Project Gutenberg's The Alphabet of Economic Science, by Philip H. Wicksteed

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: The Alphabet of Economic Science
       Elements of the Theory of Value or Worth

Author: Philip H. Wicksteed

Release Date: May 30, 2010 [EBook #32497]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK THE ALPHABET OF ECONOMIC SCIENCE ***




Produced by Andrew D. Hwang, Frank van Drogen, and the
Online Distributed Proofreading Team at http://www.pgdp.net
(This file was produced from scans of public domain works
at McMaster University's Archive for the History of Economic
Thought.)



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% Project Gutenberg's The Alphabet of Economic Science, by Philip H. Wicksteed
%                                                                         %
% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
% re-use it under the terms of the Project Gutenberg License included     %
% with this eBook or online at www.gutenberg.org                          %
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%                                                                         %
% Title: The Alphabet of Economic Science                                 %
%        Elements of the Theory of Value or Worth                         %
%                                                                         %
% Author: Philip H. Wicksteed                                             %
%                                                                         %
% Release Date: May 30, 2010 [EBook #32497]                               %
% Most recently updated: June 11, 2021                         %
%                                                                         %
% Language: English                                                       %
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Project Gutenberg's The Alphabet of Economic Science, by Philip H. Wicksteed

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: The Alphabet of Economic Science
       Elements of the Theory of Value or Worth

Author: Philip H. Wicksteed

Release Date: May 30, 2010 [EBook #32497]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK THE ALPHABET OF ECONOMIC SCIENCE ***
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Produced by Andrew D. Hwang, Frank van Drogen, and the
Online Distributed Proofreading Team at http://www.pgdp.net
(This file was produced from scans of public domain works
at McMaster University's Archive for the History of Economic
Thought.)
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

\frontmatter

\pagenumbering{roman}
\pagestyle{empty}

%% -----File: 001.png---Folio iv-------
% [Blank Page]
%% -----File: 002.png---Folio v-------

\begin{center}
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\Large THE ALPHABET \\[3\TmpLen]
\footnotesize OF \\[3\TmpLen]
\Huge ECONOMIC SCIENCE \\[4\TmpLen]
\footnotesize BY \\[\TmpLen]
\normalsize PHILIP H. WICKSTEED \\[5\TmpLen]
\footnotesize ELEMENTS OF THE THEORY OF VALUE OR WORTH
\end{center}
\vfil
\newpage
%% -----File: 003.png---Folio vi-------
\iffalse
%[** TN: No longer present in page scan]
London: Macmillan \& Company Ltd., 1888
\fi
%% -----File: 004.png---Folio vii-------
\null
\vfil
\selectlanguage{latin}%
``Est ergo sciendum, quod quædam sunt, quæ nostræ potestati
minime subjacentia, speculari tantummodo possumus, operari
autem non, velut Mathematica, Physica, et~Divina. Quædam vero
sunt quæ nostræ potestati subjacentia, non solum speculari, sed et
operari possumus; et~in iis non operatio propter speculationem, sed
hæc propter illam adsumitur, quoniam in talibus operatio est finis.
Cum ergo materia præsens politica sit, imo fons atque principium
rectarum politiarum; et~omne politicum nostræ potestati subjaceat;
manifestum est, quod materia præsens non ad speculationem
per prius, sed ad operationem ordinatur. Rursus, cum in
operabilibus principium et causa omnium sit ultimus Finis (movet
enim primo agentem), consequens est, ut omnis ratio eorum quæ
sunt ad Finem, ab ipso Fine sumatur: nam alia erit ratio incidendi
lignum propter domum construendam, et alia propter navim. Illud
igitur, si quid est, quod sit Finis ultimus Civilitatis humani Generis,
erit hoc principium, per quod omnia quæ inferius probanda sunt,
erunt manifesta sufficienter.''---\textsc{Dante.}
\vfil\vfil
\newpage
%% -----File: 005.png---Folio viii-------

\null
\vfil
\selectlanguage{english}
Be it known, then, that there are certain things, in no degree
subject to our power, which we can make the objects of speculation,
but not of action. Such are mathematics, physics and theology.
But there are some which are subject to our power, and to which
we can direct not only our speculations but our actions. And in
the case of these, action does not exist for the sake of speculation,
but we speculate with a view to action; for in such matters action
is the goal. Since the material of the present treatise, then, is
political, nay, is the very fount and starting-point of right polities,
and since all that is political is subject to our power, it is obvious
that this treatise ultimately concerns conduct rather than speculation.
Again, since in all things that can be done the final goal is
the general determining principle and cause (for this it is that first
stimulates the agent), it follows that the whole rationale of the
actions directed to the goal depends upon that goal itself. For the
method of cutting wood to build a house is one, to build a ship
another. Therefore that thing (and surely there is such a thing)
which is the final goal of human society will be the principle by
reference to which all that shall be set forth below must be made
clear.
\vfil\vfil
\newpage
%% -----File: 006.png---Folio ix-------


\Chapter{Preface}
\pagestyle{fancy}

\First{Dear Reader}---I venture to discard the more stately
forms of preface which alone are considered suitable for
a serious work, and to address a few words of direct
appeal to you.

An enthusiastic but candid friend, to whom I showed
these pages in proof, dwelt in glowing terms on the
pleasure and profit that my reader would derive from
them, ``if only he survived the first cold plunge into
`functions.'\,'' Another equally candid friend to whom
I reported the remark exclaimed, ``\emph{Survive} it indeed!
Why, what on earth is to induce him to \emph{take} it?''

Much counsel was offered me as to the best method
of inducing him to take this ``cold plunge,'' the substance
of which counsel may be found at the beginning
of the poems of Lucretius and Tasso, who have given
such exquisite expression to the theory of ``sugaring
the pill'' which their works illustrate. But I am no
Lucretius, and have no power, even had I the desire
to disguise the fact that a firm grasp of the elementary
truths of Political Economy cannot be got without the
same kind of severe and sustained mental application
which is necessary in all other serious studies.

At the same time I am aware that forty pages of
almost unbroken mathematics may seem to many readers
a most unnecessary introduction to Economics, and it
is impossible that the beginner should see their bearing
upon the subject until he has mastered and applied
%% -----File: 007.png---Folio x-------
them. Some impatience, therefore, may naturally be
expected. To remove this impatience, I can but express
my own profound conviction that the beginner who has
mastered this mathematical introduction will have solved,
before he knows that he has even met them, some of the
most crucial problems of Political Economy on which
the foremost Economists have disputed unavailingly
for generations for lack of applying the mathematical
method. A glance at the ``\hyperref[indexpage]{Index of Illustrations}'' will
show that my object is to bring Economics down from
the clouds and make the study throw light on our
daily doings and experiences, as well as on the great
commercial and industrial machinery of the world.
But in order to get this light some mathematical knowledge
is needed, which it would be difficult to pick out
of the standard treatises as it is wanted. This knowledge
I have tried to collect and render accessible to
those who dropped their mathematics when they left
school, but are still willing to take the trouble to master
a plain statement, even if it involves the use of mathematical
symbols.

The portions of the book printed in the smaller type
should be omitted on a first reading. They generally
deal either with difficult portions of the subject that
are best postponed till the reader has some idea of the
general drift of what he is doing, or else with objections
that will probably not present themselves at first, and
are better not dealt with till they rise naturally.

The student is strongly recommended to consult the
Summary of Definitions and Propositions on \Pagerange{139}{140}
at frequent intervals while reading the text.

\begin{flushright}
P. H. W.\hspace*{2em}
\end{flushright}
%% -----File: 008.png---Folio xi-------

\Chapter{Introduction}

\First{On} 1st~June 1860 Stanley Jevons wrote to his brother
Herbert, ``During the last session I have worked a good
deal at political economy; in the last few months I
have fortunately struck out what I have no doubt is \emph{the
true Theory of Economy}, so thoroughgoing and consistent,
that I cannot now read other books on the subject
without indignation.''

Jevons was a student at University College at this
time, and his new theory failed even to gain him the
modest distinction of a class-prize at the summer examination.
He was placed third or fourth in the list, and,
though much disappointed, comforted himself with the
prospect of his certain success when in a few months he
should bring out his work and ``re-establish the science
on a sensible basis.'' Meanwhile he perceived more
and more clearly how fruitful his discovery must prove,
and ``how the want of knowledge of this determining
principle throws the more complicated discussions of
economists into confusion.''

It was not till 1862 that Jevons threw the main outlines
of his theory into the form of a paper, to be read
before the British Association. He was fully and most
justly conscious of its importance. ``Although I know
pretty well the paper is perhaps worth all the others
that will be read there put together, I cannot pretend to
say how it will be received.'' When the year had but
five minutes more to live he wrote of it, ``It has seen
my theory of economy offered to a learned society~(?)
%% -----File: 009.png---Folio xii-------
and received without a word of interest or belief.
It has convinced me that success in my line of endeavour
is even a slower achievement than I had thought.''

In 1871, having already secured the respectful attention
of students and practical men by several important
essays, Jevons at last brought out his \textit{Theory of Political
Economy} as a substantive work. It was received in
England much as his examination papers at college and
his communication to the British Association had been
received; but in Italy and in Holland it excited some
interest and made converts. Presently it appeared that
Professor Walras of Lausanne had been working on the
very same lines, and had arrived independently at conclusions
similar to those of Jevons. Attention being
now well roused, a variety of neglected essays of a like
tendency were re-discovered, and served to show that
many independent minds had from time to time reached
the principle for which Jevons and Walras were contending;
and we may now add, what Jevons never
knew, that in the very year 1871 the Viennese Professor
Menger was bringing out a work which, in complete
independence of Jevons and his predecessors, and by a
wholly different approach, established the identical
theory at which the English and Swiss scholars were
likewise labouring.

In 1879 appeared the second edition of Jevons's
\textit{Theory of Political Economy}, and now it could no longer
be ignored or ridiculed. Whether or not his guiding
principle is to win its way to general acceptance and to
``re-establish the science on a sensible basis,'' it has at
least to be seriously considered and seriously dealt with.

It is this guiding principle that I have sought to
illustrate and enforce in this elementary treatise on the
Theory of Value or Worth. Should it be found to meet
a want amongst students of economics, I shall hope to
follow it by similar introductions to other branches of
the science.

I lay no claim to originality of any kind. Those
%% -----File: 010.png---Folio xiii-------
who are acquainted with the works of Jevons, Walras,
Marshall, and Launhardt, will see that I have not only
accepted their views, but often made use of their
terminology and adopted their illustrations without
specific acknowledgment. But I think they will also
see that I have copied nothing mechanically, and have
made every proposition my own before enunciating it.

I have to express my sincere thanks to Mr.\ John
Bridge, of Hampstead, for valuable advice and assistance
in the mathematical portions of my work.

I need hardly add that while unable to claim credit
for any truth or novelty there may be in the opinions
advocated in these pages, I must accept the undivided
responsibility for them.
\medskip

\asterism Beginners will probably find it conducive to the
comprehension of the argument to omit the small print
in the first reading.

\begin{Remark}
\NB---I have frequently given the formulas of the curves
used in illustration. Not because I attach any value or importance
to the special forms of the curves, but because I
have found by experience that it would often be convenient
to the student to be able to calculate for himself any point
on the actual curve given in the figures which he may wish
to determine for the purpose of checking and varying the
hypotheses of the text.

As a rule I have written with a view to readers guiltless
of mathematical knowledge (see \Chapref{1}{Preface}). But I have sometimes
given information in footnotes, without explanation,
which is intended only for those who have an elementary
knowledge of the higher mathematics.

In conclusion I must apologise to any mathematicians into
whose hands this primer may fall for the evidences which they
will find on every page of my own want of systematic mathematical
training, but I trust they will detect no errors of
reasoning or positive blunders.
\end{Remark}
%% -----File: 011.png---Folio xiv-------
% [Blank Page]
%% -----File: 012.png---Folio xv-------


\Chapter{Table of Contents}

\ToCLine{\hfill\scriptsize PAGE}{}

\ToCLine{Preface}{chap:1}% ix %[** TN: N.B. 3rd arg hard-coded]

\ToCLine{Introduction}{chap:2}% xi

\ToCLine{Theory of Value---}{}

% [** TN: Skip chap:3 = this ToC]
\ToCLine[I.]{Individual}{chap:4}% 1

\ToCLine[II.]{Social}{chap:5}% 68

\ToCLine{Summary---Definitions and Propositions}{chap:6}% 139

\ToCLine{Index of Illustrations}{indexpage}% 141

\vfill
%% -----File: 013.png---Folio xvi-------
% [Blank Page]
%% -----File: 014.png---Folio 1-------

\mainmatter
\phantomsection
\pdfbookmark[-1]{Main Matter}{Main Matter}%
\pagestyle{fancy}

\Chapter[I. Individual]{I}

\Pagelabel{1}%
\First{It} is the object of this volume in the first place to
explain the meaning and demonstrate the truth of the
proposition, that \emph{the value in use and the value in exchange
of any commodity are two distinct, but connected, functions of
the quantity of the commodity possessed by the persons or the
community to whom it is valuable}, and in the second place,
so to familiarise the reader with some of the methods
and results that necessarily flow from that proposition
as to make it impossible for him unconsciously to accept
arguments and statements which are inconsistent with
it. In other words, I aim at giving what theologians
might call a ``saving'' knowledge of the fundamental
proposition of the Theory of Value; for this, but no more
than this, is necessary as the first step towards mastering
the ``alphabet of Economic Science.''

When I speak of a ``function,'' I use the word in the
mathematical not the physiological sense; and our first
business is to form a clear conception of what such a
function is.

\emph{One quantity, or measurable thing~{\upshape($y$)}, is a function of
another measurable thing~{\upshape($x$)}, if any change in~$x$ will produce
or ``determine'' a definite corresponding change in~$y$.}
Thus the sum I pay for a piece of cloth of given quality
\index{Cloth, price of}%
is a function of its length, because any alteration in the
length purchased will cause a definite corresponding
alteration in the sum I have to pay.
%% -----File: 015.png---Folio 2-------

\begin{Remark}
\Pagelabel{2}%
If I do not stipulate that the cloth shall be of the same
quality in every case, the sum to be paid will still be a function
of the length, though not of the length alone, but of the
quality also. For it remains true that an alteration in the
length will always produce a definite corresponding alteration
in the sum to be paid, although a contemporaneous alteration
in the quality may produce another definite alteration (in the
same or the opposite sense) at the same time. In this case
the sum to be paid would be ``a function of two variables''
(see below). It might still be said, however, without qualification
or supplement, that ``the sum to be paid is a function
of the length;'' for the statement, though not complete, would
be perfectly correct. It asserts that every change of length
causes a corresponding change in the sum to be paid, and it
asserts nothing more. It is therefore true without qualification.
In this book we shall generally confine ourselves to
the consideration of one variable at a time.
\end{Remark}

So again, if a heavy body be allowed to drop from a
\index{Body, falling}%
\index{Falling@{\textsc{Falling body}}}%
height, the longer it has been allowed to fall the
greater the space it has traversed, and any change in
the time allowed will produce a definite corresponding
change in the space traversed. Therefore the space
traversed (say $y$~ft.)\ is a function of the time allowed
(say $x$~seconds).

Or if a hot iron is plunged into a stream of cold
\index{Cooling iron}%
\index{Iron, cooling}%
water, the longer it is left in the greater will be the fall
in its temperature. The fall in temperature then (say
$y$~degrees) is a function of the time of immersion (say $x$~seconds).

