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vii
Part Five Some further topics
18 Integration 591
19 Matrix algebra 618
20 Difference and differential equations 642
Contents
W21 Extensions and future directions (on the Online Resource Centre)
Detailed Contents
8.17 Profit maximization with perfect competition 270
8.18 Comparing the equilibria under monopoly
and perfect competition 273
6 Derivatives and differentiation 187
8.19 Two common fallacies concerning profit
6.1 Introduction 187
maximization 274
6.2 The difference quotient 188
8.20 The second-order condition for profit maximization 275
6.3 Calculating the difference quotient 189
Appendix 8.1: The relationship between total
6.4 The slope of a curved line 190
cost, average cost, and marginal cost 280
6.5 Finding the slope of the tangent 192
Appendix 8.2: The relationship between price,
6.6 Generalization to any function of x 194 total revenue, and marginal revenue 281
6.7 Rules for evaluating the derivative of a function 195
6.8 Summary of rules of differentiation 205 9 Elasticity 284
9.1 Introduction 284
7 Derivatives in action 208 9.2 Absolute, proportionate, and percentage changes 285
7.1 Introduction 208 9.3 The arc elasticity of supply 287
7.2 Increasing and decreasing functions 209 9.4 Elastic and inelastic supply 288
7.3 Optimization: finding maximum and 9.5 Elasticity as a rate of proportionate change 288
minimum values 211 9.6 Diagrammatic treatment 289
7.4 A maximum value of a function 212 9.7 Shortcomings of arc elasticity 291
7.5 The derivative as a function of x 213 9.8 The point elasticity of supply 291
7.6 A minimum value of a function 214 9.9 Reconciling the arc and point supply elasticities 293
7.7 The second derivative 215 9.10 Worked examples on supply elasticity 293
7.8 A rule for maximum and minimum values 216 9.11 The arc elasticity of demand 296
7.9 Worked examples of maximum and
9.12 Elastic and inelastic demand 299
minimum values 217
9.13 An alternative definition of demand elasticity 300
7.10 Points of inflection 220
9.14 The point elasticity of demand 301
7.11 A rule for points of inflection 223
9.15 Reconciling the arc and point demand elasticities 303
7.12 More about points of inflection 224
9.16 A worked example on demand elasticity 303
7.13 Convex and concave functions 232
9.17 Two simplifications 304
7.14 An alternative notation for derivatives 235
9.18 Marginal revenue and the elasticity of demand 306
7.15 The differential and linear approximation 236
9.19 The elasticity of demand under
perfect competition 310
8 Economic applications of functions
9.20 Other elasticities in economics 313
and derivatives 239
9.21 The firm’s total cost function 313
8.1 Introduction 239
9.22 The aggregate consumption function 314
8.2 The firm’s total cost function 240
9.23 Generalizing the concept of elasticity 316
8.3 The firm’s average cost function 242
8.4 Marginal cost 244
8.5 The relationship between marginal and Part Three Mathematics of
average cost 246 finance and growth
8.6 Worked examples of cost functions 248
8.7 Demand, total revenue, and marginal revenue 255 10 Compound growth and present
8.8 The market demand function 255 discounted value 321
8.9 Total revenue with monopoly 257 10.1 Introduction 321
8.10 Marginal revenue with monopoly 258 10.2 Arithmetic and geometric series 322
8.11 Demand, total, and marginal revenue 10.3 An economic application 325
functions with monopoly 260
10.4 Simple and compound interest 328
8.12 Demand, total, and marginal revenue with
10.5 Applications of the compound growth formula 331
perfect competition 261
10.6 Discrete versus continuous growth 334
x 10.7 When interest is added more than once per year 334 14.3 Examples of functions with two
10.8 Present discounted value 339 independent variables 425
10.9 Present value and economic behaviour 342 14.4 Partial derivatives 430
10.10 Present value of a series of future receipts 342 14.5 Evaluation of first-order partial derivatives 433
10.11 Present value of an infinite series 345 14.6 Second-order partial derivatives 437
10.12 Market value of a perpetual bond 346 14.7 Economic applications 1: the production
Detailed Contents
14.2 Functions with two independent variables 422 16.12 Deriving the consumer’s demand functions 551
17 Returns to scale and homogeneous functions; 19.6 Vector multiplication 623 xi
partial elasticities; growth accounting; 19.7 Scalar multiplication 624
logarithmic scales 558 19.8 Matrix algebra as a compact notation 624
17.1 Introduction 558 19.9 The determinant of a square matrix 625
17.2 The production function and returns to scale 559 19.10 The inverse of a square matrix 628
17.3 Homogeneous functions 561 19.11 Using matrix inversion to solve linear
