis.
WO R K E D E X A M P L E 3 . 1 2 TAXES AND THEIR DISTRIBUTION Find animated worked examples at www.wiley.com/college/bradley The demand and supply functions for a good are given as Demand function: Pd = 100 − 0.5Qd
Supply function: Ps = 10 + 0.5Qs
(a) Calculate the equilibrium price and quantity. (b) Assume that the government imposes a fixed tax of £6 per unit sold. (i) Write down the equation of the supply function, adjusted for tax. (ii) Find the new equilibrium price and quantity algebraically and graphically. (iii) Outline the distribution of the tax, that is, calculate the tax paid by the consumer and the producer.
Further applications such as elasticity, budget and cost constraints, equilibrium in the labour market and the national income model are available on the website www.wiley.com/college/bradley. Financial mathematics is an important application of arithmetic and geometric series and also requires the use of rules for indices and logs. Applications and analysis based on calculus in Chapters 6, 7, 8 and 10 are essential for students of economics.
r Graphs help to reinforce and provide a more comprehensive understanding by visualisation. Use Excel to plot graphs since parameters of equations are easily varied and the effect of changes are seen immediately in the graph. Exercises that use Excel are available on the website. r A key feature of this text is the verbal explanations and interpretation of problems and of their solutions. This is designed to encourage the reader to develop both critical thinking and problemsolving techniques. r The importance of practice, reinforced by visualisation, is summarised succinctly in the following quote attributed to the ancient Chinese philosopher Lao Tse: You read and you forget; You see and you remember; You do and you learn.
[ xv ] INTRODUCTION
3. Test whether goals were achieved There are several options:
r Attempt the worked examples without looking at the text. r Do the progress exercises. Answers and, in some cases, solutions are given at the back of the text. r In this new edition a large question bank, with questions classified as ‘easy’, ‘average’ or ‘hard’, is provided in WileyPLUS and MapleTA. Many of the questions are designed to reinforce problemsolving techniques and so require several inputs from the reader not just a single final numeric answer. r A test exercise is given at the end of each chapter. Answers to test exercises are available to lecturers online.
Structure of the Text Mathematics is a hierarchical subject. The core topics are covered in Chapters 1, 2, 3, 4, 6 and 8. Some flexibility is possible by deciding when to introduce further material based on these core chapters, such as linear algebra, partial differentiation and difference equations, etc., as illustrated in the following chart. Chapter 1 Preliminaries
Chapter 2 Straight line
Chapter 9 Linear Programming Matrices Determinants
Chapter 10 Difference Equations
Chapter 3 Simultaneous Equations
Chapter 4 Quadratics: Indices Logs: 1/(ax + b)
Chapter 5 Financial Mathematics
Chapter 6 Differentiation
Chapter 7 Partial Differentiation
Chapter 8 Integration
Note: An introductory course may require the earlier sections from the core chapters and a limited number of applications.
[ xvi ] INTRODUCTION
WileyPLUS is a powerful online tool that provides instructors and students with an integrated suite of teaching and learning resources, including an online version of the text, in one easy-to-use website. To learn more about WileyPLUS, request an instructor test drive or view a demo, please visit www.wileyplus.com.
WileyPLUS Tools for Instructors WileyPLUS enables you to:
r Assign automatically graded homework, practice, and quizzes from the test bank. r Track your students’ progress in an instructor’s grade book. r Access all teaching and learning resources, including an online version of the text, and student and instructor supplements, in one easy-to-use website. These include an extensive test bank of algorithmic questions; full-colour PowerPoint slides; access to the Instructor’s Manual; additional exercises and quizzes with solutions; Excel problems and solutions; and, answers to all the problems in the book. r Create class presentations using Wiley-provided resources, with the ability to customise and add your own materials.
WileyPLUS Resources for Students within WileyPLUS In WileyPLUS, students will find various helpful tools, such as an e-book, additional Progress Exercises, Problems in Context, and animated worked examples.
r e-book of the complete text is available in WileyPLUS with learning links to various features and tools to assist students in their learning.
r Additional Progress Exercises are available affording students the opportunity for further practice of key concepts.
r Problems in Context provide students with further exposition on the mathematics elements of a key economics or business topic.
r Animated Worked Examples utilising full-colour graphics and audio narration; these animations of key worked examples from the text provide the students with an added way of tackling more difficult material.