The correlative term to ``function'' is ``variable,''
or, in full, ``independent variable.'' If $y$~is a function
of~$x$, then $x$ is the variable of that function.
Thus in the case of the falling body, the time is the
variable and the space traversed the function. When
we wish to state that a magnitude is a function of~$x$,
without specifying what particular function (\ie~when
we wish to say that the value of~$y$ depends upon the
value of~$x$, and changes with it, without defining the
%% -----File: 016.png---Folio 3-------
nature or law of its dependence), it is usual to represent
the magnitude in question by the symbol~$f(x)$ or~$\phi(x)$,
etc. Thus, ``let $y=f(x)$'' would mean ``let $y$~be a
magnitude which changes when $x$~changes.'' In the
case of the falling body we know that the space traversed,
measured in feet, is (approximately) sixteen times
the square of the number of seconds during which the
body has fallen. Therefore if $x$~be the number of
seconds, then $y$~or~$f(x)$ equals~$16x^2$.

\begin{Remark}
\Pagelabel{3}%
Since the statement $y=f(x)$ implies a \emph{definite relation}
between the changes in~$y$ and the changes in~$x$, it follows
that a change in~$y$ will determine a corresponding change in~$x$,
as well as \textit{vice versâ}. Hence if $y$ is a function of~$x$ it follows
that $x$ is also a function of~$y$. In the case of the falling body,
if $y=16x^2$, then $x=\dfrac{\sqrt{y}}{4}$.\footnote
  {In the abstract $x=±\dfrac{\sqrt{y}}{4}$. For $-x$ and $x$ will give the same
  values of $y$ in $f(x)=16x^2=y$; and we shall have $±x=\dfrac{\sqrt{y}}{4}$.}
It is usual to denote inverse functions
of this description by the index~$-1$. Thus if $f(x)=y$
then $f^{-1}(y)=x$. In this case $y=16x^2$, and $f^{-1}(y)$ becomes
$f^{-1}(16x^2)$. Therefore $f^{-1}(16x^2)=x$. But $x=\dfrac{\sqrt{16x^2}}{4}$. Therefore
$f^{-1}(16x^2)=\dfrac{\sqrt{16x^2}}{4}$. And $16x^2=y$. Therefore $f^{-1}(y)=\dfrac{\sqrt{y}}{4}$.
In like manner $f^{-1}(a)=\dfrac{\sqrt{a}}{4}$; and generally $f^{-1}(x)=\dfrac{\sqrt{x}}{4}$,
whatever $x$ may be.
\begin{flalign*}
&\text{\indent Thus } & y&=f(x)=16x^2,                    && \\
&                     & x&=f^{-1}(y)=\dfrac{\sqrt{y}}{4}. &&
\end{flalign*}
(See below, \Pageref{11}.)
\end{Remark}

From the formula $y=f(x)=16x^2$ we can easily
calculate the successive values of~$f(x)$ as~$x$ increases, \ie\
the space traversed by the falling body in~one, two,
three, etc., seconds.
%% -----File: 017.png---Folio 4-------
\Pagelabel{4}%
\begin{align*}
&\underline{x\quad  f(x) = 16x^2} \\
&0\quad  f(0) = 16 × 0^2 = \Z0. \\
&1\quad  f(1) = 16 × 1^2 = \Z16 \quad\text{growth during last second } \Z16\DPtypo{}{.} \\
&2\quad  f(2) = 16 × 2^2 = \Z64 \quad\PadTo{\text{growth during }}{\Ditto}\PadTo{\text{last second }}{\Ditto} \Z48\DPtypo{}{.} \\
&3\quad  f(3) = 16 × 3^2 = 144 \quad\PadTo{\text{growth during }}{\Ditto}\PadTo{\text{last second }}{\Ditto} \Z80\DPtypo{}{.} \\
&4\quad  f(4) = 16 × 4^2 = 256 \quad\PadTo{\text{growth during }}{\Ditto}\PadTo{\text{last second }}{\Ditto} 112\DPtypo{}{.} \\
&\text{etc.\ etc.} \PadTo{{}=16 × 4^2={}}{\text{etc.}}\text{etc.}\quad\PadTo[r]{growth during last second\quad\;99}{\text{etc.}}
\end{align*}

In the case of the cooling iron in the stream the
time allowed is again the variable, but the function,
which we will denote by~$\phi (x)$, is not such a simple one,
and we need not draw out the details. Without doing
so, however, we can readily see that there will be an
important difference of character between this function
and the one we have just investigated. For the space
traversed by the falling body not only grows continually,
but grows more in each successive second than it
did in the last, as is shown in the last column of the
table. Now it is clear that though the cooling iron
will always go on getting cooler, yet it will not cool
more during each successive second than it did during
the last. On the contrary, the fall in temperature of
the red-hot iron in the first second will be much greater
than the fall in, say, the hundredth second, when the
water is only very little colder than the iron; and the
total fall can never be greater than the total difference
between the initial temperatures of the iron and the
water. This is expressed by saying that the one
function~$f(x)$, \emph{increases without limit} as the variable,~$x$,
increases, and that the other function~$\phi (x)$ \emph{approaches a
definite limit} as the variable,~$x$, increases. In either
case the function is always increased by an increase of
the variable, but only in the first case can we make the
function as great as we like by increasing the variable
sufficiently; for in the second case there is a certain
fixed limit which the function will never reach, however
long it continues to increase. If the reader finds this
conception difficult or paradoxical, let him consider the
%% -----File: 018.png---Folio 5-------
series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}$, etc., and let $f(x)$ signify the
sum of $x$~terms of this series. Then we shall have
\begin{align*}
&\underline{\PadTo{\text{etc.}}{x}\ f(x)} \\
&\PadTo{\text{etc.}}{1}\ \PadTo{f(x)}{1.} \\
&\PadTo{\text{etc.}}{2}\ \PadTo{f(x)}{\frac{3}{2}}   \left(\ie\ 1 + \tfrac{1}{2}\right). \\
&\PadTo{\text{etc.}}{3}\ \PadTo{f(x)}{\frac{7}{4}}   \left(\ie\ 1 + \tfrac{1}{2} + \tfrac{1}{4}\right). \\
&\PadTo{\text{etc.}}{4}\ \PadTo{f(x)}{\frac{15}{8}}  \left(\ie\ 1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8}\right). \\
&\PadTo{\text{etc.}}{5}\ \PadTo{f(x)}{\frac{31}{16}} \left(\ie\ 1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \tfrac{1}{16}\right). \\
&\text{etc.}\ \PadTo{f(x)}{\text{etc.}}
\end{align*}
\Pagelabel{5}%
Here $f(x)$ is always made greater by increasing~$x$, but
however great we make~$x$ we shall never make~$f(x)$
quite equal to~$2$. This case furnishes a simple instance
of a function which always increases as its variable
increases, but yet never reaches a certain fixed limit.
The cooling iron presents a more complicated case of
such a function.

The two functions we have selected for illustration
differ then in this respect, that as the variable (time)
increases, the one (space traversed by a falling body)
increases without limit, while the other (fall of temperature
in the iron) though always increasing yet approaches
a fixed limit. But $f(x)$~and~$\phi (x)$ resemble
each other in this, that they both of them always increase
(and never decrease) as the variable increases.

There are, however, many functions of which this
cannot be said. For instance, let a body be projected
\index{Projectile}%
vertically upwards, and let the height at which we find
it at any given moment be regarded as a function of
the time which has elapsed since its projection. It is
obvious that at first the body will rise (doing work
against gravitation), and the function (height) will increase
as the variable (time) increases. But the initial
energy of the body cannot hold out and do work against
gravitation for ever, and after a time the body will rise
no higher, and will then begin to fall, in obedience to the
still acting force of gravitation. Then a further increase
%% -----File: 019.png---Folio 6-------
of the variable (time) will cause, not an increase, but a
decrease in the function (height). Thus, as the variable
increases, the function will at first increase with it, and
then decrease.

To recapitulate: one thing is a function of another
if it varies with it, whether increasing as it increases or
decreasing as it increases, or changing at a certain point
or points from the one relation to the other.
\Pagelabel{6}%

We have already reached a point at which we can
attach a definite meaning to the proposition: \emph{The value-in-use
of any commodity to an individual is a function of the
quantity of it he possesses}, and as soon as we attach a
definite meaning to it, we perceive its truth. For by
the value-in-use of a commodity to an individual, we
mean the total worth of that commodity to him, for his
own purposes, or the sum of the advantages he derives
immediately from its possession, excluding the advantages
he anticipates from exchanging it for something else.
Now it is clear that this sum of advantages is greater
or less according to the quantity of the commodity the
man possesses. It is not the same for different quantities.
The value-in-use of two blankets, that is to say
\index{Blankets}%
the total direct service rendered by them, or the sum of
direct advantages I derive from possessing them, differs
from the value-in-use of one blanket. If you increase
or diminish my supply of blankets you increase or
diminish the sum of direct advantages I derive from
them. The value-in-use of my blankets, then, is a
function of the number (or quantity) I possess. Or if
we take some commodity which we are accustomed to
think of as acquired and used at a certain rate rather than
in certain absolute quantities, the same fact still appears.
The value-in-use of one gallon of water a day, that is to
say the sum of direct advantages I derive from commanding
it, differs from the value-in-use of a pint a day
or of two gallons a day. The sum of direct advantages
which I derive from half a pound of butcher's meat a
%% -----File: 020.png---Folio 7-------
day is something different from that which I should
derive from either an ounce or a whole carcase per day.
In other words, \emph{the sum of the advantages I derive from
the direct use or consumption of a commodity is a function
of its quantity, and increases or decreases as that quantity
changes}.

\begin{Remark}
Two points call for attention here. In the first place,
there are many commodities which we are not in the habit
of thinking of as possessed in varying quantities; or at any
rate, we usually think of the services they render as functions
of some other variable than their quantity. For instance,
a watch that is a good time-keeper renders a greater
sum of services to its possessor than a bad one; but it seems
an unwarrantable stretch of language to say that the owner
of a good watch has ``a greater amount or quantity of watch''
than the owner of a bad one. It is a little more reasonable,
though still hardly admissible, to say that the one has ``more
time-keeping apparatus'' than the other. But, as the reader
will remember, we have already seen that a function may
depend on two or more variables (\Pageref{2}), and if we consider
watches of different qualities as one and the same commodity,
\index{Watches}%
then we must say that the most important variable is the
quality of the watch; but it will still be true that two
watches of the same quality would, as a rule, perform a
different (and a greater) service for a man than one watch;
for most men who have only one have experienced temporary
inconvenience when they have injured it, and would have
been very glad of another in reserve. Even in this case,
therefore, the sum of advantages derived from the commodity
``watches'' is a function of the quantity as well as the quality.
Moreover, the distinction is of no theoretical importance, for
the propositions we establish concerning value-in-use as a
function of quantity will be equally true of it as a function
of quality; and indeed ``quality'' in the sense of ``excellence,''
being conceivable as ``more'' or ``less,'' is obviously
itself a quantity of some kind.

The second consideration is suggested by the frequent use
of the phrase ``\emph{sum of advantages}'' as a paraphrase of ``\emph{worth}''
or ``\emph{value-in-use}.'' What are we to consider an ``advantage''?
%% -----File: 021.png---Folio 8-------
It is usual to say that in economics everything which a man
wants must be considered ``useful'' to him, and that the
word must therefore be emptied of its moral significance.
In this sense a pint of beer is more ``useful'' than a gimlet
\index{Beer}%
\index{Gimlet}%
to a drunken carpenter. And, in like manner, a wealthier
person of similar habits would be said to derive a greater
``sum of advantages'' from drinking two bottles of wine at
\index{Wine}%
dinner than from drinking two glasses. In either case, we
are told, that is ``useful'' which ministers to a desire, and it
is an ``advantage'' to have our desires gratified. Economics,
it is said, have nothing to do with ethics, since they
deal, not with the legitimacy of human desires, but with the
means of satisfying them by human effort. In answer to
this I would say that if and in so far as economics have nothing
to do with ethics, economists must refrain from using ethical
words; for such epithets as ``useful'' and ``advantageous''
will, in spite of all definitions, continue to carry with them
associations which make it both dangerous and misleading to
apply them to things which are of no real use or advantage.
I shall endeavour, as far as I can, to avoid, or at least to
minimise, this danger. I am not aware of any recognised
word, however, which signifies the quality of being desired.
``Desirableness'' conveys the idea that the thing not only is
but deserves to be desired. ``Desiredness'' is not English,
but I shall nevertheless use it as occasion may require.
``Gratification'' and ``satisfaction'' are expressions morally
indifferent, or nearly so, and may be used instead of ``advantage''
when we wish to denote the result of obtaining a
thing desired, irrespective of its real effect on the weal or
woe of him who secures it.\Pagelabel{8}%
\end{Remark}

Let us now return to the illustration of the body
\index{Projectile}%
projected vertically upwards at a given velocity. In
this case the time allowed is the variable, and the
height of the body is the function. Taking the
rough approximation with which we are familiar, which
gives sixteen feet as the space through which a body
will fall from rest in the first second, and supposing
that the velocity with which the body starts is $a$~ft.\
per~second, we learn by experiment, and might deduce
%% -----File: 022.png---Folio 9-------
from more general laws, that we shall have $y=ax-16x^2$,
where $x$ is the number of seconds allowed, and $y$ is the
height of the body at the end of $x$~seconds. If $a=128$,
\ie~if the body starts at a velocity of $128$~ft.\ per~second,
we shall have
\[
y=128x-16x^2.
\]

\begin{Remark}
In such an expression the figures $128$~and~$-16$ are called
the \emph{constants}, because they remain the same throughout the
investigation, while $x$ and $y$ change. If we wish to indicate
the general type of the relationship between $x$ and $f(x)$ or $y$
without determining its details, we may express the constants
by letters. Thus $y=ax+bx^2$ would determine the general character
of the function, and by choosing $128$ and~$-16$ as the constants
we get a definite specimen of the type, which absolutely
determines the relation between $x$~and~$y$. Thus $y=ax+bx^2$
is the general formula for the distance traversed in $x$~seconds
by a body that starts with a given velocity and works directly
with or against a constant force. If the constant force is
gravitation, $b$ must equal~$16$; if the body is to work against
(not with) gravitation the sign of~$b$ must be negative. If
the initial velocity of the body is $128$~ft.\ per~second, $a$~must
equal~$128$.
\end{Remark}

By giving successive values of $1$, $2$, $3$, etc.\ to~$x$ in
the expression $128x-16x^2$, we find the height at which
the body will be at the end of the $1$, $2$, $3$, etc.\ seconds.
\begin{align*}
&\underline{\PadTo{\text{etc.}}{x}\ f(x) = 128x - 16x^2\qquad} \\
&\PadTo{\text{etc.}}{0}\ f(0) = 128 × 0 - 16 × 0^2 = 0   \\
&\PadTo{\text{etc.}}{1}\ f(1) = 128 × 1 - 16 × 1^2 = 112 \\
&\PadTo{\text{etc.}}{2}\ f(2) = 128 × 2 - 16 × 2^2 = 192 \\
&\PadTo{\text{etc.}}{3}\ f(3) = 128 × 3 - 16 × 3^2 = 240 \\
&\text{etc.\quad etc.} \PadTo{{} = 128 × 3 - {}}{\text{etc.}}\PadTo{16 × 3^2 =}{} \text{etc.}\\
\end{align*}

Now this relation between the function and the
variable may be represented graphically by the well-known
method of measuring the \emph{variable} along a base
line, starting from a given point, and measuring the
\emph{function} vertically upwards from that line, negative
%% -----File: 023.png---Folio 10-------
quantities in either case being measured in the opposite
direction to that selected for positive quantities. To
apply this method we must select our unit of length
and then give it a fixed interpretation in the quantities
we are dealing with. Suppose we say that a unit
measured along the base line~$OX$ in \Figref{1} shall represent
one second, and that a unit measured vertically from~$OX$
in the direction~$OY$ shall represent $10$~ft. We
may then represent the connection between the height
at which the body is to be found and the lapse of time
since its projection by a curved line. We shall proceed
thus. Let us suppose a movable button to slip along
the line~$OX$, bearing with it as it moves along a vertical
line (parallel to~$OY$) indefinitely extended both upwards
and downwards. The movement of this button (which
we may regard as a point, without magnitude, and
which we may call a ``bearer'') along~$OX$ will represent
the lapse of time. The lapse of one second, therefore,
will be represented by the movement of the bearer one
unit to the right of~$O$. Now by this time the body
will have risen $112$~ft., which will be represented by
$11.2$~units, measured upwards on the vertical line
carried by the bearer. This will bring us to the point
indicated on \Figref{1} by~$P_1$. Let us mark this point and
then slip on the bearer through another unit. This will
represent a total lapse of two seconds, by which time
the body will have reached a height of $192$~ft., which
will be represented by $19.2$~units measured on the
vertical. This will bring us to~$P_2$. In $P_1$ and~$P_2$ we
have now representations of two points in the history of
the projectile. $P_1$~is distant one unit from the line~$OY$
and $11.2$~units from~$OX$, \ie~it represents a movement
from~$O$ of $1$~unit in the direction~$OX$ (time, or~$x$), and
of $11.2$~units in the direction of~$OY$ (height, or~$y$). This
indicates that $11.2$ is the value of~$y$ which corresponds
to the value~$1$ of~$x$. In like manner the position of~$P_2$
indicates that $19.2$ is the value of~$y$ that corresponds
to the value~$2$ of~$x$. Now, instead of finding an
%% -----File: 024.p n g----------
%[Blank Page]
%% -----File: 025.p n g----------
\begin{figure}[p]
\Pagelabel{9}%
  \begin{center}
    \begin{minipage}[c]{2.25in}
      \Fig{1}
      \Input[2.25in]{025a}
    \end{minipage}\hfil
    \begin{minipage}[c]{2.25in}
      \Fig{3}
      \Input[2.25in]{025b}
    \end{minipage}
  \end{center}
\end{figure}
%[To face page 11.]
%% -----File: 026.png---Folio 11-------
indefinite number of these points, let us suppose that as
the bearer moves continuously (\ie~without break) along~$OX$
a pointed pencil is continuously drawn along the
vertical, keeping exact pace, to scale, with the moving
body, and therefore always registering its height,---a unit
of length on the vertical representing $10$~ft. Obviously the
point of the pencil will trace a continuous curve, the course
of which will be determined by two factors, the horizontal
factor representing the lapse of time and the vertical
factor representing the movement of the body, and if we
take any point whatever on this curve it will represent
a point in the history of the projectile; its distance
from~$OY$ giving a certain point of time and its distance
from~$OX$ the corresponding height.