Detailed Contents
17.4 Properties of homogeneous functions 564 simultaneous equations 630
17.5 Partial elasticities 571 19.12 Cramer’s rule 632
17.6 Partial elasticities of demand 572 19.13 A macroeconomic application 633
17.7 The proportionate differential of a function 574 19.14 Input–output analysis 635
17.8 Growth accounting 577 19.15 Conclusions 639
17.9 Elasticity and logs 579
17.10 Partial elasticities and logarithmic scales 580 20 Difference and differential equations 642
17.11 The proportionate differential and logs 582 20.1 Introduction 642
17.12 Log linearity with several variables 584 20.2 Difference equations 643
20.3 Qualitative analysis 646
20.4 The cobweb model of supply and demand 651
20.5 Conclusions on the cobweb model 656
Part Five Some further topics
20.6 Differential equations 658
20.7 Qualitative analysis 661
18 Integration 591 20.8 Dynamic stability of a market 662
18.1 Introduction 591 20.9 Conclusions on market stability 665
18.2 The definite integral 592
18.3 The indefinite integral 594 W21 Extensions and future directions
18.4 Rules for finding the indefinite integral 595 (on the Online Resource Centre)
18.5 Finding a definite integral 602 21.1 Introduction
18.6 Economic applications 1: deriving the total 21.2 Functions and analysis
cost function from the marginal cost function 605 21.3 Comparative statics
18.7 Economic applications 2: deriving total 21.4 Second order difference equations
revenue from the marginal revenue function 607
APPENDIX 21.1: Proof of Taylor’s theorem
18.8 Economic applications 3: consumers’ surplus 609
APPENDIX 21.2: Using Taylor’s formula to
18.9 Economic applications 4: producers’ surplus 611 relate production function forms
18.10 Economic applications 5: present value of a APPENDIX 21.3: The firm’s maximum profit
continuous stream of income 613 function with two products
APPENDIX 21.4: Removing the imaginary number
19 Matrix algebra 618
19.1 Introduction 618
19.2 Definitions and notation 619 Answers to progress exercises 669
19.3 Transpose of a matrix 620 Answers to chapter 1 self-test 701
19.4 Addition/subtraction of two matrices 621 Glossary 702
19.5 Multiplication of two matrices 621 Index 711
xii
About the author
Geoff Renshaw was formerly a lecturer and is now an associate
fellow in the Economics Department at Warwick University.
He has lectured mainly in the areas of international econom-
ics, national and international economic policy, and political
economy, but also taught maths to economists for more than
thirty years. His teaching philosophy has always been to re-
member his first encounters with new ideas and techniques,
and keep in mind how difficult they seemed then, even though
they may seem obvious now. Geoff has always endeavoured to
keep things simple and down to earth and infect his students
with his own enthusiasm for economics.
Geoff was educated at Oxford and the London School of
Economics. Before becoming an academic he worked in the research department of the Trades
Union Congress. Most of his career has been spent at Warwick University, but he has also taught
at Washington University, St Louis, and at Birmingham University. He has also been a consul-
tant to the International Labour Organization (a UN agency) and spent two years in Geneva
working on international trade and economic relations between industrialized and developing
countries, in addition to a year in Budapest, where he headed a project on the Hungarian labour
market. Geoff has also consulted for the United Nations Industrial Development Organization,
and has spent time in Vienna and Warsaw working on the Polish economy.
Geoff has published several books on industrial adjustment, north–south trade and develop-
ment, and multinational corporations.
Outside of economics and politics Geoff enjoys studying the English language, practising
DIY on houses and cars, and thinking up new inventions—none successful as yet. He is married
with three children.
1 Confidence building
Recognizing that many economics students found maths difficult and unrewarding at school,
and have often forgotten much of what they once knew, part 1 of the book is devoted entirely
to revision and consolidation of basic skills in arithmetic, algebraic manipulation, solving
equations, and curve sketching. Part 1 starts at the most elementary level and terminates at
GCSE level or a little above. It should be possible for every student to find a starting point in
part 1 that matches his or her individual needs, while more advanced students can of course
proceed directly to part 2. More guidance on finding the appropriate starting point is given in
the chapter map on pages xviii-xix.