[ xvii ] INTRODUCTION
Ancillary Teaching and Learning Materials All materials are housed on the companion website, which you can access at www.wiley.com/ college/bradley. Students’ companion website containing:
r Animations of key worked examples from the text. r Problems in Context covering key mathematic problems in a business or economics context. r Excel exercises, which draw upon data within the text. r Introduction to using Maple, which has been updated for the new edition. r Additional learning material and exercises, with solutions for practice. Instructor’s companion website containing:
r Instructor’s Manual containing comments and tips on areas of difficulty; solutions to test exercises; additional exercises and quizzes, with solutions; sample test papers for introductory courses and advanced courses. r PowerPoint presentation slides containing full-colour graphics to help instructors create stimulating lectures. r Test bank including algorithmic questions, which is provided in Microsoft Word format and can be accessed via Maple T.A. r Additional exercises and solutions that are not available in the text.
M AT H E M AT I C A L PRELIMINARIES
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
Some Mathematical Preliminaries Arithmetic Operations Fractions Solving Equations Currency Conversions Simple Inequalities Calculating Percentages The Calculator. Evaluation and Transposition of Formulae Introducing Excel
Chapter Objectives At the end of this chapter you should be able to:
r Perform basic arithmetic operations and simplify algebraic expressions r Perform basic arithmetic operations with fractions r Solve equations in one unknown, including equations involving fractions r Understand the meaning of no solution and infinitely many solutions r Convert currency r Solve simple inequalities r Calculate percentages. In addition, you will be introduced to the calculator and a spreadsheet.
Some Mathematical Preliminaries
Brackets in mathematics are used for grouping and clarity. Brackets may also be used to indicate multiplication. Brackets are used in functions to declare the independent variable (see later). Powers: positive whole numbers such as 23 , which means 2 × 2 × 2 = 8: (anything)3 = (anything) × (anything) × (anything) (x)3 = x × x × x (x + 4)5 = (x + 4)(x + 4)(x + 4)(x + 4)(x + 4) Note: Brackets: (A)(B) or A × B or AB all indicate A multiplied by B. Variables and letters: When we don’t know the value of a quantity, we give that quantity a symbol, such as x. We may then make general statements about the unknown quantity or variable, x. For example, ‘For the next 15 weeks, if I save x per week I shall have $4500 to spend on a holiday’. This statement may be expressed as a mathematical equation: 15 × weekly savings = 4500 15 × x = 4500 Now that the statement has been reduced to a mathematical equation (see Section 1.4 for more on equations), we may solve the equation for the unknown, x: 15x = 4500 4500 15x = 15 15 x = 300
divide both sides of the equation by 15
Algebra: The branch of mathematics that deals with the manipulation of symbols (letters) is called algebra. An algebraic term consists of letters/symbols: therefore, in the above example, x is referred to as an algebraic term (or simply a term). An algebraic expression (or simply an expression) is a formula that consists of several algebraic terms (constants may also be included), e.g., x + 5x + 8. Square roots: The square root of a number is the reverse of squaring: √ (2)2 = 4 → 4 = 2 √ (2.5)2 = 6.25 → 6.25 = 2.5
Accuracy: rounding numbers correct to x decimal places When you use a calculator you will frequently end up with a string of numbers after the decimal point. For example, 15/7 = 2.142 857 1. . . . For most purposes you do not require all these numbers.
 M AT H E M AT I C A L P R E L I M I N A R I E S However, if some of the numbers are dropped, subsequent calculations are less accurate. To minimise this loss of accuracy, there are rules for ‘rounding’ numbers correct to a specified number of decimal places, as illustrated by the following example. Consider: (a) 15/7 = 2.142 857 1; (b) 6/7 = 0.857 142 8. Assume that three numbers after the decimal point are required. To round correct to three decimal places, denoted as 3D, inspect the number in the fourth decimal place:
r If the number in the fourth decimal place is less than 5, simply retain the first three numbers after the decimal place: (b) 6/7 = 0.857 142 8: use 0.857, when rounded correct to 3D.
r If the number in the fourth decimal place is 5 or greater, then increase the number in the third decimal place by 1, before dropping the remaining numbers: (a) 15/7 = 2.142 857 1, use 2.143, when rounded correct to 3D. To get some idea of the greater loss of accuracy incurred by truncating (chopping off after a specified number of decimal places) rather than rounding to the same number of decimal places, consider (a) the exact value of 15/7, using the calculator, (b) 15/7 truncated after 3D and (c) 15/7 rounded to 3D. While these errors appear small they can become alarmingly large when propagated by further calculations. A simple example follows in which truncated and rounded values of 15/7 are each raised to the power of 20. Note: Error = exact value − approximate value.