Such a curve is represented by \Figref{1}. We have
seen how it is to be formed; and when formed it is to
be read thus: If we push the bearer along~$OX$, then for
every length measured along~$OX$ the curve cuts off a corresponding
length on the vertical, which we will call the
``vertical intercept.'' That is to say, for every value of $x$~(time)
the curve marks a corresponding value of $y$~(height).

$OX$ is called ``the axis of~$x$,'' because $x$ is measured
along it or in its direction. $OY$~is, for like reason,
called ``the axis of~$y$.''

\begin{Remark}
\Pagelabel{11}%
We have seen that if $y$ is a function of~$x$ then it follows
that $x$~is also a function of~$y$ (\Pageref{3}). Hence the curve we
have traced may be regarded as representing $x = f^{-1}(y)$ no
less than $y = f(x)$. If we move our bearer along~$OY$ to
represent the height attained, and make it carry a line
parallel to~$OX$, then the curve will cut off a length indicating
the time that corresponds to that height. It will be seen
that there are two such lengths of $x$ corresponding to every
length of $y$ between $0$~and~$25.6$, one indicating the moment
at which the body will reach the given height as it ascends,
and the other the moment at which it returns to the same
height in its descent.

As an exercise in the notation, let the student follow this
series of axiomatic identical equations: given $y = f(x)$, then
%% -----File: 027.png---Folio 12-------
$xy=f(x)x=f^{-1}(y)f(x)=f^{-1}(y)y$. Also $f^{-1}\left[f(x)\right]=x$ and
$f\left[f^{-1}(y)\right]=y$.
\end{Remark}

\Pagelabel{12}%
It must be carefully noted that the curve \emph{does not
give us a picture of the course of the projectile}. We have
supposed the body to be projected vertically upwards,
and its course will therefore be a straight line, and
would be marked by the movement of the pencil up and
down the vertical, taken alone, and not in combination
with the movement of the vertical itself; just as the
time would be marked by the movement of the pencil,
with the bearer, along~$OX$, taken alone. In fact the
best way to conceive of the curve is to imagine one
bearer moving along~$OX$ and marking the time, to scale,
while a second bearer moves along~$OY$ and marks the
height of the body, to scale, while the pencil point \emph{follows
the direction and speed of both of them at once}. The
pencil point, it will be seen, will always be at the intersection
of the vertical carried by one bearer and the
horizontal carried by the other. Thus it will be quite
incorrect and misleading to call the curve ``a curve
of height,'' and equally but not more so to call it ``a
curve of time.'' Both height and time are represented
by straight lines, and the curve is a ``curve
of height-and-time,'' or ``a curve of time-and-height,''
that is to say, \emph{a curve which shows the history of the connection
between height and time}.

And again the scales on which time and height are
measured are altogether indifferent, as long as we read our
curve by the same scale on which we construct it. The
student should accustom himself to draw a curve on a
number of different scales and observe the wonderful
changes in its appearance, while its meaning, however
tested, always remains the same.

All these points are illustrated in \Figref{2}, where the
very same history of the connection between time and
height in a body projected vertically upwards at $128$~ft.\
per~second is traced for four seconds and $256$~ft., but the
%% -----File: 028.png---Folio 13-------
height is drawn on the scale $50$~ft.\ $\frac{1}{6}$~in.\ instead of $10$~ft.\
$\frac{1}{6}$~in. It shows us that the lines representing space
\Pagelabel{13}%
and those representing time
\begin{wrapfigure}[13]{r}{2in}
  \Fig{2}
  \Input[2in]{028a}
\end{wrapfigure}
enter into the construction of
the curve on precisely the
same footing. The curve, if
drawn, would therefore be
neither a curve of time nor
a curve of height, but a curve
of time-and-height.

The curve then, is not a
picture of the course of the
projectile in space, and a
similar curve might equally
well represent the history of a phenomenon that has no
course in space and is independent of time.

For instance, the expansion of a metal bar under
tension is a function of the degree of tension; and a
testing machine may register the connection between
\index{Testing@{\textsc{Testing Machine}}}%
the tension and expansion upon a curve. The tension
is the variable~$x$ (measured in tons, per inch cross-section
of specimen tested, and drawn on axis of~$x$ to
the scale of, say, seven tons to the inch), and the expansion
is $f(x)$ or~$y$ (measured in inches, and drawn on
axis of~$y$, say to the natural scale, $1:1$).\footnote
  {If we take tension (the variable) along~$y$, and expansion (the
  function) along~$x$, the theory is of course the same. As a fact,
  it is usual in testing-machines to regard the tension as measured
  on the vertical and the expansion on the horizontal. It is only a
  question of how the paper is held in the hand, and the reader will do
  well to throw the curve of time-and-height also, on its side, read its
  $x$ as~$y$ and its $y$ as~$x$, and learn with ease and certainty to read off the
  same results as before. This will be useful in finally dispelling the
  illusion (that reasserts itself with some obstinacy) that the figure represents
  the course of the projectile. The figures may also be varied by
  being drawn from right to left instead of from left to right,~etc. It is
  of great importance not to become dependent on any special convention
  as to the position,~etc.\ of the curves.}

The tension and expansion, then, are indicated by
straight lines, constantly changing in length, but the
history of their connection is a curve. It is not a curve
%% -----File: 029.png---Folio 14-------
of expansion or a curve of tension, but a curve of tension-and-expansion.

Or again, the pleasurable sensation of sitting in a
Turkish bath is a function, amongst other things, of
\index{Turkish bath}%
the temperature to which the bath is raised. If we
treat that temperature as the variable, and measure its
increase by slipping the bearer along the base line~$OX$,
then the whole body of facts concerning the varying
degrees of pleasure to be derived from the bath, according
to its varying degrees of heat, might be represented
by a curve, which would be in some respects analogous
to that represented on \Figref{1}; for, as we measure the
rise of temperature by moving the bearer along our
base line, we shall, up to a certain point, read our increasing
sense of luxury on the increasing length of the
vertical intercepted by a rising curve, after which the
increasing temperature will be accompanied by a decreasing
sense of enjoyment, till at last the enjoyment
will sink to zero, and, if the heat is still raised, will
become a rapidly increasing negative quantity. Thus:

\emph{If we have a function (of one variable), then whatever
the nature of the function may be, the connection between the
function and the variable is theoretically capable of representation
by a curve.} And since we have seen that the
total satisfaction we derive from the enjoyment or use
of any commodity is a function of the quantity we
possess (\ie~changes in magnitude as the quantity increases
or decreases), it follows that \emph{a curve must theoretically
exist which assigns to every conceivable quantity of
a given commodity the corresponding total satisfaction to be
derived by a given man from its use or possession}; or, in
other words, \emph{the connection between the total satisfaction
derived from the enjoyment of a commodity and the quantity
of the commodity so enjoyed is theoretically capable of being
represented by a curve}. Now this ``total satisfaction
derived'' is what economists call the ``total utility,'' or
the ``value-in-use'' of a commodity. The conclusion
we have reached may therefore be stated thus: Since
%% -----File: 030.png---Folio 15-------
the value-in-use of a commodity varies with the quantity
of the commodity used, \emph{the connection between the quantity
of a commodity possessed and its value-in-use may, theoretically,
be represented by a curve}.

\begin{Remark}
\Pagelabel{15}%
Here an initial difficulty presents itself. To imagine the
construction of such a curve as even theoretically possible, we
should have to conceive the theoretical possibility of fixing
a unit of satisfaction, by which to measure off satisfactions
two, three, four times as great as the standard unit, on our
vertical line, just as we measured tens of feet on it in \Figref{1}.
We shall naturally be led in the course of our inquiry to deal
with this objection, which is not really formidable (see \Pageref{52});
and it is only mentioned here to show that it has not been
overlooked. Meanwhile, it may be observed that since satisfaction
is certainly capable of being ``more'' or ``less,'' and
since the mind is capable of estimating one satisfaction as
``greater than'' or ``equal to'' another, it cannot be theoretically
impossible to conceive of such a thing as an accurate
measurement of satisfaction, even though its practical measurement
should always remain as vague as that of heat was when
the thermometer was not yet invented.
\index{Thermometer}%
\end{Remark}

We may go a step farther, and may say that,
if curves representing the connection between these
economic functions (values-in-use) and their variables
(quantities of commodity) could be actually drawn out,
they would, at any rate in many cases, present an important
point of analogy with our curve in \Figref{1}; for
they would first ascend and then descend, and ultimately
pass below zero. As the quantity of any commodity in
our possession increases we gradually approach the point
at which it has conferred upon us the full satisfaction
we are capable of deriving from it; after this a larger
stock is not in any degree desired, and would not add
anything to our satisfaction. In a word, we have as
much as we want, and would not take any more at
a gift. The function has then reached its maximum
value, corresponding to the highest point on the curve.
%% -----File: 031.png---Folio 16-------
If the commodity is still thrust upon us beyond this
point of complete satisfaction, the further increments
become, as a rule, \emph{discommodious}, and the excessive
quantity \emph{diminishes} the total satisfaction we derive from
possessing the commodity, till at length a point is
reached at which the inconvenience of the excessive
supply neutralises the whole of the advantage derived
from that part which we can enjoy, and we would just
as soon go without it altogether as have so far too
much of a good thing. If the supply is still increased,
the net result is a balance of inconvenience, and (if shut
up to the alternative of \emph{all} or \emph{none}) we should, on the
whole, be the gainers if relieved of the advantage and
disadvantage alike. The heat of a Turkish bath has
already given us one instance; and for another we may
take butcher's meat. Most of us derive (or suppose
\index{Meat@{\textsc{Meat}, butcher's}}%
ourselves to derive) considerable satisfaction from the
consumption of fresh meat. The sum of satisfaction
increases as the amount of meat increases up to a point
roughly fixed by the popular estimate at half to three-quarters
of a pound per diem. Then we have enough,
and if we were required to consume or otherwise personally
dispose of a larger amount, the inconvenience
of eating, burying, burning, or otherwise getting rid of
the surplus, or the unutterable consequences of failing
to do so, would partially neutralise the pleasure and
advantage of eating the first half pound, till at some
point short of a hundredweight of fresh meat per head
per diem we should (if shut in to the alternative of all
or none) regretfully embrace vegetarianism as the lesser
evil. In this case the curve connecting the value-in-use
of meat with its quantity would rise as the supply of
meat, measured along the base line, increased until, say
at half a pound a day, it reached its maximum elevation,
indicating that up to that point more meat meant more
satisfaction, after which the curve would begin to descend,
indicating that additional supplies of meat would
be worse than useless, and would tend to neutralise the
%% -----File: 032.png---Folio 17-------
satisfaction derived from the portion really desired, and
to reduce the total gratification conferred, till at a
certain point the curve would cross the base line, indicating
that so much meat as that (if we were obliged to
take all or none) would be just as bad as none at all,
and that if more yet were thrust upon us it would on
the whole be \emph{worse} than having none.

\begin{Remark}
Though practically we are almost always concerned with
commodities our desire for which is not fully satisfied, that
is to say, with the portions of our curves which are still ascending,
yet it is highly important, as a matter of theory, to realise
the fact that curves of quantity-and-value-in-use must always
\emph{tend} to reach a maximum somewhere, and that as a rule they
would actually reach that maximum if the variable (measured
along the axis of~$x$) were made large enough, and would then
descend if the variable were still further increased; or in
other words, that there is hardly any commodity of which
we might not conceivably have enough and too much, and
even if there be such a commodity its increase would still
\emph{tend} to produce satiety (compare \Pageref{5}). Some difficulty is
often felt in fully grasping this very simple and elementary
fact, because we cannot easily divest our minds in imagination
of the conditions to which we are practically accustomed.
Thus we may find that our minds refuse to isolate the \emph{direct}
use of commodities and to contemplate that alone (though it
is of this direct use only that we are at present speaking),
and persist, when we are off our guard, in readmitting the
idea that we might exchange what we cannot use ourselves
for something we want. A man will say, for instance, if
confronted with the illustration of fresh meat which I have
used above, that he would very gladly receive a hundredweight
of fresh meat a-day and would still want more,
because he could sell what he did not need for himself.
This is of course beside the mark, since our contention is that
the \emph{direct value-in-use} of an article always tends to reach a
maximum; but in order to assist the imagination it may be
well to take a case in which a whole community may suffer
from having too much of a good thing, so that the confusing
side-lights of possible exchange may not divert the attention.
%% -----File: 033.png---Folio 18-------
\emph{Rain}, in England at least, is an absolute necessary of life,
but if the rainfall is too heavy we derive less benefit from it
\index{Rainfall}%
than if it is normal. Every extra inch of rainfall then
becomes a very serious discommodity, reducing the total
utility or satisfaction-derived to something lower than it
would have been had the rain been less; and it is conceivable
that in certain districts the rain might produce floods
that would drown the inhabitants or isolate them, in
inaccessible islands, till they died of starvation, thus cancelling
the whole of the advantages it confers and making their
absolute sum zero.

Another class of objections is, however, sometimes raised.
We are told that there are some things, notably money, of
which the ordinary man could never have as much as he
wanted; and daily experience shows us that so far from an
increased supply of money tending to satisfy the desire for
it, the more men have the more they want. This objection
is based on a loose use of the phrase ``more money.'' Let
us take any definite sum, say~£1, and ask what effort or
privation a man will be willing to face in order that he may
secure it. We shall find, of course, that if a man has a
hundred thousand a-year he will be willing to make none
but the very smallest effort in order to get a pound more,
whereas if the same man only has thirty shillings a-week he
will do a good deal to get an extra pound. It is true that
the millionaire may still exert himself to get more money;
but to induce him to do so the prospect of gain must be
much greater than was necessary when he was a comparatively
poor man. He does not want \emph{the same sum of money} as
much as he did when he was poor, but he sees the possibility
of getting a very large sum, and wants that as much as he
used to want a small one. All other objections and apparent
exceptions will be found to yield in like manner to careful
and accurate consideration.

It is true, however, that a man may form instinctive
habits of money-making which are founded on no rational
principle, and are difficult to include in any rationale of
action; but even in these cases the action of our law is only
complicated by combination with others, not really suspended.

It is also true that the very fact of our having a thing
may develop our taste for it and make us want more; but
%% -----File: 034.png---Folio 19-------
this, too, is quite consistent with our theory, and will be
duly provided for hereafter (\Pageref{63}).
\end{Remark}

Enough has now been said in initial explanation of
a curve in general, and specifically a curve that first
ascends and then descends, as an appropriate means of
representing the connection between the quantity of a
commodity and its value-in-use, or the total satisfaction
it confers.

But if we return once more to \Figref{1}, and recollect
\index{Projectile}%
\Pagelabel{19}%
that the curve there depicted is a curve of time-and-height,
representing the connection between the elevation
a body has attained (function) and the time that has
elapsed since its projection (variable), we are reminded
that there is another closely-connected function of the
same variable, with which we are all familiar. We are
accustomed to ask of a body falling from rest not only
how far it will have travelled in so many seconds, but
\emph{at what rate it will be moving} at any given time. And so,
of a body projected vertically upwards we ask not only
at what height will it be at the end of $x$~seconds, but
also \emph{at what rate will it then be rising}. Let us pause for
a moment to inquire exactly what we mean by saying
that at a given moment a body, the velocity of which
is constantly changing, is moving ``at the rate'' of, say,
$y$~feet per~second. We mean that if, at that moment,
all causes which \emph{modify} the movement of the body were
suddenly to become inoperative, and it were to move on
solely under the impulse already operative, it would then
move $y$~feet in every second, and, consequently, $ay$~feet
in $a$~seconds. In the case of \Figref{1} the modifying
force is the action of gravitation, and what we mean by
the rate at which the body is moving at any moment is
the rate at which it would move, from that moment onwards,
if from that moment the action of gravitation
ceased to be operative.

As a matter of fact it never moves through any space,
however small, at the rate we assign, because modifying
%% -----File: 035.png---Folio 20-------
causes are at work \emph{continuously} (\ie~without intervals
and without jerks), so that the velocity is never uniform
over any fraction of time or space, however small.

When we speak of rate of movement ``at a point,''
then, we are using an abbreviated expression for the
rate of movement which would set in at that point if all
modifying causes abruptly ceased to act thenceforth.

For instance, if we say that a body falling from rest
has acquired a velocity of $32$~feet per~second when it
has been falling for one second, we mean that if, after
acting for one second, terrestrial gravitation should then
cease to act, the body would thenceforth move $32$~feet
in every second.