3 Comprehensive explanation
Many textbooks skim briefly over a wide range of mathematical techniques and their economic
applications, leaving students able to solve problems in a mechanical way but feeling frustrated
by their lack of real understanding.
In this book I explain concepts and techniques in a relatively leisurely and detailed way,
using an informal style, trying to anticipate the misconceptions and misunderstandings that
the reader can so easily fall victim to, and avoiding jumps in the chain of reasoning, however
small. Wherever possible, every step is illustrated by means of a graph or diagram, based on the
adage that ‘one picture is worth a thousand words’. Many of the explanations are by means of
worked examples, which most students find easier to understand than formal theoretical expla-
nations. There is extensive cross-referencing both within and between chapters, making it easy
for the reader to quickly refresh their understanding of earlier concepts and rules when they
are re-introduced later. There is also a glossary which defines all of the key terms in maths and
economics used in the book.
To the student
(A) You have forgotten (B) You have passed GCSE (C) You have passed AS/A2
almost all of the maths you maths or an equivalent exam maths or equivalent exams
ever knew and want to make taken at age 16+, but you taken at age 17+ and 18+ and
a completely fresh start. have done no maths since are fairly confident in your
and now feel the need for maths knowledge at
some revision. this level.
Chapter 1. This starts from the lowest Take the self-test at the end of chapter
possible level and aims to rebuild basic 1; answers are at the end of the book.
knowledge and self-confidence. Be sure If you struggle with this, read chapter 1
to complete the progress exercises and and complete the progress exercises
the self-test at the end of the chapter. before going on.
Chapters 2–5. These revise the algebra component of GCSE maths or equivalent maths
exam taken at age 16+. Chapters 3–5 contain in addition some economic applications.
In chapter 5, sections 5.5–5.9 go a little beyond GCSE maths and you can skip them
if you wish, but be sure to study sections 5.10–5.12 on inequalities as these are important
in economics.
Chapters 8 and 9. These apply to economics the techniques of differentiation learned in chapters 6
and 7. Chapter 8 is concerned with a firm’s costs, the demand for its product and its profit-seeking
behaviour. Chapter 9 is devoted to the concept of elasticity.
If you are joining the book at this point because you have passed AS/A2 maths or equivalent exams,
you will find that you are already familiar with all the pure maths used in these chapters. However,
you may feel the need to browse chapters 6 and 7 for revision purposes. You should also study the
economic applications in chapters 3–5 (see detailed contents pages). This will also help you to tune in
to the book’s notation and style.
xvii
Part three Mathematics of finance and growth
Chapter 10. This important chapter introduces the Chapters 11–13. These chapters explain
key concept of present discounted value, and also the maths of logarithmic and exponential
Chapter map
how to calculate growth rates, effective interest rates functions, which are used widely in
and repayments of a loan. The maths is fairly simple economics. These concepts are covered in
and mostly covered in the GCSE syllabus, though AS/A2 maths, though less fully. If you find
its economic application will of course be new. you know the maths already, skip to the
This chapter is not closely linked to any other economic applications in sections 11.8,
chapters and can be read at any time. 12.9–12.11, and 13.8–13.10.
Chapters 14–17. These four chapters are, in a sense, the core of the book. The maths in these
chapters will be new to all students, but is a natural extension of part 3 and you should find it no more
difficult than earlier chapters.
Chapters 14 and 15 introduce functions with two or more independent variables, their derivatives,
and maximum/minimum values. This material, although new to all students, is a natural extension of
chapters 6 and 7 (and earlier chapters).
Chapter 16 explains the Lagrange multiplier, an optimization technique with many important
uses in economics. Chapter 17 introduces some new but quite simple mathematical concepts and
techniques: homogeneous functions, Euler’s theorem, and the proportionate differential.
The economic applications—to cost minimization, profit maximization, and consumer choice
among others—take up about one-half of chapters 14–16, and most of chapter 17.
There are four chapters in this part, each of which can be studied independently of one another and
of the rest of the book. All chapters contain economic applications.