Raise to power of 20 Error
(a) Exact value 15/7 = 2.142 857 143 . . .
(b) Truncated to 3D 15/7 = 2.142
(c) Rounded to 3D 15/7 = 2.143
Truncation error = 0.000 857 143 . . .
Rounding error = 0.000 142 857 . . .
(2.142)20 = 4 134 180 (integer part of result)
(2.143)20 = 4 172 952 (integer part of result)
4 167 392 − 4 134 180 = 33 213
4 167 392 − 4 172 952 = −5560
= (2.142 857 143 . . .)20
= 4 167 392 (integer part of result) 0
Addition and subtraction Adding: If all the signs are the same, simply add all the numbers or terms of the same type and give the answer with the common overall sign. Subtracting: When subtracting any two numbers or two similar terms, give the answer with the sign of the largest number or term.
If terms are of the same type, e.g., all x-terms, all xy-terms, all x2 -terms, then they may be added or subtracted as shown in the following examples: Add/subtract with numbers, mostly
Add/subtract with variable terms
5 + 8 + 3 = 16 similarity → 5 + 8 + 3 + y = 16 + y similarity → The y-term is different, so it cannot be added to the others
5x + 8x + 3x = 16x (i) 5x + 8x + 3x + y = 16x + y (ii) 5xy + 8xy + 3xy + y = 16xy + y The y-term is different, so it cannot be added to the others (i) 7x − 10x = −3x (ii) 7x2 − 10x2 = −3x2 7x2 – 10x2 – 10x = –3x2 – 10x The x-term is different from the x 2 -terms, so it cannot be subtracted from the x 2 -terms
7 − 10 = −3
7 − 10 − 10x = −3 − 10x similarity → The x-term is different, so it cannot be subtracted from the others
WO R K E D E X A M P L E 1 . 1 ADDITION AND SUBTRACTION For each of the following, illustrate the rules for addition and subtraction: (a) 2 + 3 + 2.5 = (2 + 3 + 2.5) = 7.5 (b) 2x + 3x + 2.5x = (2 + 3 + 2.5)x = 7.5x (c) −3xy − 2.2xy − 6xy = (−3 − 2.2 − 6)xy = −11.2xy (d) 8x + 6xy − 12x + 6 + 2xy = 8x − 12x + 6xy + 2xy + 6 = −4x + 8xy + 6 (e) 3x2 + 4x + 7 − 2x2 − 8x + 2 = 3x2 − 2x2 + 4x − 8x + 7 + 2 = x2 − 4x + 9
Multiplication and division Multiplying or dividing two quantities with like signs gives an answer with a positive sign. Multiplying or dividing two quantities with different signs gives an answer with a negative sign.
WO R K E D E X A M P L E 1 . 2 a MULTIPLICATION AND DIVISION Each of the following examples illustrates the rules for multiplication. (a) 5 × 7 = 35
(d) −5 × 7 = −35
(b) −5 × −7 = 35 (c) 5 × −7 = −35
(e) 7/5 = 1.4 (f) (−7)/(−5) = 1.4
 M AT H E M AT I C A L P R E L I M I N A R I E S
Remember It is very useful to remember that a minus sign is a −1, so −5 is the same as −1 × 5 (g) (−7)/5 = −1.4 (h) 7/(−5) = −1.4 (i) 5(7) = 35
multiply each term inside the bracket by the term outside the bracket multiply the second bracket by x, then multiply the second bracket by (+4) and add multiply each bracket by the term outside it; add or subtract similar terms, such as 2x + 4x = 6x
(o) (x + y)2 multiply the second bracket by x and then by y; add the = (x + y)(x + y) similar terms: x y + yx = 2x y = x(x + y) + y(x + y) = xx + xy + yx + yy = x2 + 2xy + y2 The following identities are important: 1. (x + y)2 = x2 + 2xy + y2 2. (x − y)2 = x2 − 2xy + y2 3. (x + y)(x − y) = (x2 − y2 )
Remember Brackets are used for grouping terms together to Enhance clarity. Indicate the order in which arithmetic operations should be carried out. The precedence of arithmetic operators is summarised as follows: (i) Simplify or evaluate the terms within brackets first. (ii) When multiplying/dividing two terms: (1) determine the overall sign first; (2) multiply/divide the numbers; (3) finally, multiply/divide the variables (letters). (iii) Finally, add and/or subtract as appropriate.