It follows, then, that the departures from this ideal
rate spring from the continuous action of the modifying
cause, and will be greater or smaller according as the
action of that cause has been more or less considerable;
and since the cause (in this instance) acts uniformly in
time, it will act more in more time and less in less.
Hence, the less the time we allow after the close of one
second the more nearly will the rate at every moment
throughout that time (and therefore the average rate
during that time) conform to the rate of $32$~feet per~second.
And in fact we find that if we calculate (by
the formula $s=16x^2$) the space traversed between the
close of the first second and some subsequent point of
time, then the smaller the time we allow the more
nearly does the average rate throughout that time
become $32$~ft.\ per~second. Thus---\\
\Pagelabel{20}%
\[
\begin{array}{c@{ }r@{ }c@{ }l@{ }c@{\quad}cc}
        &    &          &               &
  &\settowidth{\TmpLen}{\small Body falls}
   \parbox[b]{\TmpLen}{\small Body falls}
  &\settowidth{\TmpLen}{\small Average rate}
   \parbox[b]{\TmpLen}{\small\centering Average rate\\ per sec.} \\
\text{Between } &1 &\text{ sec.\ and } &2 & \text{ sec.} & 48 \text{ ft.}&48 \text{ ft.}\\
\Ditto  &1 &  \Ditto  &1\frac{1}{2} & \Ditto & 20 \Ditto            &40 \Ditto\\
\Ditto  &1 &  \Ditto  &1\frac{1}{4} & \Ditto & \Z9 \Ditto           &36 \Ditto\\
\Ditto  &1 &  \Ditto  &1\frac{1}{8} & \Ditto & \frac{17}{4} \Ditto  &34 \Ditto\\
\Ditto  &1 &  \Ditto  &1\frac{1}{16}& \Ditto & \frac{33}{16} \Ditto &33 \Ditto\\
\Ditto  &1 &  \Ditto  &1\frac{1}{32}& \Ditto & \frac{65}{64} \Ditto &32\DPtypo{\,}{.}5
\end{array}
\]
%% -----File: 036.png---Folio 21-------
and the average rate between $1$~second and $1 + \dfrac{1}{z}$~second
may be made as near $32$~ft.\ a second as we like, by making
$z$ large enough. This is usually expressed by saying
that the average rate between $1$~second and $\dfrac{(z+1)}{z}$~seconds
\Pagelabel{21}%
becomes $32$~ft.\ per second \emph{in the limit}, as $z$ becomes greater,
or the time allowed smaller.

We may, therefore, define ``rate at a point'' as
the ``\emph{limit of the average rate between that point and
a subsequent point, as the distance between the two points
decreases}.''

With this explanation we may speak of the rate at
which the projected body is moving as a function of the
time that has elapsed since its projection; for obviously
the rate changes with the time, and that is all that is
needed to justify us in regarding the time that elapses as
a variable and the rate of movement as a function of that
variable. Let us go on then, to consider the relation of
this new function of the time elapsed to the function we
have already considered. We will call the first function
$f(x)$ and the second function~$f'(x)$. Then we shall have
$x=$~the lapse of time since the projection of the body,
measured in seconds; $f(x)=$~the height attained by the
body in $x$~seconds, measured in feet; $f'(x) =$~the rate
at which the body is rising after $x$~seconds, measured in
feet per~second.

\begin{Remark}
It will be observed that $x$~must be positive, for we have
no data as to the history of the body \emph{before} its projection,
and if $x$ were negative that would mean that the lapse of
time since the projection was negative, \ie~that the projection
was still in the future. On the other hand, $f(x) = 128x-16x^2$
will become negative as soon as $16x^2$ is greater than~$128x$,
\ie~as soon as $16x$ is greater than~$128$, or $x$~greater than
$\frac{128}{16}= 8$; which means that after eight seconds the body will
not only have passed its greatest height but will already
have fallen below the point from which it was originally
%% -----File: 037.png---Folio 22-------
projected, so that the ``height'' at which it is now found, \ie~$f(x)$,
will be negative. Again $f'(x)$, or the rate at which the
body is ``rising,'' will become negative as soon as the maximum
height is passed, for then the body will be rising
negatively, \ie~falling.
\end{Remark}

We have now to examine the connection between
$f(x)$~and~$f'(x)$. Our common phraseology will help us
to understand it. Thus: $f(x)$~expresses the height of
the body at any moment, $f'(x)$~expresses the rate at which
the body is rising; but the rate at which it is rising is
\emph{the rate at which its height, or~$f(x)$, is increasing}. That is,
$f'(x)$~represents the rate which $f(x)$ is increasing. A glance
at \Figref{1} will suffice to show that this rate is not uniform
throughout the course of the projectile. At first the
moving body rises, or increases its height, rapidly, then
less rapidly, then not at all, then negatively---that is to
say, it begins to fall. This, as we have seen, may be
expressed in two ways. We may say $f(x)$ [$={}$the
height] first increases rapidly, then slowly, then negatively,
or we may say $f'(x)$ [$={}$the rate of rising] is first
great, then small, then negative.

Formula: \emph{$f'(x)$~represents the rate at which $f(x)$~grows}.

It is obvious then that some definite relation exists
between $f(x)$ and~$f'(x)$, and Newton and Leibnitz discovered
the nature of that relation and established rules
by which, if any function whatever,~$f(x)$, be given, another
function~$f'(x)$ may be derived from it which shall
indicate the rate at which it is growing.

\begin{Remark}
This second function is called the ``\emph{first derived function},''
or the ``\emph{differential coefficient}''\footnote{See \Pageref{31}.} of the original function, and if
the original function is called~$f(x)$, it is usual to represent the
first derived function by~$f'(x)$. In some cases it is possible
to perform the reverse operation, and if a function be given,
say~$\phi(x)$, to find another function such that $\phi(x)$ shall
%% -----File: 038.png---Folio 23-------
represent the rate of its increase.\footnote
  {Such a function always exists, but we cannot always ``find'' it,
  \ie~express it conveniently in finite algebraical notation.}
This function is then
\Pagelabel{23}%
called the ``\emph{integral}'' of~$\phi(x)$ and is written ${\displaystyle \int_0^x \phi(x)\, dx}$. Thus
if we start with~$f(x)$, find the function which represents the
rate of its growth and call it~$f'(x)$, and then starting with~$f'(x)$
find a function whose rate of growth is~$f'(x)$ and call
it ${\displaystyle \int_0^x f'(x)\, dx}$, we shall obviously have ${\displaystyle \int_0^x f'(x)\, dx = f(x)}$.

The only flaw in the argument is that it assumes there to
be only one function of~$x$ which increases at the rate indicated
by~$f'(x)$, and therefore assumes that if we find \emph{any} function
${\displaystyle \int_0^x f'(x)\, dx}$ which increases at that rate, it must necessarily be
the function,~$f(x)$, which we already know does increase at that
rate. This is not strictly true, and ${\displaystyle \int_0^x f'(x)\, dx}$ is, therefore, an
indeterminate symbol, which represents~$f(x)$ and also certain
other functions of~$x$, which resemble~$f(x)$ in all respects save
one, which one will not in any way affect our inquiries. As
far as any properties we shall have to consider are concerned,
we may regard the equation
\[
\int_0^x f'(x)\, dx = f(x)
\]
as absolute.
\end{Remark}

In the case we are now considering, $f(x)$ is $128x - 16x^2$,
and an application of Newton's rules will tell us that
$f'(x)$ is $128 - 32x$. That is to say, if we are told that
$x$ being the number of seconds since the projection, the
height of the body in feet is always $128x - 16x^2$ for all
values of~$x$, then we know by the rules, without further
experiment, that the rate at which its height is increasing
will always be $128 - 32x$ ft.-per-second, for all
values of~$x$. But the rate at which the height is
increasing is the rate at which the body is rising, so
that $128 - 32x$ is the formula which will tell us the
rate at which the body is rising after the lapse of $x$~seconds.\footnote
  {See table on \Pageref{24}.---\textit{Trans.}}%[** TN: Added footnote]
%% -----File: 039.png---Folio 24-------
\begin{table}[hbt]%[** TN: Floating to avoid noticeably underfull page]
\Pagelabel{24}%
\[
\begin{array}{c@{}l}
\settowidth{\TmpLen}{\small $x =$ number of seconds}
\parbox[c]{\TmpLen}{\centering\small $x =$ number of seconds\\ since the projection.}
 &\quad\settowidth{\TmpLen}{\small Rate at which the}
  f'(x) = 128 - 32x = \left\{
  \parbox[c]{\TmpLen}{\centering\small Rate at which the\\ body is rising, in\\ feet-per-second.}\right.\\
&\\[-12pt]
\hline
\Strut
0 & f'(0) = 128 - 32 × 0 = \PadTo[r]{-128}{128} \\
1 & f'(1) = 128 - 32 × 1 =  \PadTo[r]{-128}{96} \\
2 & f'(2) = 128 - 32 × 2 =  \PadTo[r]{-128}{64} \\
3 & f'(3) = 128 - 32 × 3 =  \PadTo[r]{-128}{32} \\
4 & f'(4) = 128 - 32 × 4 =   \PadTo[r]{-128}{0} \\
5 & f'(5) = 128 - 32 × 5 = \PadTo[r]{-128}{-32} \\
6 & f'(6) = 128 - 32 × 6 = \PadTo[r]{-128}{-64} \\
7 & f'(7) = 128 - 32 × 7 = \PadTo[r]{-128}{-96} \\
8 & f'(8) = 128 - 32 × 8 = \PadTo[r]{-128}{-128}\\
\text{etc.} & \ \text{etc.}\PadTo{{}= 128 - 32 × 8 = {}}{\text{etc.}} \;\PadTo[r]{-128}{\text{etc.}}
\end{array}
\]
\end{table}

Now the connection between $f'(x)$~and~$x$ can be
represented graphically, just as the connection between
$f(x)$~and~$x$ was. It must be represented by a curve (in
this case a straight line), which makes the vertical
intercept $12.8$ (representing $128$~ft.\ per~second), when
the bearer is at the origin (\ie~when $x$~is~$0$), making it $9.6$
when the bearer has been moved through one unit to the
right of the origin (or when $x$~is~$1$), and so forth. It is
given in \Figref{3} (\Pageref{9}), and registers all the facts drawn out
in our table, together with all the intermediate facts
connected with them. If we wish to read this curve,
and to know at what rate the body will be rising after,
say, one and a half seconds, we suppose our bearer to
be pushed half-way between $1$~and~$2$ on our base line,
and then running our eye up the vertical line it carries
till it is intercepted by the curve, we find that the
vertical intercept measures $8$~units. This means that
the rate at which the body is rising, one and a half
seconds after its projection, is $80$~ft.\ per~second.

\begin{Remark}
No attempt will be made here to demonstrate, even in a
simple case, the algebraical rules by which the derived
functions are obtained from the original ones; but it may be
well to show in some little detail, by geometrical methods,
%% -----File: 040.png---Folio 25-------
the true nature of the connection between a function and its
derived function, and the possibility of passing from the one
to the other.\footnote
  {The student who finds this note difficult to understand is recommended
  not to spend much time over it till he has studied the rest of
  the book.}

Suppose $OP_1P_2P_3$ in \Figref{4} to be a curve representing the
connection of $f(x)$~and~$x$. We may again suppose $f(x)$ to
represent the amount of work done against some constant
force, in which case it will conform to the type $y=f(x)=ax-bx^2$.
The curve in the figure is drawn to the formula
\[
	f(x) = 2x - \frac{x^2}{8}, \text{ where } a=2, b=\tfrac{1}{8}.
\]
This will give the following pairs of corresponding values:---
\[
\begin{array}{c@{\quad}r@{\;}l@{\;}l@{}c}
x &f(x)=& 2x-\dfrac{x^2}{8} & =y.
  &\settowidth{\TmpLen}{\small Growth for last}%
   \parbox[c]{\TmpLen}{\centering\small Growth for last\\ unit of in-\\crease of~$x$.\medskip} \\
\hline
\Strut
0 &f(0)=&  2 × 0 - \frac{0}{8}  &= 0. \\
1 &f(1)=&  2 × 1 - \frac{1}{8}  &= 1\frac{7}{8}  &\frac{15}{8} \\
2 &f(2)=&  2 × 2 - \frac{4}{8}  &= 3\frac{1}{2}  &\frac{13}{8} \\
3 &f(3)=&  3 × 2 - \frac{9}{8}  &= 4\frac{7}{8}  &\frac{11}{8} \\
4 &f(4)=&  4 × 2 - \frac{16}{8} &= 6             &\frac{9}{8}  \\
5 &f(5)=&  5 × 2 - \frac{25}{8} &= 6\frac{7}{8}  &\frac{7}{8}  \\
6 &f(6)=&  6 × 2 - \frac{36}{8} &= 7\frac{1}{2}  &\frac{5}{8}  \\
7 &f(7)=&  7 × 2 - \frac{49}{8} &= 7\frac{7}{8}  &\frac{3}{8}  \\
8 &f(8)=&  8 × 2 - \frac{64}{8} &= 8             &\frac{1}{8}  \\
9 &f(9)=&  9 × 2 - \frac{81}{8} &= 7\frac{7}{8}  &\makebox[0pt][r]{$-$}\frac{1}{8} \\
\text{etc.}  &\text{etc.}\quad  &\multicolumn{2}{c}{\PadTo{9 × 2 - \frac{81}{8}= 7\frac{7}{8}}{\text{etc.}}}
                                                 & \text{etc.}
\end{array}
\]
It is clear from an inspection of the curve and from the
last column in our table that the rate at which $f(x)$ or~$y$
increases per unit increase of~$x$ is not uniform throughout its
history. While $x$~increases from $0$ to~$1$, $y$~grows nearly two
units, but while $x$~increases from $7$ to~$8$, $y$~only grows one
eighth of a unit. Now we want to construct a curve on
which we can read off the rate at which $y$ is growing at any
point of its history. For instance, if $y$~represents the height
%% -----File: 041.png---Folio 26-------
of a body doing work against gravitation (say rising), we want
to construct a curve which shall tell us at what rate the height
is increasing at any moment, \ie~at what rate the body is rising.

Now since the increase of the function is represented by
the rising of the curve, the rate at which the function is
increasing is the same thing as the rate at which the curve is
rising, and this is the same thing as the steepness of the curve.

Again, common sense seems to tell us (and I shall presently
show that it may be rigorously proved) that the steepness of
the tangent, or line touching the curve, at any point is the
same thing as the steepness of the curve at that point. Thus
in \Figref{4}, $R_{1}P_{1}$ (the tangent at~$P_{1}$) is steeper than~$R_{2}P_{2}$
(the tangent at~$P_{2}$), and that again is steeper than~$R_{3}P_{3}$ (the
tangent at~$P_{3}$), which last indeed has no steepness at all; and
obviously the curve too is steeper at~$P_{1}$ than at~$P_{2}$, and
has no steepness at all at~$P_{3}$.

\Pagelabel{26}%
But we can go farther than this and can get a precise numerical
expression for the steepness of the tangent at any point~$P$,
by measuring how many times the line~$QP$ contains the line~$RQ$
($Q$~being the point at which the perpendicular from any
point,~$P$, cuts the axis of~$x$, and~$R$ the point at which
the tangent to the curve, at the same point~$P$, cuts the same
axis). For since $QP$ represents the total upward movement
accomplished by passing from~$R$ to~$P$, while $RQ$ represents
the total forward movement, obviously $QP:RQ = {}$ratio of upward
movement to forward movement${}={}$steepness of tangent.

But steepness of tangent at~$P = {}$steepness of curve at~$P = {}$rate
at which $y$~is growing at~$P$. To find the rate at which
$y$~is growing at $P_{1}$,~$P_{2}$, $P_{3}$,~etc.\ we must therefore find the
ratios $\dfrac{Q_{1}P_{1}}{R_{1}Q_{1}}$, $\dfrac{Q_{2}P_{2}}{R_{2}Q_{2}}$, $\dfrac{Q_{3}P_{3}}{R_{3}Q_{3}}$~etc.  But if we take $r_{1}$,~$r_{2}$,~$r_{3}$, etc.\
each one unit to the left of $Q_{1}$,~$Q_{2}$, $Q_{3}$,~etc.\ and draw
$r_{1}p_{1}$,~$r_{2}p_{2}$, $r_{3}p_{3}$~etc.\ parallel severally to $R_{1}P_{1}$,~$R_{2}P_{2}$, $R_{3}P_{3}$~etc.,
then by similar triangles we shall have
\[
\frac{Q_{1}P_{1}}{R_{1}Q_{1}} = \frac{Q_{1}p_{1}}{r_{1}Q_{1}},\quad
\frac{Q_{2}P_{2}}{R_{2}Q_{2}} = \frac{Q_{2}p_{2}}{r_{2}Q_{2}},\quad
\frac{Q_{3}P_{3}}{R_{3}Q_{3}} = \frac{Q_{3}p_{3}}{r_{3}Q_{3}},\ \text{etc.,}
\]
but the denominators of the fractions on the right hand of
the equations are all of them, by hypothesis, unity. Therefore
the steepness of the curve at the points $P_{1}$,~$P_{2}$, $P_{3}$~etc.\
is numerically represented by $Q_{1}p_{1}$,~$Q_{2}p_{2}$, $Q_{3}p_{3}$,~etc.

