Essential mathematics for economics and business

Essential Mathematics for
Economics and Business
Fourth Edition



Essential Mathematics for
Economics and Business

Fourth Edition

Teresa Bradley


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Senan and Ferdia



CONTENTS

Introduction

xiii

CHAPTER 1

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

567

Worked Examples

653

Index

659



INTRODUCTION

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

(3.17)

Supply function: Ps = 10 + 0.5Qs

(3.18)

(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.

1


[2]
CHAPTER

1.1

1

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.


[3]
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.

Error

Raise to
power of
20
Error

1.2

(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

0

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

15
7

20

= (2.142 857 143 . . .)20

= 4 167 392
(integer part of result)
0

Arithmetic Operations

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.


[4]
CHAPTER

1

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

similarity →

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


[5]
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

(j) (−5)(−7) = 35
(k) (−5)y = −5y
(l) (−x)(−y) = xy

Remember
0 × (any real number) = 0
0 ÷ (any real number) = 0
But you cannot divide by 0
(m) 2(x + 2) = 2x + 4
(n) (x + 4)(x + 2)
= x(x + 2) + 4(x + 2)
= x2 + 2x + 4x + 8
= x2 + 6x + 8

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.