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MATHEMATICAL ANALYSIS FOR ECONOMISTS BY R. G. D. ALLEN The general
science of mathematics is concerned with the investigation of
patterns of connectedness, in abstrac tion from the particular
relata and the particular modes of connection. ALFRED NORTH
WHITEHEAD, Adventures of Ideas To connect elements in laws
according to some logical or mathematical pattern is the ultimate
ideal of science. MORRIS R. COHEN, Reason and Nature MAQMILLAN AND
CO., LIMITED ST. MARTINS STREET, LONDON 1938 PRINTED IN GREAT
I3KITAIN FOREWORD TEUS book, which is based on a series of lectures
given at the London School of Economics annually since 1931, aims
at providing a course of pure mathematics developed in the
directions most useful to students of economics. At each stage the
mathematical methods described are used in the elucidation of
problems of economic theory. Illustrative examples are added to all
chapters and it is hoped that the reader, in solving them, will
become familiar with the mathematical tools and with their
applications to concrete economic problems. The method of treatment
rules out any attempt at a systematic development of mathematical
economic theory but the essentials of such a theory are to be found
either in the text or in the examples. I hope that the book will be
useful to readers of different types. The earlier chapters are
intended primarily for the student with no mathematical equipment
other than that obtained, possibly many years ago, from a
matriculation course. Such a student may need to accustom himself
to the application of the elementary methods before proceeding to
the more powerful processes described in the later chapters. The
more advanced reader may use the earlysections for purposes of
revision and pass on quickly to the later work. The experienced
mathematical economist may find the book as a whole of service for
reference and discover new points in some of the chapters. I have
received helpful advice and criticism from many mathe maticians and
economists. I am particularly indebted to Professor A. L. Bowley
and to Dr. J. Marschak and the book includes numerous modifications
made as a result of their suggestions on reading the original
manuscript. I am also indebted to Mr. G. J. Nash who has read the
proofs and has detected a number of slips in my construction of the
examples. R. G. D. ALLEN THE LONDON SCHOOL OF ECONOMICS October,
1937 CONTENTS CHAP. PAGE FOREWORD ----------v A SHORT BIBLIOGRAPHY
- ..... xiv THE USE OF GREEK LETTERS IN MATHEMATICAL ANALYSIS - -
...... xvi I. NUMBERS AND VARIABLES -------1 1.1 Introduction
---------1 1.2 Numbers of various types ------3 1.3 The real number
system -------6 1.4 Continuous and discontinuous variables ... - 7
1.5 Quantities and their measurement ..... 9 1.0 Units of
measurement - - - - - - - 13 1.7 Derived quantities - - - - - - - -
14 1.8 The location of points in space - - - - - 1G 1.9 Va viable
points and their co-ordinates 20 EXAMPLES 1 The measurement of
quantities graphical methods ---------23 . JpOJ ACTIONS AND THEIR
DIAGRAMMATIC REPRESENTATION 28 2.1 Definition and examples of
functions 28 2.2 The graphs of functions - - - - - - - 32 2.3
Functions and curves - - - - - - - 3 5 2.4 Classification of
functions - - - - - - 38 2.5 Function types - - - - - - - - 41 2.6
The symbolic representation of functions of any form - 45 2.7 The
diagrammatic method - - - - - - 48 2.8 The solution ofequations in
one variable 50 2.9 Simultaneous equations in two variables 54
EXAMPLES II Functions and graphs the solutionjof equa- tions
......... 57 III. ELEMENTARY ANALYTICAL GEOMETRY 61 3.1
Introduction ......... 61 3.2 The gradient of a straight line .....
03 3.3 The equation of a straight line - - - 66 viii CONTENTS CHAP.
3.4 The parabola 09 3.5 The rectangular hyperbola - - - - - - 72
3.6 The circle 75 3.7 Curve classes and curve systems . - ... 76
3.8 An economic problem in analytical geometry 80 EXAMPLES III--The
straight line curves and curve systems 82 IV...
There is no book currently available that gives a comprehensive
treatment of the design, construction, and use of index numbers.
However, there is a pressing need for one in view of the increasing
and more sophisticated employment of index numbers in the whole
range of applied economics and specifically in discussions of
macroeconomic policy. In this book, R. G. D. Allen meets this need
in simple and consistent terms and with comprehensive coverage. The
text begins with an elementary survey of the index-number problem
before turning to more detailed treatments of the theory and
practice of index numbers. The binary case in which one time period
is compared with another is first developed and illustrated with
numerous examples. This is to prepare the ground for the central
part of the text on runs of index numbers. Particular attention is
paid both to fixed-weighted and to chain forms as used in a wide
range of published index numbers taken mainly from British official
sources. This work deals with some further problems in the
construction of index numbers, problems which are both troublesome
and largely unresolved. These include the use of sampling
techniques in index-number design and the theoretical and practical
treatment of quality changes. It is also devoted to a number of
detailed and specific applications of index-number techniques to
problems ranging from national-income accounting, through the
measurement of inequality of incomes and international comparisons
of real incomes, to the use of index numbers of stock-market
prices. Aimed primarily at students of economics, whatever their
age and range of interests, this work will also be of use to those
who handle index numbers professionally.
There is no book currently available that gives a comprehensive
treatment of the design, construction, and use of index numbers.