In our figure the points~$P_{1}$, $P_{2}$,~$P_{3}$ correspond to the
%% -----File: 042.p n g----------
%[Blank Page]
%% -----File: 043.p n g----------
\begin{figure}[p]
\Pagelabel{25}%
  \begin{center}
    \Fig{4}
    \Input{043a}
    \vfil
    \null\hfill\Fig{5}
    \Input[2.5in]{043b}
  \end{center}
\end{figure}
%[To face page 27.]
%% -----File: 044.png---Folio 27-------
values $x=2$, $x=4$, $x=8$, and the lines $Q_{1}p_{1}$, $Q_{2}p_{2}$, $Q_{3}p_{3}$ are
found on measurement to be $\frac{3}{2}$,~$1$,~$0$.

We may now tabulate the three degrees of steepness of
the curve (or rates at which the function is increasing), corresponding
to the three values of~$x$:---
\[
\begin{array}{c@{\qquad}c}
x & \settowidth{\TmpLen}{\small Steepness of curve${}={}$rate}
\parbox[c]{\TmpLen}
  {\centering\small Steepness of curve${}={}$rate \\ at which $y$ is growing.\medskip} \\
\hline
\Strut
2 & \frac{3}{2} \\
4 & 1           \\
8 & 0
\end{array}
\]

By the same method we may find as many more pairs of
corresponding values as we choose, and it becomes obvious
that the rate at which $y$ or~$f(x)$ is growing is itself a function
of~$x$ (since it changes as $x$~changes); and we may indicate this
function by~$f'(x)$. Then our table gives us pairs of corresponding
values of $x$~and~$f'(x)$, and we may represent the connection
between them by a curve, as usual. In this particular
instance the curve turns out to be a straight line, and it is
drawn out in \Figref{5}.\footnote
  {Its formula is $y=2-\frac{x}{4}$.}
Any vertical intercept on \Figref{5},
therefore, represents the rate at which the vertical intercept
for the same value of~$x$ on \Figref{4} is growing.

Thus we see that, given a curve of any variable and
function, a simple graphical method enables us to find as
many points as we like upon the curve of the same variable
and a second function, which second function represents the
rate at which the first function is growing; \textit{e.g.}, given a
curve of time-and-height that tells us what the height of a
body will be after the lapse of any given time, we can construct
a curve of time-and-rate which will tell us at what rate
that height is increasing, \ie~at what rate the body is rising,
at any given time.

It remains for us to show that the common sense notion
of the steepness of the curve at any point being measured by
the steepness of the tangent is rigidly accurate. In proving
this we shall throw further light on the conception of ``rate
%% -----File: 045.png---Folio 28-------
of increase at a point'' as applied to a movement, or other
increase, which is constantly varying.

If I ask what is the average rate of increase of~$y$ between
the points $P_{2}$~and~$P_{3}$ (\Figref{4}), I mean: If the increase of
$y$ bore a uniform ratio to the increase of~$x$ between the
points $P_{2}$~and~$P_{3}$, what would that ratio be? or, if a point
moved from $P_{2}$ to~$P_{3}$ and if throughout its course its upward
movement bore a uniform ratio to its forward movement,
what would that ratio be? The answer obviously is $\dfrac{S_3P_3} {P_2S_3}$.
Completing the figure as in \Figref{4} we have, by similar
triangles, average ratio of increase of~$y$ to increase of~$x$
between the points $P_{2}$ and $P_{3}=\dfrac{S_3P_3}{P_2S_3}=\dfrac{Q_3P_3}{MQ_3}$.

Now, keeping the same construction, we will let $P_{3}$ slip
along the curve towards~$P_{2}$, making the distance over which
the average increase is to be taken smaller and smaller.
Obviously as $P_{3}$~moves, $Q_{3}$,~$S_{3}$, and~$M$ will move also, and
the ratio $\dfrac{S_3P_3}{P_2S_3}$ will change its value, but the ratio $\dfrac{Q_3P_3}{MQ_3}$ will
likewise change its value in precisely the same way, and will
always remain equal to the other. This is indicated by the
dotted lines and the thin letters in \Figref{4}.

Thus, however near $P_{3}$ comes to $P_{2}$ the average ratio of
the increase of~$y$ to the increase of~$x$ between $P_2$~and~$P_3$ will
always be equal to $\dfrac{Q_3P_3}{MQ_3}$.  But this ratio, though it changes
as $P_{3}$ approaches~$P_{2}$, does not change indefinitely, or without
limit; on the contrary, it is always approaching a definite,
fixed value, which it can never quite reach as long as $P_{3}$
remains distinct from~$P_{2}$, but which it can approach within
any fraction we choose to name, however small, if we make
$P_{3}$ approach $P_{2}$ near enough. It is easy to see what this
ratio is. For as $P_{3}$ approaches~$P_{2}$, $S_{3}$ approaches~$P_{2}$, $Q_{3}$ approaches~$Q_{2}$,
$M$ approaches~$R_{2}$, and therefore the ratio $\dfrac{Q_3P_3}{MQ_3}$
approaches the ratio $\dfrac{Q_2P_2}{R_2Q_2}$,  which is the ratio that measures
the steepness of the tangent at~$P_{2}$. We must realise exactly
what is meant by this. The lengths $Q_{2}P_{2}$ and~$R_{2}Q_{2}$ have
definite magnitudes, which do not change as $P_{3}$ approaches~$P_{2}$,
whereas the lengths $S_{3}P_{3}$ and $MR_{2}+Q_{2}Q_{3}$, which distinguish
%% -----File: 046.png---Folio 29-------
$Q_2P_2$ and $R_2Q_2$ from $Q_3P_3$ and $MQ_3$ respectively,
may be made as small as we please, and therefore as
small fractions of the fixed lengths $Q_2P_2$ and $R_2Q_2$ as
we please.  Therefore the numerator and denominator of
$\dfrac{Q_3P_3}{MQ_3}$ may be made to differ from the numerator and denominator
of $\dfrac{Q_2P_2}{R_2Q_2}$  by \emph{as small fractions of $Q_2P_2$ and $R_2Q_2$ themselves}
as we please. That is to say, the former fraction, or
ratio, may be made to approach the latter without limit.
But the ratio $\dfrac{S_3P_3}{P_2S_3}$  is always the same as the ratio $\dfrac{Q_3P_3}{MQ_3}$, and
therefore the ratio $\dfrac{S_3P_3}{P_2S_3}$ (or the average ratio of the increase of~$y$
to the increase of~$x$ between $P_2$~and~$P_3$) may be made to
approach the ratio $\dfrac{Q_2P_2}{R_2Q_2}$ without limit.  Thus, though $S_3P_3$
and $P_2S_3$ can be made as small as we please absolutely, neither
of them can be made as small as we please with reference to
the other. On the contrary, they tend towards the fixed ratio
$\dfrac{Q_2P_2}{R_2Q_2}$ as they severally approach zero. This is the limit of
the average ratio of the increase of~$y$ to the increase of~$x$
between $P_2$~and~$P_3$, and may be approached as nearly as we
please by taking that average over a small enough part of the
curve, that is by taking $P_3$ near enough to~$P_2$. If we take
the average over no space at all and make $P_3$~coincide with~$P_2$,
we may if we like say that the ratio of the increase of~$y$
to the increase of~$x$ \emph{at} the point $P_2$ actually \emph{is} $\dfrac{Q_2P_2}{R_2Q_2}$, or $Q_2p_2$
per unit. [\NB---Let special note be taken of the conception
of \emph{rate per unit} as a limit to which a ratio approaches, as
the related quantities diminish without limit.] But we must
remember that since neither $y$~nor~$x$ increases at all \emph{at} a
point, and since $S_3P_3$ and $P_2S_3$ both alike disappear when $P_3$
coincides with~$P_2$, there is not really any ratio between them
\emph{at} the limit. But this is exactly in accordance with our
original definition of the ``rate of growth of~$y$ \emph{at} a given
point in its history'' (\Pageref{19}), which we discovered to mean
``the rate at which $y$ would grow if all modifying circumstances
ceased to operate,'' or ``the limit of the average rate
of growth of~$y$ between $P_2$~and~$P_3$, as $P_3$ approaches~$P_2$.'' As a
%% -----File: 047.png---Folio 30-------
matter of fact $y$ never grows at that rate at all, for as soon as it
grows ever so little it becomes subject to modifying influence.

We see, then, that as $P_3$ approaches $P_2$ the limiting position
of the line $P_3P_2M$ is~$P_2R_2$, the tangent at~$P_2$ (as indeed
is evident to the eye), and the limiting ratio of the increase
of~$y$ to the increase of~$x$ is $\dfrac{Q_2P_2}{R_2Q_2}$, or the steepness of the
tangent at~$P_2$. Thus ``the steepness of the tangent at~$P_2$'' is
the only exact interpretation we can give to ``the steepness
of the curve at~$P_2$,'' and our common sense notion turns out
to be rigidly scientific.

We see, then, that by drawing the tangents we can read
$f'(x)$ as well as~$f(x)$ from \Figref{4}. But this is not easy. On
the other hand, in \Figref{5}, it is easy to read~$f'(x)$, but not so
easy to read~$f(x)$. This latter may also be read, however. Let
the student count the units of area included in the triangle~$OPP_3$
(\Figref{5}). He will find that they equal the units of
length in $Q_3P_3$ (\Figref{4}). Or if he take $Q_2$ in \Figref{5}, corresponding
to $Q_2$ in \Figref{4}, he will find that the area~$OPP_2Q_2$
(\Figref{5}) contains as many units as the length~$Q_2P_2$ (\Figref{4}).
Or again, taking $Q_1$~and~$Q_2$, the area $Q_1P_1P_2Q_2$ (\Figref{5}) contains
as many units as the length~$S_2P_2$ (\Figref{4}), which gives
the growth of~$y$ between $P_1$~and~$P_2$.

Thus in \Figref{4} the absolute value of~$y$, or~$f(x)$, is indicated
by \emph{length} and the rate of growth of~$y$, or~$f'(x)$, by \emph{slope} of
the tangent; whereas in \Figref{5} $f'(x)$ is indicated by \emph{length}
and $f(x)$ by \emph{area}. In either case the different character of the
units in which $f(x)$~and~$f'(x)$ are estimated indicates the difference
in their nature, the one being \emph{space} and the other \emph{rate}.

The reason why the areas in \Figref{5} correspond to the
lengths in \Figref{4} is not very difficult to understand, for we
shall find that the units of length in~$S_2P_2$ (\Figref{4}), for example,
and the units of area in~$Q_1P_1P_2Q_2$ (\Figref{5}) both represent
exactly the same thing, viz.\ the product of the average
rate of growth of~$y$ between $P_1$~and~$P_2$ into the period over
which that average growth is taken, which is obviously equivalent
to the total actual growth of~$y$ between the two points.

To bring this out, let us call the average rate of growth
of~$y$, between $P_1$~and~$P_2$, $r$, and the period over which that
growth is taken,~$t$. Then we shall have $rt={}$average rate of
growth${}×{}$period of growth${}={}$total growth.
%% -----File: 048.png---Folio 31-------

Now, in \Figref{4}, taking $OQ_1=x_1$, $OQ_2=x_2$, $Q_1P_1=y_1$, $Q_2P_2=y_2$,
we shall have $r=\dfrac{P_2S_2}{P_1S_2}=\dfrac{y_2-y_1}{x_2-x_1}$, and $t=Q_1Q_2=x_2-x_1$, and
$rt = \dfrac{y_2-y_1}{x_2-x_1}(x_2-x_1) = y_2-y_1 = P_2S_2$.

We must now find the representative of~$rt$ in \Figref{5}, and
to do so we must look for some line that represents~$r$ or
$\dfrac{y_2-y_1}{x_2-x_1}$ or the average rate of growth of~$y$ between $P_1$~and~$P_2$.
Now the rate of growth of~$y$ at~$P_1$ is represented by~$y'_1$, and
its rate of growth at~$P_2$ by~$y'_2$; and an inspection of the
figure shows that it declines \emph{uniformly} between the two
points, so that the average rate will be half way between $y'_1$~and~$y'_2$.
This is represented by the line~$AB$, which equals
$\dfrac{Q_1P_1+Q_2P_2}{2}$ or $\dfrac{y'_1+y'_2}{2}$. We have then, in \Figref{5}, $r=AB$.
But $t=x_2-x_1$ or $Q_1Q_2$ as before. Therefore $rt = AB × Q_1Q_2$.
Again, a glance at \Figref{5} will show that, by equality of
triangles, the area $AB × Q_1Q_2$ is equal to the area~$Q_1P_1P_2Q_2$.
Combining our results then, we have
\[
Q_1P_1P_2Q_2 \text{ (\Figref{5})} =rt=P_2S_2 \text{ (\Figref{4})}
\]
or units of length in $P_2S_2=$ units of area in~$Q_1P_1P_2Q_2$.
\QED

Had the curve in \Figref{5} not been a straight line, the proof
would have been the same in principle, though not so simple;
and the areas would still have corresponded exactly to the
lengths in the figure of the original function.\footnote
  {We have seen that the increment of~$y$ (or~$y_2-y_1$) equals the increment
  of~$x$ (or~$x_2-x_1$) multiplied by $\dfrac{y'_1+y'_2}{2}$  $\left(\text{or } \dfrac{y_2-y_1}{x_2-x_1}\right)$.\Pagelabel{31}%

  Thus: increment of $y={}$increment of $x × \dfrac{y'_1+y'_2}{2}$; and $\dfrac{y'_1+y'_2}{2}=
  \dfrac{f'(x_1)+f'(x_2)}{2}$; now the increment of~$y$ is the magnitude that differentiates
  $y_2$ from~$y_1$, and is, therefore, called by Leibnitz the ``quantitas
  differentialis'' of~$y$, though this term is only applied when $y_1$ and
  $y_2$ are taken very near together, so that the ``quantitas differentialis''
  of $y_1$ and $y_2$ bears only a very small ratio to the ``quantitas integralis,''
  or integral magnitude of $y_1$~itself.

  Thus when $y_2$~and~$x_2$ approach $y_1$~and~$x_1$ very nearly, we have
  differential of $y_1={}$differential of $x_1 × \dfrac{ f'(x_1)+f'(x_2)}{2}$, and as we approach
  the limit, and the difference between $f'(x_1)$ and~$f'(x_2)$ becomes not
  only smaller itself, but a smaller fraction of~$f'(x_1)$, we find that
  $\dfrac{f'(x_1)+f'(x_2)}{2}$ approaches $\dfrac{f'(x_1)+f'(x_1)}{2}=f'(x_1)$.

  In the limit, then, we have differential of $y_1 ={}$differential of $x_1 × f'(x_1)$;
  or generally, differential of $y ={}$differential of $x × f'(x)$, where $f'(x)$ is
  \emph{the coefficient which turns the differential of~$x$ into the differential
  of~$y$}. Hence $f'(x)$ or~$y'$ is called the ``differential coefficient'' of
  $f(x)$ or~$y$, and $y$ or~$f(x)$ is called the ``integral'' of $f'(x)$ or~$y'$.

  I insert this explanation in deference to the wish of a friend, who
  declares that he ``can never properly understand a term scientifically
  until he understands it etymologically,'' and asks ``why it is a
  coefficient and why it is differential.''  I believe his state of mind is
  typical.}
\end{Remark}

It is essential that the reader should familiarise
himself perfectly with the precise nature of the relation
%% -----File: 049.png---Folio 32-------
subsisting between the two functions we have been investigating,
and I make no apology, therefore, for dwelling
on the subject at some length and even risking
repetitions.

We have seen that $f'(x)$ is the rate at which $f(x)$ is
increasing, or rate of growth of~$f(x)$. And we measure
the rate at which a function is increasing by the
number of units which would be added to the function
while one unit is being added to the variable if all the
conditions which determine the relation should remain
throughout the unit exactly what they were at its commencement.

Again, when we denote a certain function of~$x$ by the
symbol~$f(x)$, we have~$y=f(x)$, and for $x=a$ $y=f(a)$,
for $x=1$ $y=f(1)$, for $x=0$ $y=f(0)$, etc. This has been
fully illustrated in previous tables (compare \Pageref{24}).
\begin{flalign*}
&\text{\indent Thus if } & f(x)&=128x-16x^{2},        & \phantom{Thus if} \\
&\text{then}             & f(2)&=[128 × 2-16 × 2^{2}] & \\
&                        &     &= 192.
\end{flalign*}
In \DPtypo{}{the} future, then, we may omit the intermediate stage
and write at once $f(x)=128x-16x^2$; $f(2)=192$, etc.