Chapter 18 introduces the mathematical technique of integration, with some applications to
economics. The maths will be familiar if you have taken AS/A2 maths, but will also be well within the
capacity of any student who has progressed this far in the book.
Chapter 19 is concerned with matrix algebra, which some students of AS/A2 maths will have met
before, but which again will be fairly readily understood by any sufficiently motivated student.
Chapter 20 introduces difference and differential equations, which will be new to all students but
which are in part merely an extension of work in chapter 13.
Finally, chapter 21 develops three relatively advanced topics as a taster for students who want
to carry their study of mathematical economics further. Owing to space constraints this chapter is
located on the book’s Online Resource Centre www.oxfordtextbooks.co.uk/orc/renshaw4e/.
xviii
Guided tour of the textbook features
Maths can seem like a daunting topic if you have not studied it for a while, and you may be
somewhat surprised to find how much maths there seems to be in university economics
courses. However, once you have overcome your initial fears you will find that the maths tech-
niques used in mainstream economics are quite straightforward and that using them can even
be enjoyable! This guided tour shows you how best to utilize this textbook and get the most out
of your study, whatever your level of maths.
Objectives
Each chapter begins with a bulleted list of learning objec-
OBJECTIVES
tives outlining the main concepts and ideas you will en-
Having completed this chapter you should be able to:
counter in the chapter. These serve as helpful signposts for
● Add, subtract, multiply, and divide with positive and negativ
● Use brackets to find a common factor and a common denominato
learning and revision.
● Add, subtract, multiply, and divide fractions and decimal numbers
● Convert decimal numbers into fractions and vice versa.
● Convert fractions into proportions and percentages and vice v
Progress exercises
At the end of each main section of each chapter you will
PROGRESS EXERCISE 1.1
have the opportunity to complete a progress exercise, de-
Calculate the following without using your calculator. Then signed to test your understanding of key concepts before
answers.
(a) 14 − ( −16 ) + 4 – 3 (b) 5 – 8 + ( ) )
moving on. You are strongly recommended to complete
(d) 15 + ( ) ) (e) −2 + ( −2 ) − 2 + 4
these
(
exercises to help reinforce your understanding and
identify any areas requiring further revision. Solutions
to the progress exercises are at the end of the book, with
further materials on the Online Resource Centre at www.
oxfordtextbooks.co.uk/orc/renshaw4e/.
Examples
You understand the theory, but how is it used in practice?
EXAMPLE 1.1
Examples play a key role in the book, from short illustra-
Consider:
3 × ( 4 + 5)
tive examples that demonstrate a formula in use to more
Applying the B-E-D-M-A-S rule, we evaluate this as 3 × 9 = involved worked examples that show step by step how an
answer if we evaluate 3 × ( 4 + 5 ) as individual problem is solved.
3 × ( 4 + 5 ) = ( 3 × 4 ) + ( 3 × 5 ) = 12 + 15 = 27
Thus in order to remove the brackets we must take each of the
4 and the 5) and multiply it by the multiplicative term in fron
Hints
Hint boxes have been included throughout the text to alert
HInT A very common mistake is made in this type of calculation. If we are told that the price
you to common mistakes and misunderstandings, so that
including VAT is €176.25, then it is very tempting to think that if we deduct 17.5% from this we
will get back to the price before VAT. So following example 1.30 we might calculate the VAT as
17.5% of 176.25, giving you can proceed with your studies with confidence.
17.5
× 176.25 = 30.84
100
and thus conclude that the price before VAT is 176.25 − 30.84 = 145.41
But this is wrong, because we know from example 1.32 that the amount of VAT is 26.25, and the
price before VAT is 150. What have we done wrong? Our mistake was to calculate the VAT as
Summaries
sUMMARy Of sectiOns 3.1–3.7
The central points and concepts covered in each chapter
are distilled into summaries at the end of chapters. These
In sections 3.1–3.4 we showed how any equation could be manipulated by performing the
provide
elementary operations (adding, multiplying, and so on) to both sides a mechanism
of the equation. We stated for you to reinforce your under-
standing and can
the key distinction between variables and parameters in any equation. Any linear equation with
one unknown can be solved (rule 3.2).
be used as a revision tool.
In sections 3.5 and 3.6 we introduced the idea of a function, involving two or more variables.