However, there is a pressing need for one in view of the increasing
and more sophisticated employment of index numbers in the whole
range of applied economics and specifically in discussions of
macroeconomic policy. In this book, R. G. D. Allen meets this need
in simple and consistent terms and with comprehensive coverage.
The text begins with an elementary survey of the index-number
problem before turning to more detailed treatments of the theory
and practice of index numbers. The binary case in which one time
period is compared with another is first developed and illustrated
with numerous examples. This is to prepare the ground for the
central part of the text on runs of index numbers. Particular
attention is paid both to fixed-weighted and to chain forms as used
in a wide range of published index numbers taken mainly from
British official sources.
This work deals with some further problems in the construction
of index numbers, problems which are both troublesome and largely
unresolved. These include the use of sampling techniques in
index-number design and the theoretical and practical treatment of
quality changes. It is also devoted to a number of detailed and
specific applications of index-number techniques to problems
ranging from national-income accounting, through the measurement of
inequality of incomes and international comparisons of real
incomes, to the use of index numbers of stock-market prices. Aimed
primarily at students of economics, whatever their age and range of
interests, this work will also be of use to those who handle index
numbers professionally. "R. G. D. Allen" (1906-1983) was Professor
Emeritus at the University of London. He was also once president of
the Royal Statistical Society and Treasurer of the British Academy
where he was a fellow. He is the author of "Basic Mathematics,"
"Mathematical Analysis for Economists," "Mathematical Economics"
and "Macroeconomic Theory."
MATHEMATICAL ANALYSIS FOR ECONOMISTS BY R. G. D. ALLEN The general
science of mathematics is concerned with the investigation of
patterns of connectedness, in abstrac tion from the particular
relata and the particular modes of connection. ALFRED NORTH
WHITEHEAD, Adventures of Ideas To connect elements in laws
according to some logical or mathematical pattern is the ultimate
ideal of science. MORRIS R. COHEN, Reason and Nature MAQMILLAN AND
CO., LIMITED ST. MARTINS STREET, LONDON 1938 PRINTED IN GREAT
I3KITAIN FOREWORD TEUS book, which is based on a series of lectures
given at the London School of Economics annually since 1931, aims
at providing a course of pure mathematics developed in the
directions most useful to students of economics. At each stage the
mathematical methods described are used in the elucidation of
problems of economic theory. Illustrative examples are added to all
chapters and it is hoped that the reader, in solving them, will
become familiar with the mathematical tools and with their
applications to concrete economic problems. The method of treatment
rules out any attempt at a systematic development of mathematical
economic theory but the essentials of such a theory are to be found
either in the text or in the examples. I hope that the book will be
useful to readers of different types. The earlier chapters are
intended primarily for the student with no mathematical equipment
other than that obtained, possibly many years ago, from a
matriculation course. Such a student may need to accustom himself
to the application of the elementary methods before proceeding to
the more powerful processes described in the later chapters. The
more advanced reader may use the earlysections for purposes of
revision and pass on quickly to the later work. The experienced
mathematical economist may find the book as a whole of service for
reference and discover new points in some of the chapters. I have
received helpful advice and criticism from many mathe maticians and
economists. I am particularly indebted to Professor A. L. Bowley
and to Dr. J. Marschak and the book includes numerous modifications
made as a result of their suggestions on reading the original
manuscript. I am also indebted to Mr. G. J. Nash who has read the
proofs and has detected a number of slips in my construction of the
examples. R. G. D. ALLEN THE LONDON SCHOOL OF ECONOMICS October,
1937 CONTENTS CHAP. PAGE FOREWORD ----------v A SHORT BIBLIOGRAPHY
- ..... xiv THE USE OF GREEK LETTERS IN MATHEMATICAL ANALYSIS - -
...... xvi I. NUMBERS AND VARIABLES -------1 1.1 Introduction
---------1 1.2 Numbers of various types ------3 1.3 The real number
system -------6 1.4 Continuous and discontinuous variables ... - 7
1.5 Quantities and their measurement ..... 9 1.0 Units of
measurement - - - - - - - 13 1.7 Derived quantities - - - - - - - -
14 1.8 The location of points in space - - - - - 1G 1.9 Va viable
points and their co-ordinates 20 EXAMPLES 1 The measurement of
quantities graphical methods ---------23 . JpOJ ACTIONS AND THEIR
DIAGRAMMATIC REPRESENTATION 28 2.1 Definition and examples of
functions 28 2.2 The graphs of functions - - - - - - - 32 2.3
Functions and curves - - - - - - - 3 5 2.4 Classification of
functions - - - - - - 38 2.5 Function types - - - - - - - - 41 2.6
The symbolic representation of functions of any form - 45 2.7 The
diagrammatic method - - - - - - 48 2.8 The solution ofequations in
one variable 50 2.9 Simultaneous equations in two variables 54
EXAMPLES II Functions and graphs the solutionjof equa- tions
......... 57 III. ELEMENTARY ANALYTICAL GEOMETRY 61 3.1
Introduction ......... 61 3.2 The gradient of a straight line .....
03 3.3 The equation of a straight line - - - 66 viii CONTENTS CHAP.
3.4 The parabola 09 3.5 The rectangular hyperbola - - - - - - 72
3.6 The circle 75 3.7 Curve classes and curve systems . - ... 76
3.8 An economic problem in analytical geometry 80 EXAMPLES III--The
straight line curves and curve systems 82 IV...
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