We may therefore epitomise the information given us
\index{Projectile}%
by the curves in Figs.\ \Figref[]{1}~and~\Figref[]{3} (combined in \Figref{6})\DPtypo{}{.}
Thus---
%% -----File: 050.p n g----------
%[Blank Page]
%% -----File: 051.p n g----------
\begin{figure}[p]
  \begin{center}
    \Fig{6}
    \Input[2in]{051a}
  \end{center}
\end{figure}
%
%[To face page 33.]
%% -----File: 052.png---Folio 33-------
%[** TN: Size-dependent hack to get table below to stay on the same page.]
{\small
\Pagelabel{33}%
\[
\begin{array}{r@{\;}l@{\quad}r@{\;}c}
f(x) =& 128x - 16x^2 & f'(x) = & 128 - 32x \\
\hline
\Strut
f(0) = & \Z\Z0       & f'(0) = &  \PadTo[r]{-128}{128} \\
f(1) = &   112       & f'(1) = &   \PadTo[r]{-128}{96} \\
f(2) = &   192       & f'(2) = &   \PadTo[r]{-128}{64} \\
f(3) = &   240       & f'(3) = &   \PadTo[r]{-128}{32} \\
f(4) = &   256       & f'(4) = &    \PadTo[r]{-128}{0} \\
f(5) = &   240       & f'(5) = &  \PadTo[r]{-128}{-32} \\
f(6) = &   192       & f'(6) = &  \PadTo[r]{-128}{-64} \\
f(7) = &   112       & f'(7) = &  \PadTo[r]{-128}{-96} \\
f(8) = & \Z\Z0       & f'(8) = & \PadTo[r]{-128}{-128} \\
\end{array}
\]}%
which may be read in \Figref{6} from the lengths cut off
by the two curves respectively on the vertical carried
by the bearer as it passes points $0$,~$1$, $2$, $3$,~etc.

This table states the following facts:---At the commencement
the height of the body~[$f(x)$] is~$0$, but the
rate at which that height is increasing~[$f'(x)$] is $128$~ft.\ per~second.
That is to say, the height would increase by $128$~ft.,
while the time increased by one second, if the conditions
which regulate the relations between the time that elapses
and space traversed remained throughout the second
exactly what they are at the beginning of it. But those
conditions are continuously changing and never remain
the same throughout any period of time, however small.
At the end of the first second then, the height attained
[$f(x)$] is, not $128$~ft.\ as it would have been had there
been no change of conditions, but $112$~ft., and the rate
at which that height is now growing is $96$~ft.\ per~second.
That is to say, if the conditions which determine the
relation between the time allowed and the space traversed
were to remain throughout the second exactly what they
are at the beginning of it, then the height of the body
[$f(x)$] would \emph{grow} $96$~ft., while the time grew one second.
Since these conditions change, however, the height
grows, not $96$~ft., but $80$~ft.\ during the next second, so
that after the lapse of two seconds it has reached the
height of $(112 + 80) = 192$~ft., and is now \emph{growing} at
the rate of $64$~ft.\ per~second. After the lapse of four
%% -----File: 053.png---Folio 34-------
seconds the height of the body is $256$~ft., and that height
\emph{is not growing at all}. That is to say, if the conditions
remained exactly what they are at this moment, then
the lapse of time would not affect the height of the
body at all. But in this case we realise with peculiar
vividness the fact that these conditions never do
remain exactly what they are for any space of time,
however brief. The movement of the body is the
resultant of two tendencies, the constant tendency to
\emph{rise} $128$~ft.\ per~second in virtue of its initial velocity,
and the growing tendency to \emph{fall} in virtue of the continuous
action of gravitation. At this moment these
two tendencies are exactly equal, and \emph{if they remained}
equal then the body would rise $0$~ft.\ per~second, and
the lapse of time would not affect its position. But of
the two tendencies now exactly equal to each other,
one is continuously increasing while the other remains
constant. Therefore they will not remain equal during
any period, however short. Up to this moment
the body rises, after this moment the body falls.
There is no period, however short, \emph{during} which it is
neither rising nor falling, but there is a point of time \emph{at}
which the conditions are such that if they were continued
(which they are not) it \emph{would} neither rise nor fall. This
is expressed by saying that \emph{at} that moment the rate at
which the height is growing is~$0$. If the reader will
pause to consider this special case, and then apply the
like reasoning to other points in the history of the projectile,
it may serve to fortify his conception of ``rate.''
After $6$~seconds the height is~$192$, and the rate at
which it is growing is $-64$~ft.-per-second. That is to
say, the body is \emph{falling} at the rate of $64$~ft.-per-second.
At the end of $8$~seconds the height is~$0$, and the rate
at which the height is growing is $-128$~ft.-per-second.

All this is represented on the table, which may be
continued indefinitely on the supposition that the body
is free to fall below the point from which it was
originally projected.
%% -----File: 054.png---Folio 35-------

The instance of the vertically projected body must
be kept for permanent reference in the reader's mind,
so that if any doubt or confusion as to the relation
between $f'(x)$ and~$f(x)$ should occur, he may be able
to use it as a tuning fork: $f'(x)$ is the rate at which
$f(x)$ is growing, so that if $f(x)$ is the space traversed,
then $f'(x)$ is the rate of motion, \ie~the rate at which
the space traversed,~$f(x)$, is being increased.

Now, when we are regarding time solely as a regulator
of the height of the body, we may without any
great stretch of language speak of the \emph{effect} of the
lapse of time in allowing or securing a definite result
in height. Thus the effect of $1$~second would be
represented by $112$~ft., the effect of $4$~seconds by $256$~ft.,
the effect of $7$~seconds by $112$~ft., the effect of
$8$~seconds by $0$~ft. And to make it clear that we mean
to register only the net result of the whole lapse of
time in question, we might call this the ``total effect''
of so many seconds. In this case $f(x)$ will represent
the total effect of the lapse of $x$~seconds, regarded as
a condition affecting the height of the body. What,
then, will $f'(x)$ signify? It will signify, as always,
the rate at which $f(x)$ is increasing. That is to say,
it will signify the rate at which additions to the time
are at this point increasing the effect, \ie~the rate at
which the effect is growing. Now, since more time
must always be added on at the margin of the time
that has already elapsed, we may say that $f(x)$~represents
the \emph{total effect} of $x$~seconds of time in giving height
to the body, and that $f'(x)$~represents the \emph{effectiveness}
of time, added at the margin of $x$~seconds, in \emph{increasing}
the height. Or, briefly, $f(x) ={}$total effect, $f'(x) ={}$marginal
effectiveness.

Here the change of terms from ``effect'' to ``effectiveness''
may serve to remind us that in the two cases
we are dealing with two different kinds of magnitude---in
the one case \emph{space} measured in feet absolutely (effect),
in the other case \emph{rate} measured in feet-per-second.
%% -----File: 055.png---Folio 36-------

Before passing on to the economic interpretation of
all that has been said, we will deal very briefly with
another scientific illustration, which may serve as a
transition.

Suppose we have a carbon furnace in which the
carbon burns at a temperature of $1500°$~centigrade, and
suppose we are using it to heat a mass of air under
\begin{figure}[hbt]
  \begin{center}
    \Fig{7}
    \Input[4.5in]{055a}
  \end{center}
\end{figure}
given conditions. Obviously the temperature to which
we raise the air will be a function of the amount of
carbon we burn, and will be a function which will
increase as the variable increases; but not without
limit, for it can never exceed the temperature of~$1500°$.
Suppose the conditions are such that the first pound
%% -----File: 056.png---Folio 37-------
of carbon burnt raises the temperature of the air from
\index{Carbon@{\textsc{Carbon Furnace}}}%
$0°$ to $500°$, \ie~raises it one-third of the way from its
present temperature to that of the burning carbon, then
(neglecting certain corrections) the second pound of
carbon burnt will again raise the temperature one-third
of the way from its present point ($500°$) to that of
the carbon ($1500°$). That is to say, it will raise it to
$833.3°$; and so forth. Measuring the pounds of carbon
consumed along the axis of~$x$ and the degrees centigrade
to which the air is raised along the axis of~$y$
($100°$ to a unit), we may now represent the connection
between $f(x)$~and~$x$ by a curve.\footnote
  {The formula will be $y = f(x) = 15 \left\{1-(\frac{2}{3})^x \right\}$}
Its general
form may be seen in \Figref{7}, and we shall have the
total effect of the carbon in raising the temperature
represented by $f(x)$, and assuming the following values:---
\begin{align*}
f(0) &= 0           & f(4)  &= 12.04   & f(8)  &= 14.42 \\
f(1) &= 5[ = 500°]  & f(5)  &= 13.02   & f(9)  &= 14.61 \\
f(2) &= 8.3         & f(6)  &= 13.68   & f(10) &= 14.74 \\
f(3) &= 10.5        & f(7)  &= 14.12   & f(11) &= 14.83 \\
     &              & f(12) &= 14.88
\end{align*}

Now here, as before, we may proceed (either graphically,
see \Pageref{26}, or by aid of the rules of the calculus)
to construct a second curve, the curve of $x$~and~$f'(x)$,
which shall set forth the connection between $x$ and the
steepness of the first curve, \ie~the connection between
the value of~$x$ and the rate at which $f(x)$ is growing.\footnote
  {Its formula will be $15(\frac{2}{3})^x \log_e (\frac{3}{2})$.}
Again allowing $100°$ to the unit, measured on the axis
of~$y$, we shall obtain (\Figref{8})---
\begin{align*}
f'(0) &= 6.08  & f'(4) &=  1.2    & f'(8)  &= .24 \\
f'(1) &= 4.05  & f'(5) &= \Z.8    & f'(9)  &= .16 \\
f'(2) &= 2.7   & f'(6) &= \Z.53   & f'(10) &= .1 \\
f'(3) &= 1.8   & f'(7) &= \Z.35   & \rlap{\text{etc.}}\Z &
\end{align*}

What then will $f'(x)$ represent?  Here as always
%% -----File: 057.png---Folio 38-------
we have $f'(x) ={}$the rate at which $f(x)$~is growing. But
$f(x) ={}$the heat to which the air is raised, \ie~the total
effect of the carbon. Therefore $f'(x)$~is the rate at which
carbon, added at the margin, will increase the heat, or the
marginal effectiveness of carbon in raising the heat.
We have $x ={}$quantity of carbon burnt, $f(x) ={}$total effect
of~$x$ in raising the heat of the air, $f'(x) ={}$marginal effectiveness
of additions to~$x$.

Comparing the illustration of the heated air with
\begin{figure}[hbt]
  \begin{center}
    \Fig{8}
    \Input[3in]{057a}
  \end{center}
\end{figure}
that of the falling body we find that pounds of carbon
have taken the place of seconds of time as the variable,
total rise of temperature has taken the place of total
space traversed as the first function of the variable, rate
at which additions to carbon are increasing the temperature
has taken the place of rate at which additions
to the time allowed are increasing the space traversed,
as the derived function; but in both cases the derived
function represents the rate at which the first function
is growing, in both cases the first function represents
%% -----File: 058.png---Folio 39-------
the total efficiency of any given quantity of the variable,
and the derived function represents its effectiveness at
any selected margin, so that in both cases the relation
$f'(x)$~to~$f(x)$ is identical.\Pagelabel{39}%

And now at last we may return to the economic
interpretation of the curves.

Assuming that \Figref{1} (\Pageref{9}) represents the connection
between some economic function and its variable, as, for
example, the connection between the quantity of coal I
\index{Coal}%
burn and the sum of advantages or gratifications I
derive from it, and assuming further that one unit along
the axis of~$x$ is taken to mean one ton of coal per month,
we shall have no difficulty in reading \Figref{1} as follows:
$f(0) = 0$, \ie~if I burn no coal I get no benefit from
burning it; $f(1) = 11.2$, \ie~the total effect of burning
one ton of coal per month is represented by $11.2$~units
of satisfaction; $f(2) = 19.2$, \ie~the total effect of burning
two tons of coal a month is greater than that of
burning one ton a month, but not twice as great. The
difference to my comfort between burning no coal and
burning a ton a month is greater than the difference
between burning one and burning two tons. So again,
$f(4) = 25.6$, \ie~the total effect of four tons of coal per
month in adding to my comfort is represented by $25.6$~units
of gratification, and at this point its total effect is
at its maximum; for now I have as much coal as I
want, and if I were forced to burn more the total effect
of that greater quantity would be less than that of a
smaller quantity, or $f(5)$~is less than~$f(4)$. At last the
point would arrive at which if I were forced to choose between
burning, say, eight tons of coal a month and burning
none at all, I should be quite indifferent in the matter.
The total effect of eight tons of coal per month as a
direct instrument of comfort would then be nothing.
And if more yet were forced upon me at last I should
prefer the risk of dying of cold to the certainty of
being burned to death, and $f(x)$ would be a negative
quantity.
%% -----File: 059.png---Folio 40-------

\begin{Remark}
It must be observed that I am not here speaking of the
\emph{construction} of economic curves, but of their \emph{interpretation} supposing
we had them (see \Pageref{15}). But it will be seen presently
that the construction of such curves is quite conceivable
ideally, and that there is no absurdity involved in speaking
of so many units of gratification. It is extremely improbable,
however, that any actual economic curve would coincide with
that of \Figref{1} (see \Pageref{48}).
\end{Remark}

Such would be the interpretation of \Figref{1}, $f(x)$~being
read as the curve of quantity-and-total-effect of coal as
a producer of comfort under given conditions of consumption.
What then would be the interpretation of
\Figref{3} or~$f'(x)$? Obviously $f'(x)$, signifying the rate of
growth of~$f(x)$, or the ratio of the increase of~$f(x)$ to the
increase of~$x$ at any point, would mean the rate at which
an additional supply of coal is increasing my comfort,
or the marginal effectiveness of coal as a producer of
comfort to me. This marginal effectiveness of course
varies with the amount I already enjoy. That is to
say, $f'(x)$~assumes different values as $x$~changes. When
I have no coal, the marginal effectiveness is very high.
That is to say, increments of coal would add to my comfort
at a great rate, $f'(0)= 12.8$. When I already command
a ton a month further increments of coal would
add to my comfort at a less rapid rate, $f'(1) = 9.6$;
when I have four tons a month further increments would
not add to my comfort at all, $f'(4) = 0$, after that yet
further increments would detract from my comfort,
$f'(5)=-32$.

In thus interpreting Figs.\ \Figref[]{1}~and~\Figref[]{3} we have substituted
consumption of coal per month (measured in
tons), for lapse of time (measured in seconds), as our
variable; sum of advantages derived from consuming
the coal, for space traversed by the projectile, as $f(x)$,
or the total effect of the variable; and rate per unit
at which coal is increasing comfort, for rate per unit
at which time is increasing the space traversed, as
$f'(x)$, or the marginal effectiveness of the variable.
%% -----File: 060.png---Folio 41-------

If we call $f(x)$ the ``total utility'' of $x$~tons of coal
per month, we might call $f'(x)$ the ``marginal usefulness''
of coal when the supply is $x$~tons per month.

The reader should now turn back to \Pageref{33}, and
read the table of successive values of $f(x)$ and~$f'(x)$
with the subsequent comments and interpretations,
substituting the economic meanings of $x$, $f(x)$, and
$f'(x)$ for the physical ones throughout.

A similar re-reading of Figs.\ \Figref[]{7}~and~\Figref[]{8} will also be
instructive.

Before going on to the further consideration of the
total effect and marginal effectiveness of a commodity
as functions of the quantity possessed, it will be well
to point out a method of reading $f'(x)$ which will bring
it more nearly within the range of our ordinary experiences,
and make it stand for something more definitely
realisable by the practical intellect than can be the
case with the abstract idea of rate.\Pagelabel{41}%

Reverting to our first interpretation of \Figref{3}, we
remember that $f'(2)=64$ means that after the lapse
of $2$~seconds the body will be rising \emph{at the rate} of
$64$~ft.\ per~second; but it is entirely untrue that it will
actually rise $64$~ft.\ during the next following second.
We see by \Figref{1} that it will only rise $48$~ft.\ in that
second. This is because the rate, which was $64$~ft.\
per~second at the beginning of the second, has constantly
changed during the lapse of the second itself.
But the rate of $64$~ft.\ per~second is the same thing as the
rate of $6.4$~ft.\ per~tenth of a second (or per $.1$~second),
and this again is the same as the rate $.64$~ft.\ per $.01$~second,
or $.000064$~ft.\ per $.000001$~second, and I may
therefore read \Figref{3} thus: $f'(2)=64$, \ie~after the lapse
of $2$~seconds the body will be rising at the rate of
$64$~millionths of a foot per millionth of a second. Now,
we should have to allow many millionths of a second
to elapse before the rate of movement materially
altered, and therefore we may with a very close approximation
to the truth say that the rate of motion will
%% -----File: 061.png---Folio 42-------
be the same at the end as it was at the beginning
of the first millionth of a second, \ie\ $64$~millionths
of a foot per millionth of a second. Hence it will
be approximately true to say that during the next
millionth of a second the body will actually rise $64$~millionths
of a foot (compare \Pageref{20}).\Pagelabel{42}\footnote
  {It would be [assuming the formula to be absolutely true]
  $63.999984$ millionths of a foot.  The error, therefore, would be
  $\frac{16}{1000000}$ or $\frac{1}{62500}$ in~$64$.}
But a rise of
$64$~millionths of a foot would be a concrete \emph{effect; hence
if we translate the \textsc{effectiveness} of the variable into terms
of a small enough unit, it tells us within any degree of
accuracy we may demand the actual \textsc{effect} of the next small
increment of the variable}. This is expressed by saying
that ``in the limit'' each small increment actually produces
this effect; which means that by making the
increments small enough we may make the proposition
as nearly true as we like.