The graph of any linear function y = ax + b is a straight line, which is why it is referred to as a
linear function. The slope is given by the constant, a. The intercept of the graph on the
given by the constant b
End-of-chapter
a=0
checklists
Ax + By + C = 0
A=0
The topics in each chapter are presented in checklist form
B=0
CheCklist
y= −A x − C
B B
x= −B y − C
A A at the end of every chapter to allow you to reflect on your
Be sure to test your understanding of this chapter learning
Fractions, proportions, and ‘tick’ each topic as you master it, before mov-
and percentages
by attempting the progress exercises (answers are at verting fractions into proportions and percentages
the end of the book). The Online Resource Centre ing on if
and vice versa. Increasing/decreasingyou wish
a number by to the further exercises on the Online
contains further exercises and materials relevant
to this chapter: www.oxfordtextbooks.co.uk/orc/
Resource Centre.
a given percentage; calculating percentage changes.
Index numbers
renshaw4e/
Expressing time series data in index number form.
The overall objective of this chapter was to refresh Powers and roots
xx
Guided tour of the Online Resource Centre
www.oxfordtextbooks.co.uk/orc/renshaw4e/
The Online Resource Centre that accompanies this book contains a further chapter available
online. The chapter, written by Norman Ireland of Warwick University, provides an introduc-
tion to some more advanced topics which should help undergraduate students intending to
take further modules in mathematical economics in their second or later years of study, as well
as postgraduate students. The Online Resource Centre also provides students and adopting
lecturers with ready-to-use teaching and learning resources. These are free of charge and are
designed to maximize the learning experience. Below is a brief outline of what you will find.
For students
Solutions to progress exercises
Once you’ve attempted the progress exercises in the text
you can check the solutions at the end of the book. Some
of the exercises also have expanded solutions at the Online
Resource Centre to enhance your understanding.
Further exercises
The best way to master a topic area is through practice,
practice, and more practice! A bank of questions, with
answers, additional to the progress exercises in the book
itself, has been provided for each chapter in the book to
allow you to further test your understanding of the topics.
‘Ask the author’ forum xxi
If you are struggling with a particular problem, or just can-
not seem to get your head around a specific technique or
idea, then you can submit your question to the author via
the interactive online forum created for this text. As well
as replying directly to you by email, Geoff will post his re-
PowerPoint® slides
Accompanying each chapter is a suite of customizable and
illustrated PowerPoint® slides for you to use in your lec-
tures. Arranged by chapter theme, the slides may also be
used as handouts in class and can be easily adapted to suit
your teaching style.
SIMPLE: With a highly intuitive design, it will take you less than 15 minutes to learn and master
the system.
MOBILE: You can access Dashboard from every major platform and device connected to the
internet, whether that’s a computer, tablet or smartphone.
INFORMATIVE: Your assignment and assessment results are automatically graded, giving
your instructor a clear view of the class’s understanding of the course content.
Student resources: xxiii
Dashboard offers all the features of the Online Resource Centre, but comes with additional
questions to take your learning further.
Lecturer resources:
Gradebook
Dashboard will automatically grade the homework assignments that you set for your students.
The Gradebook also provides heat maps for you to view your students’ progress which helps
you to quickly identify areas of the course where your students may need more practice, as well
as the areas they are most confident in. This feature helps you focus your teaching time on the
areas that matter.
The Gradebook also allows you to administer grading schemes, manage checklists and ad-
minister learning objectives and competencies.
xxiv
Acknowledgements
In preparing the fourth edition of this book I am again greatly indebted to the OUP editing
and production team for their limitless encouragement and advice and their unfailing enthusi-
asm for the project. Specifically I warmly thank Amber Stone-Galilee, my commissioning edi-
tor; my production editor, Sian Jenkins; Elisabeth Heissler, text designer; and Scott Greenway
who designed the book’s cover. I would also like to thank Kathryn Rylance for her enthusiasm,
energy, and patience in the development of this new edition. For their immensely hard work
and relentless attention to detail I am very grateful to the copy-editor, Bobbie Nichols, and the
proofreader, Jayne MacArthur. I also thank June Morrisson for compiling a very comprehensive
and well-structured index, and Sweta Gupta for checking the accuracy of the maths. For her past
and, I hope, future management of the book’s Online Resource Centre, I am grateful to Fiona
Goodall. I should also like to repeat my thanks to the many others who have been involved in
various ways and at various times in the production of this book. In particular I thank Tim Page
and Jane Clayton, two former OUP staff without whom this book would almost certainly never
have seen the light of day.