Thus [assuming the ordinary formula $y=16x^2$ to
be absolutely correct] it is nearly true to say that when
a body has been falling $2$~seconds it will fall $64$~millionths
of a foot in the next millionth of a second,
$128$~millionths of a foot in the next $2$~millionths of a
second, $64n$~millionths of a foot in the next $n$~millionths
of a second, so long as $n$~is an insignificant
number in comparison to one million. What is nearly
true when the unit is small and more and more nearly
true as the unit grows smaller is said to be ``true in
the limit, as the unit decreases.''

Marginal \emph{effectiveness} of the variable, then, may always
be read as marginal \emph{effect} per unit of very small units
of increment. And in this sense we shall generally
understand it. Total effect and unitary marginal effect
will then be magnitudes of the same nature or character;
and indeed the unitary marginal effect will itself
be a total effect in a certain sense, the total effect
namely of one small unit, added at that particular place.
Even when we are not dealing with small units we
%% -----File: 062.png---Folio 43-------
may still speak of the marginal effect of a unit of the
commodity, but in that case the effect of a unit of the
commodity at the margin of~$x$ will no longer correspond
closely to the marginal effectiveness of the commodity
at~$x$. It will correspond to the \emph{average} marginal effectiveness
of the commodity between~$x$, at which its
application begins, and $x + 1$, at which it ends. And if
the effect of the next unit after the~$a$\textsuperscript{th} is~$z$, it will probably
not be true (as it is in the case of small units)
that the effect of the next two units will be nearly~$2z$.
A reference to Figs.~\Figref[]{1}, \Figref[]{3}, \Figref[]{7},~\Figref[]{8}, and a comparison of
the last column and the last but one in the table of
\Pageref{4}, will sufficiently illustrate this point; and the
economic illustration of the next paragraph will furnish
an instance of the correspondence, in the limit, between
the effectiveness of the commodity and the effect of a
small unit.

\begin{Remark}
Reverting to Figs.\ \Figref[]{4}~and~\Figref[]{5} (\Pageref{25}) we have $Q_1 p_1$ in \Figref{4}
$= Q_1 P_1$ in \Figref{5}. But we have seen that if we start from $P_1$ in
\Figref{4} and move a very little way along the curve, the ratio of
the increment of~$x$ to the increment of~$y$ will be very nearly
$\dfrac{r_1 Q_1}{Q_ 1p_1}$; or in the limit $\dfrac{\text{increment of } x}{\text{increment of } y} = \dfrac{r_1 Q_1}{Q_1p_1}$. But $r_1 Q_1 = 1$,
therefore in the limit $\dfrac{\text{increment of } x}{\text{increment of } y} = \dfrac{1}{\DPtypo{Q}{Q_1} p_1}$ (\Figref{4}) $= \dfrac{1}{Q_1 P_1}$ (\Figref{5}),
or, in the limit, $Q_1 P_1 × \text{ increment of } x = \text{increment of } y$.
Now in \Figref{5} increments of~$x$ are measured along~$OX$, and
therefore (if we follow the ordinary system of interpretation)
we shall regard $Q_1 P_1 × \text{ increment of } x$, as an area, and it will
be seen that as $x$ decreases the area in question approximates
to a thin slice cut vertically from the triangle~$Q_1 P_1 P_3$. But we
have seen that areas cut in vertical slices out of this triangle
correspond to lengths in \Figref{4}, or portions of the total effect
of the variable. Thus if a small unit is taken, the \emph{effect} of
units of a commodity applied at any margin (\Figref{4}) is approximately
represented by the \emph{effectiveness} of the commodity
at that margin (\Figref{5}) multiplied by the number of units.
And in the limit this relation is said to hold absolutely
(compare pp.~\Pageref[]{21},~\Pageref[]{42}).
\end{Remark}
%% -----File: 063.png---Folio 44-------

The method of reading curves of quantity-and-marginal-effectiveness
as though they were curves of
quantity-and-marginal-effect may be illustrated by the
following example.

\Figref{9} represents part of the curve of quantity-and-marginal-effectiveness
\Pagelabel{44}%
of wheat in Great Britain, based
\index{Wheat}%
upon a celebrated estimate made about the beginning of
the eighteenth century.\footnote
  {The estimate is generally known as ``Gregory King's,'' and its
  formula is
  \[
  60y = 1500 - 374x + 33x^2 - x^3.
  \]
  }
In the figure the unit of~$x$ is
(roughly speaking) about $20$~millions of bushels; and if
\begin{figure}[hbt]
  \begin{center}
    \Fig{9}
    \Input[4.5in]{063a}
  \end{center}
\end{figure}
we place our quantity-index eleven units from the origin,
that will mean that we suppose the supply of wheat in
Great Britain to be $220$~millions of bushels per annum.
Our curve asserts that when we have that supply
additions of wheat will have an ``effectiveness'' in supplying
our wants represented by $.8$~per $20$~million
bushels; but we cannot translate the ``effectiveness''
into the actual ``effect'' which $20$~millions of bushels
%% -----File: 064.png---Folio 45-------
would have; because the ``effectiveness'' would not continue
the same if so large an addition were made to our
supply. On the contrary it would drop from $.8$ to~$.6$.
But $.8$~per $20,000,000$ bushels is $.00000008$~per $2$~bushels
and $.00000004$ per~bushel, and since the addition
of another bushel to the $220$~millions already
possessed will not materially affect the usefulness or
effectiveness of wheat at the margin, we may say that
that effectiveness remains constant during the consumption
of the bushel of wheat, and therefore, given
a supply of $20,000,000$ bushels a year, not only is the
``marginal effectiveness'' of wheat $.8$~per $20,000,000$
bushels or $.0000004$ per~bushel, but the ``marginal
effect'' of a bushel is~$.00000004$. Thus, if we had two
commodities, $W$~and~$V$, and curves of their quantity-and-marginal-usefulness
or effectiveness similar to that in
\Figref{9}, the vertical intercepts on the quantity-indices
would indicate the marginal usefulness per unit of the
two commodities, and if we then selected ``small'' units
of each commodity bearing in each case the same proportion
(say $1 : z$) to the unit to which the curve of the
commodity was drawn, we should then have the marginal
utility or effect of the small units of the two commodities
proportional to the length of the vertical intercepts, and
calling the small unit of~$W$, $w$, and the small unit of~$V$, $v$,
and the ratio of the marginal usefulness of~$W$ to that of~$V$,
$r$, we should have
\begin{align*}
\text{marginal utility of }
  w &= \Z r × \text{marginal utility of } v \\
\PadTo{\text{marginal}}{\Ditto} \PadTo{\text{utility of}}{\Ditto}
 2w &= 2r × \text{marginal utility of } v. \\
\PadTo{\text{marginal utility of }}{\text{etc.}}
    & \PadTo{2r × \text{marginal utility of } v}
            {\text{etc.}\ \makebox[0pt][l]{\text{(compare \Pageref{56})}}}
\end{align*}

We shall make it a convention henceforth to use
Roman capitals $A$,~$X$, $W$,~etc., to signify commodities,
italic minuscules $a$,~$x$, $w$,~etc., to signify units of these
commodities (generally ``small'' units in the sense explained),
and italic capitals, \Person{A}, \Person{B}, etc., to signify persons.
Thus we shall speak of the marginal \emph{usefulness} or \emph{effectiveness}
of $A$,~$W$,~etc., and the marginal \emph{utility} or \emph{effect} of
$a$,~$w$,~etc.
%% -----File: 065.png---Folio 46-------

What precise interpretation we are to give to our
``units of satisfaction'' or ``utility'' measured on the
axis of~$y$ is another matter, the consideration of which
must be reserved for a later stage of our inquiry (see
pp.~\Pageref[]{52},~\Pageref[]{78}).

\begin{Remark}
Jevons uses the terms ``total utility'' and ``final degree of
utility,'' meaning by the latter what I have termed ``marginal
usefulness'' or ``marginal effectiveness.'' His terminology
hardly admits of sufficient distinction between ``marginal
effectiveness,'' \ie~the \emph{rate} per unit at which the commodity is
satisfying desire, and the ``marginal effect'' of a unit of the
commodity, \ie~the actual result which it produces when
applied at the margin. I think this has sometimes confused
his readers, and I hope that my attempt to preserve the distinction
will not be found vexatious. Note that the curves
are always curves of quantity-and-marginal-usefulness, but
that we can read them with more or less accuracy according
to the smallness of the supposed increment into curves of
quantity-and-marginal-utility for small increments.\Pagelabel{46}%
\end{Remark}

If the reader has now gained a precise idea of the
total utility or effect and the marginal usefulness of
commodities, he will see without difficulty that when
we take a broad general view of life we are chiefly
concerned with those commodities the total utility of
which (or their total effect in securing comfort, giving
pleasure, averting suffering, etc.)\ is high. In considering
from a general point of view our own material
welfare or that of a nation, our first inquiries will
concern the necessaries of life, food, water, clothing,
shelter, fuel. For these are the things a moderate
supply of which has the highest total utility. The
sum of advantages we derive from them collectively
is, indeed, no other than the advantage of the life they
support. This is what economists have in view when
they speak of the ``value in use'' of such a commodity
as water, and say that nothing is more ``useful'' than
it. They mean that the total advantage derived from
%% -----File: 066.p n g----------
%[Blank Page]
%% -----File: 067.p n g----------
\begin{figure}[p]
\Pagelabel{47}%
\begin{center}
  \Fig{10}
  \Input[4.5in]{067a}
  \vfil
%[** TN: Book's graph of 30/(15 + x) - 1 not perfectly accurate]
  \Fig{11}
  \Input[3.75in]{067b}
\end{center}
\end{figure}
%To face page 47.
%% -----File: 068.png---Folio 47-------
even a small supply of water, the total difference
\index{Water}%
between a little water and no water, is enormously
great. The graphical expression of this would be a
curve (connecting the total utility of water with its
quantity) which would rise rapidly and to a great height.

But if it is obvious that when we look upon life as
a whole, and in the abstract, we are chiefly concerned
with total utilities, and ask what are the commodities
we could least afford to dispense with altogether, it is
equally obvious that in detail and in concrete practice
we are chiefly concerned not with the total utility but
the marginal usefulness of things, or rather, their marginal
utility; and we ask, not what is my whole stock
of such a commodity worth to me, but how much would
a little \emph{more} of it \emph{add} to my satisfaction or a little less
of it detract therefrom. For instance, we do not ask,
What is the total advantage I derive from all the water
I can command, but what additional advantage should I
derive from the extra supply of water for a bath-room,
\index{Bath-room@{\textsc{Bath-room}}}%
or for a garden hose? Materfamilias does not ask
\index{Garden-hose}%
what advantage she derives from having a kitchen fire,
\index{Kitchen@{\textsc{Kitchen Fire}}}%
but she asks, what additional advantage she would
derive by keeping up her kitchen fire after dinner, by
heating the oven every day, or by always letting the
\index{Fire@{Fire in ``practising'' room}}%
girls have a fire in the room when they are ``practising.''
Or inversely, we do not ask what disadvantage we
should incur by ceasing to burn coal, but what disadvantage
\index{Coal}%
we should incur by letting our fires go down
earlier in the day, or having fewer of them. And note
that this inquiry as to marginal usefulness of a commodity
is made on its own merits, and wholly without
reference to the total utilities of the articles in question.
The fact that I should be much worse off without
clothes than without books does not make me spend
fifteen shillings on a new waistcoat instead of on
\index{Waistcoat@{\textsc{Waistcoat}}}%
Rossetti's works, if I think that the latter will \emph{add} more
\index{Rossetti's Works}%
to my comfort and enjoyment than the former. For
$f(\text{clothes})$ may be as much bigger than $\phi(\text{books})$ as it
%% -----File: 069.png---Folio 48-------
likes, but if $f'(\text{clothes})$ is smaller than~$\phi'(\text{books})$ I shall
spend the money on the books. So much is this the
case that we habitually lose sight of the connection
between $\phi'(\text{books})$ and~$\phi(\text{books})$, between $f'(\text{clothes})$
and~$f(\text{clothes})$, and do not think, for instance, of
$\phi'(\text{books})$ as marking the rate at which additional books
increase the gratification \emph{we derive from books}, but simply
as marking the rate at which they increase our gratification
in general.

\Pagelabel{48}%
Before developing certain consequences of the principles
we have been examining, let us try to get a
better representation of our supposed economic functions
than is supplied by the diagram of a projected body.
It will be remembered that we saw reason to think
that a large class of economic functions, representing
total utilities, would bear an analogy to our \Figref{1} in
so far as they would first increase and then decrease
as the variable (\ie~the supply of the commodity)
increased. But it is highly improbable that any
economic curve would increase and decrease in the
symmetrical manner there represented. It is not
likely, for instance, that the inconvenience of having a
unit too much of a commodity would be exactly equivalent
to the inconvenience of having a unit too little.
As a rule it would be decidedly less. Our economic
functions, then, will, in many instances, rise more rapidly
than they fall. The connection of such a function and
its variable is represented by the upper curve on
\Figref{10},\footnote
  {The conditions stated in the text will be complied with by a
  function of the form $a \log_e {(x + b)} - \log b - x$; and there are some
  theoretical reasons for thinking that such a function may be a fair approximation
  to some classes of actual economic functions. The
  upper curve in \Figref{\DPtypo{9}{10}} is drawn to the formula $y=11 \log_e{(x+1)}-x$.}
which rises rapidly at first, then rises slowly,
and then falls more slowly still. Household linen
\index{Linen}%
might give a curve something of this character. It is
not exactly a necessary of life, but the sum of advantages
conferred by even a small stock is great. The
rate at which additions to the stock add to its total
%% -----File: 070.png---Folio 49-------
utility is at first rapid, but it declines pretty quickly.
At last we should have as much as we wanted and
should find it positively inconvenient to stow away any
more. The excess, however, would have to be very
great indeed in order to reduce us to a condition as
deplorable as if we had no linen at all. By way of
practice in interpreting economic curves, let us suppose
the unit of household linen, measured along the base
line, to be such an amount as might be purchased for~£3.
The curve would then represent the following
case, which might well be that of a young housekeeper
\index{Housekeeper}%
with a four or five roomed cottage, and not much
space for storage: Household linen (sheets, tablecloths,
towels, etc.)\ to the amount of some £6~or~£10 worth
($x = 2$ or~$3\frac{1}{3}$) is little short of a necessity. After this
additions to the stock, though very acceptable, are not
so urgently needed, and when the stock has reached
£18~or~£20 worth ($x = 6$ or~$6\frac{2}{3}$) our housekeeper will
consider herself very well supplied, and will scarcely
desire more. Still, if she could get it for nothing, she
would be glad to find room for it up to, say, £30~worth
($x = 10$). If after this any one should offer her a present
of more she would prefer to find a polite excuse for not
accepting it, but would not be much troubled if she had to
take it, unless the amount were very large;\footnote
  {We are supposing throughout that the conditions exclude sale or
  barter of the unvalued part of the stock.}
but when
the total stock had reached, say,~£45 ($x = 15$), the inconvenience
would become serious, and our heroine, on the
whole, would be nearly as hard put to it by having £15~worth
too much as she would have been by having £12~worth
too little. If her stock were still increased till
it reached £60~worth ($x = 20$) she would be as badly
off as if she had only £11~:~8s.\ worth ($x = 3\frac{4}{5}$). At this
point our ``epic of the hearth'' breaks off.

We may, of course, apply to this curve the process with
which we are already familiar, and may find the derived
function which represents the marginal effectiveness or
%% -----File: 071.png---Folio 50-------
usefulness of linen, that is to say, the rate at which
increments of linen are increasing the sum of advantages
derived from it. This marginal effectiveness or
usefulness of linen is set forth on the higher curve in
\Figref{11};\footnote
  {Its formula is $\dfrac{11}{x + 1} - 1$.}
on which may be read the facts already
elaborated in connection with the curve on \Figref{10},
the only difference being that the specific increase
between any values of~$x$ is more easily read on \Figref{10},
and the \emph{rate} of increase at any point more easily read
on \Figref{11}.