Amongst my colleagues at Warwick University and elsewhere, I am especially grateful to
Norman Ireland who, having regrettably declined to become a co-author, agreed to write a
lengthy and extremely valuable chapter, as well as setting numerous exercises and offering much
general encouragement and support. I also owe a huge debt to Peter Law, whose meticulous
checking and painstaking comments on many of the chapters saved me from a large number of
small errors and a small number of large errors. Jeff and Ann Round also gave me valuable and
very patient advice. Peter Hammond, despite having co-authored a book with which this one
attempts to compete, was also very patient and helpful on a number of points. Alex Dobson has
helped me greatly to keep up to date with the needs of today’s lecturers and students. I am also
grateful to those users of previous editions who have taken the trouble to email me, sometimes
in praise and sometimes to point out errors. Both types of communication are very welcome.
As ever I am profoundly grateful to my wife Irene, who has been unfailingly patient and sup-
portive throughout the four editions of this book. The late Mary Pearson greatly encouraged
my labours, as did the late Lavinia McPherson—her 100 years notwithstanding. As always, the
remaining shortcomings of this book are entirely my responsibility.
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One more trait signalizes the Early Neolithic: the hewn stone ax.
This was a chipped implement, straight or slightly convex along the
cutting edge, tapering from that to the butt, about twice as long as
broad, rather thick, unperforated and ungrooved; in fact perhaps
often unhandled and driven by blows upon the butt: a sharp stone
wedge as much as an ax, in short. The whole Palæolithic shows no
such implement: even the Azilian has only bone or horn “axes.”
It is hardly necessary to repeat for the Neolithic what has already
been said of the Palæolithic periods: the older types, such as
chipped flint tools, continued very generally to be made. Such
persistence is natural: a survival of a low type among higher ones
does not mean much. It is the appearance of new and superior
inventions that counts.
The Early Neolithic can be summed up, then, in these five traits:
pottery; the bow and arrow; abundant use of bone and horn; the dog;
and the hewn ax.
Fig. 41. Prehistoric domed tombs built on the principle of corbelling (§ 116): a
probable example of the spread of a culture device over a continent. Above,
Mycenæ, Greece; middle, Alcalar, Portugal; below, New Grange, Ireland. The
Mycenæan structure, 1500 B.C. or after, at the verge of the Iron Age, is
probably later by some 1,000 years than the others, which are late Neolithic
with copper first appearing; and its workmanship is far superior. (After Sophus
Müller and Déchelette.)
227. Iron
Iron was worked by man about two thousand years later than
bronze. It is a far more abundant metal than copper, and though it
melts at a higher temperature, is not naturally harder to extract from
some of its ores. The reason for its lateness of use is not wholly
explained. It is likely that the first use of metals was of those, like
gold and copper, that are found in the pure metallic state and, being
rather soft, could be treated by hammering without heat—by
processes more or less familiar to stone age culture. It is known that
fair amounts of copper were worked in this way by many tribes of
North American Indians, who got their supplies from the Lake
Superior deposits and the Copper River placers in Alaska. If the
same thing happened in the most progressive parts of the Eastern
Hemisphere some 6,000 years ago, acquaintance with the metal
may before long have been succeeded by the invention of the arts of
casting and extracting it from its ores. When, not many centuries
later, the hardening powers of an admixture of tin were discovered
and bronze with its far greater serviceability for tools became known,
a powerful impetus was surely given to the new metallurgy, which
was restricted only by the limitations of the supply of metal,
especially tin. Progress went on in the direction first taken; the alloy
became better balanced, molds and casting processes superior, the
forms attempted more adventurous or efficient. For many centuries
iron ores were disregarded; the bronze habit intensified. Finally,
accident may have brought the discovery of iron; or shortage of
bronze led to experimenting with other ores; and a new age dawned.
Whatever the forces at work, the actual events were clearly those
outlined. And it is interesting that the New World furnishes an exact
parallel with its three areas and stages of native copper, smelted
copper and gold, and bronze (§ 108, 196), and with only the final
period of iron unattained at the time of discovery.