\Pagelabel{50}%
An analogous pair of curves, with other constants,\footnote
  {See \Pageref{9}.}
may be found in the lower lines in Figs.\ \Figref[]{10}~and~\Figref[]{11}.\footnote
  {They are drawn to the formulæ $y = 30 \log_e (x + 15) - \log_e 15 - x$
  and $y = \dfrac{30}{x + 15} - 1$ respectively.}
They might represent respectively the total utility and
the marginal usefulness of china, for example. In \Figref{10}
\index{China}%
the lower curve does not rise so rapidly or so high as
the other. That is to say, we suppose the total advantage
derived from as much china as one would care to
have to be far less than that derived from a similarly
full supply of household linen. To be totally deprived
of china (not including coarse crockery in the term)
would be a less privation than to be totally deprived
of linen. But we also observe that at a certain point,
when the curve of linen is rising very slowly, the curve
of china is rising rather more rapidly. That is to say,
if our supplies of both linen and china increase \textit{pari
passu}, unit for unit (£3~worth is the unit we have supposed),
then there comes a point at which increments of
china would add to our enjoyment at a greater rate than
similar increments of linen, although in the mass the
linen has done much more to make us comfortable than
the china.

On the curves of \Figref{11} this point is indicated by
the point at which the curve of the marginal usefulness
%% -----File: 072.png---Folio 51-------
of china crosses, and thenceforth runs above, the curve
of the marginal usefulness of linen.

Now if I possess a certain stock of linen and a
certain stock of china, and am in doubt as to the
use to make of an opportunity which presents itself
for adding in certain proportions to either or both,
how will the problem present itself to me? I shall
not concern myself at all with the total utilities,
but shall simply ask, ``Will the quantity of linen or the
quantity of china I can now secure \emph{add} most to my
satisfaction.'' The total gratification I derive from the
two articles together is made up of their two total utilities
(represented by two straight lines, viz.\ the vertical intercepts
made by the two curves on \Figref{10}), and it is
indifferent to me whether I increase the one already
greatest or the other, as long as the increase is the
the same. I therefore ask not which curve is the \emph{highest},
but which is the \emph{steepest} at the points I have reached on
them respectively, or since the curves on \Figref{11} represent
the steepness of those on \Figref{10}, I ask which of
these is highest. In other words, I examine the~$f'(x)$'s,
not the~$f(x)$'s; I compare the marginal usefulness and
not the total utilities of the two commodities. If
the choice is between one small unit of china and one
similar unit of linen, I shall ask ``Which of the two has
the higher marginal utility.'' If my stock of both is
low, the answer will be ``linen.'' If my stock of both is
high, it will be ``china.'' If, on the other hand, the
choice is between one small unit of china and \emph{two} similar
units of linen, the question will be ``Is the marginal
effectiveness of china \emph{twice} as great as that of linen,'' if
not I shall choose the linen, since double the amount at
anything more than half the effectiveness gives a balance
of effect over what the other alternative would yield.
If it seems difficult to imagine the mental process by
which one thing shall be pronounced exactly \emph{twice} as
useful as another, we may express the same thing in
other terms by asking whether half a small unit of china
%% -----File: 073.png---Folio 52-------
is as useful to us (or is worth as much to us) as one
small unit of linen, thus transferring the inequality from
the utilities to the quantities, and the equality from the
quantities to the utilities.\footnote
  {Observe that this transfer can only be made in the case of \emph{small}
  units, for it assumes that half a unit of china is half as useful as a
  whole unit, which implies that the marginal usefulness of china
  remains the same throughout the unit.}

\Pagelabel{52}%
Such considerations as these spontaneously solve the
problem that suggested itself at the threshold of our
inquiries (\Pageref{15}) as to the theoretical possibility of fixing
a unit of utility or satisfaction, and so theoretically
constructing economic curves. We now see clearly
enough that though our psychological arithmetic is so
little developed that the simplest sums in hedonistic
multiplication or division seem impossible and even
absurd, yet, as a matter of fact, we are constantly comparing
and weighing against each other the most heterogeneous
satisfactions and determining which is the
greater. The enjoyment of fresh air and friendship, of
\index{Air, fresh}%
\index{Friendship}%
fresh eggs and opportunities of study, all in definite
\index{Eggs@{\textsc{Eggs}, fresh}}%
quantities, are weighed against each other when we
canvass the advantages of residence in London within
reach of our friends and the British Museum and residence
\index{Museum, British}%
in the country with fresh air and fresh eggs.
Nay, we may even regard space and time as commodities
each with its varying marginal usefulness. This year I
eagerly accept a present of books which will occupy a
\index{Books}%
great deal of space in my house, but will save me an
occasional journey to the library; for the marginal
usefulness of my space and of my time are such that I
find an advantage in losing space and gaining time
under given conditions of exchange. Next year my
space is more contracted, and its marginal usefulness is
therefore higher; so I decline a similar present, preferring
the occasional loss of half an hour to the permanent
cramping of my movements in my own study.

Thus we see that the most absolutely heterogeneous
%% -----File: 074.png---Folio 53-------
satisfactions are capable of being practically equated
against one another, and therefore may be regarded as
theoretically \emph{reducible to a common measure}, and consequently
capable of being measured off in lengths, and
connected by a curve with the lengths representing the
quantities of commodity to which they correspond.
We might, for instance, take the effort of doing a given
amount of work as the standard unit by which to estimate
the magnitude of satisfaction. Hence the truth of
the remark, ``Pleasures cannot be measured in feet, and
they cannot be measured in pounds; but they can be
measured in foot-pounds'' (Launhardt). If I only had
\index{Foot-tons}%
one ton of coal per month, how much lifting work should
\index{Coal}%
I be willing to do for a hundredweight of coal? If I
had two tons a month, how much lifting work would
I then do for a hundredweight? Definite answers to
these two questions and other similar ones are conceivable;
and they would furnish material for a curve
on which the utility of one, two, three,~etc.\ hundredweight
of coal per month would be estimated in foot-pounds.
In academical circles it is not unusual to take an hour of
correcting examination papers as the standard measure
\index{Examination papers}%
of pleasures and pains. A pleasure to secure which a
man would be willing to correct examination papers for
six hours (choosing his time and not necessarily working
continuously) must be regarded as six times as great
as one for which he would only correct papers for an
hour. If we wished to reduce satisfactions so estimated
to the foot-pound standard, we should only have to
ascertain in the case of each of the university dignitaries
in question how many foot-pounds of heaving work he
would undertake in order to escape an hour's work at
the examination mill. Obviously this change of measure
would not affect the \emph{relative} magnitudes of the satisfactions
already estimated on the other scale. It does not,
then, matter what we suppose the standard unit of satisfaction
to be, provided we retain it unchanged throughout
any set of investigations.
%% -----File: 075.png---Folio 54-------

\begin{Remark}
It should be noted that to be theoretically accurate we
must not suppose the quantity of work offered for the same
quantity of the commodity to change over different parts of
the curve, but rather the quantity of the commodity for
which the same fixed quantity of work is offered. For if
we change the quantity of work, we thereby generally
change its hedonistic value per unit also, inasmuch as $400$~foot-tons
\index{Foot-tons}%
of work, for instance, would generally be more than
twice as irksome as $200$~foot-tons.

In working out an imaginary example, however, we will
ignore this fact, and will suppose the hedonistic value of $100$~foot-tons
to be constant. Let us, then, suppose that a householder
would be willing to do $3300$ foot-tons of work\footnote
  {An ordinary day's work is reckoned at $300$~foot-tons; a dock
  labourer does~$325$ (Mulhall).}
for a
certain amount of linen, if he could not get it any other way.
\index{Linen}%
We will reckon that amount of linen the unit, and calling~$x$
the amount of linen and $y$ its total utility, we shall have for
$x=1$ $y=3300$, or allowing $500$~foot-tons to the unit of~$y$,
$x=1$ $y=6.6$. Now suppose that having secured one unit,
our householder would be willing to do $1750$ foot-tons of
work for a second unit, but not more. This would be represented
on our scale by~$3.5$, which, added to the previous~$6.6$,
would give $y=10.1$ for~$x=2$. For yet another unit of linen,
perhaps no more than $1125$ foot-tons would be offered, represented
by~$2.2$ on our scale, or $y=12.3$ for~$x=3$, etc. On comparing
these suppositions with \Figref{10} (\Pageref{47}), it will be found
that this case would be graphically represented by the upper
curve of that figure. It will be seen that though we have
imagined an ideally perfect and exact power of estimating
what one would be willing to do under given circumstances
in order to secure a certain object of desire, yet there is
nothing theoretically absurd in the imaginary process; so
that the construction of economic curves may henceforth be
regarded as theoretically possible.

The reader may find it interesting to attempt to construct
the economic curves that depict the history of some of his
own wants. Taking some such article as coffee or tobacco,
let him ask himself how much work he would do for a single
cup or pipe per week or per day sooner than go entirely
%% -----File: 076.png---Folio 55-------
without, how much for a second, etc., and dotting down
the results, see whether they seem to follow any law and
form any regular curve. If they do not, it probably shows
that his imagination is not sufficiently vivid and accurate to
enable him to realise approximately what he would be willing
to do under varying circumstances. In any case he will
probably soon convince himself of the perfect theoretical
legitimacy of thus supposing actual concrete economic curves
to be constructed. But even if he cannot tell what amount
of work he would be willing to do under the varying circumstances,
obviously \emph{there is} a given amount, which, as a matter
of fact, he would be willing to do under any given circumstances.
Thus the curve \emph{really exists}, whether he is able to
trace it or not.\Pagelabel{55}%
\end{Remark}

We may now return to our curves with a clear conscience,
knowing that for any object of desire at any
moment there actually exists a curve (could we but get
at it) representing the complete history of the varying
total utility that would accompany the varying quantity
possessed. The man who knows most nearly what that
curve is, in each case, has the most powerful and
accurate economic imagination, and is best able to predict
what his expenditure, habits of work, etc.\ would
be under changed circumstances.

We have now actually constructed some hypothetical
curves (pp.\ \Pageref[]{48}, \Pageref[]{50}), and have shown that there are certain
properties, easy to represent, which a large class of
economic curves must have (pp.~\Pageref[]{15},~\Pageref[]{48}); and we have
further shown that we are practically engaged, from
day to day, in considering and comparing the marginal
utilities of units of heterogeneous articles, that is to say,
in constructing and comparing fragments of economic
curves.

We have seen, too, that if I had a chance of getting
more china or more linen I should not consider the total
utilities of these commodities, but the marginal utilities
of the respective quantities between which the option
lay.
%% -----File: 077.png---Folio 56-------

And so, too, if I had the opportunity of exchanging a
\Pagelabel{56}%
given quantity of china for a given quantity of linen, or the
\index{China}%
\index{Linen}%
reverse, I should consider the marginal utilities of those
quantities. Thus we see that the \emph{equivalence in worth} to
me of units of two commodities is measured by their marginal,
not their total, utilities, and in the limit (\Pageref{44}) is
directly proportional to their marginal effectiveness or usefulness.
If, for the stocks I possess, the marginal usefulness of
linen is twice as great as that of china, \ie~if $f'(\text{linen}) = 2\phi'
(\text{china})$, then I shall be glad to sacrifice small units of china
in order to secure similar units of linen at anything up to the
rate of two to one. But this very process, by decreasing my
stock of china and increasing my stock of linen, will depress
the marginal usefulness of the latter and increase that of the
former, so that now we have
\[
f'(\text{linen})<2\phi'(\text{china}).
\]
If, however,
\[
f'(\text{linen})>\phi'(\text{china})
\]
is still true, I shall still wish to sacrifice china for the sake
of linen, unit for unit, until by the action of the same principle
we have reached the point at which we have
\[
f'(\text{linen})=\phi'(\text{china}).
\]
After this I shall not be willing to sacrifice china for the
sake of obtaining linen unless I can obtain a unit of linen by
foregoing \emph{less} than a unit of china. All this may be represented
very simply and clearly on our diagrams. Drawing
out separately, for convenience, the curves given in \Figref{11},
and making any assumptions we choose as to quantities of
linen and china possessed, we may read at once (\Figref{12}) the
\emph{equivalents in worth} (to the possessor) of linen and china. Thus
if I have eight units of china [$\phi'(\text{china})=.3$] and four units
of linen [$f'(\text{linen})=1.2$]; then in the limit one small unit
of linen at the margin is equivalent in worth to four small
units of china at the margin. If I have seven units of linen
and two of china, then one small unit of china at the margin is
equivalent in worth to two small units of linen at the margin.

Hitherto we have spoken of foot-tons, or generally of
work, merely as a standard by which to measure a man's
%% -----File: 078.png---Folio 57-------
estimate of the various objects of his desire; but we
know, as a matter of fact, that work is often a \emph{means of
securing} these objects, and it by no means follows that
\begin{figure}[htbp]
  \begin{center}
    \Fig{12}
    \Input[3.5in]{078a} \\
    \Input[4.5in]{078b}
  \end{center}
\end{figure}
the precise amount of work a man would be willing to do
rather than go without a thing is also the precise amount
of work he will have to do in order to make it. Indeed
there is no reason in general why a man should have to
%% -----File: 079.png---Folio 58-------
do either more or less work for the first unit of a commodity
with its high utility than for the last with its
comparatively low utility. The question then arises:
On what principle will a man distribute his work
between two objects of desire? In other words, If a
man can make two different things which he wants, in
what proportions will he make them?

\Pagelabel{58}%
We must begin by drawing out the curves of quantity-and-marginal-usefulness
of the two commodities, and we
will select as the unit on the axis of~$x$ in each case that
quantity of the commodity that can be made or got by
an hour's work. Suppose Robinson Crusoe\footnote
  {``Political economists have always been addicted to Robinsoniads''
  (Marx).}
\index{Robinson Crusoe}%
\index{Root-digging}%
\index{Rush-gathering}%
has provided
himself with the absolute necessaries of life, but
finds that he can vary his diet by digging for esculent
roots, and can add to the comfort and beauty of his hut
by gathering fresh rushes to strew on the floor two or
three times a week. Adopting any arbitrary standard
unit of satisfaction, let us suppose that the marginal
usefulness of the roots begins at six and would be extinguished
(for the week, let us say) when eight hours'
work had been done. That is to say, the quantity
which Robinson could dig in eight hours would absolutely
satisfy him for a week, so that he would not care
for more even if he could get them for nothing. In like
manner let the marginal usefulness of rushes begin at
four and be extinguished (for the week) by five hours'
work; and let the other data be such as are depicted on
the two curves in \Figref{13}.\footnote
  {They are drawn to the formulæ---
  \[
  y=\frac{24-3x}{4+x} \text{ and } y=\frac{40-8x}{10+7x} \text{ respectively}.
  \]}
Now suppose further that
Robinson can give seven hours a week to the two tasks
together. How will he distribute his labour between
them? If he gives four hours' work to digging for roots
and three to gathering rushes, the marginal usefulness
of the two articles will be measured by the vertical
intercepts on $a$~and~$a'$ respectively.  Clearly there has
%% -----File: 080.png---Folio 59-------
been waste, for the latter portions of the time devoted
to rush-gathering have been devoted to producing a
thing less urgently needed than a further supply of roots.
Again, if six hours be given to digging and one to rush-gathering,
the marginal usefulness will be measured by
the vertical intercepts on $b$~and~$b'$, and again there
has been waste, this time from excessive root digging.
But if five hours are given to digging for roots and two
to rush-gathering, the usefulness will be measured
by the vertical intercepts on $c$~and~$c'$, and there is no
loss, for obviously any labour subtracted from either
\begin{figure}[hbt]
  \begin{center}
    \Fig{13}
    \Input{080a}
  \end{center}
\end{figure}
occupation and added to the other would result in the
sacrifice of a greater satisfaction than the one it secured.

It is obvious that for any given time, such as three
hours or two hours, there is a similar ideal distribution
between the two occupations which secures the maximum
result in gratification of desires; and the method
of distribution may be represented by a very simple and
beautiful graphic device, exemplified in \Figref{14}.

First draw the two curves one within the other,\footnote
  {If the curves should cross, as in \Figref{10}, the principle is entirely
  unaffected.}
then add them together sideways, so as to make a
%% -----File: 081.png---Folio 60-------
third curve (dotted in figure), after the following fashion:
For $y=1$ the corresponding value of~$x$ for the inner
curve is~$2$, and that for the outer curve~$5$. Adding
these two together we obtain~$7$; and for our new curve
we shall have
\[
y=1 \qquad x=7.
\]
Every other point of the new curve may be found in
the same way, and we shall then have a dotted curve such
that if any line~$pp_{1}p_{2}p_{3}$ be drawn parallel to the axis
of~$x$, and cutting the three curves, the line~$p_{2}p_{3}$ shall
be equal to the line~$pp_{1}$. We shall then have $pp_{3}=pp_{1}+pp_{2}$;\footnote
  {If the curves are drawn to the formulæ $y=f(x)$ and $y=\phi(x)$ we
  may express them also as $x=f^{-1}(y)$ and $x=\phi^{-1}(y)$. It is obvious
  that our new curve will then be $x=f'^{-1}(y)+\phi^{-1}(y)$, which in this
  case will give $x=\dfrac{312+146y-38y^{2}}{24+29y+7y^{2}}$ to which formula the curve is drawn
  between the values $y=4$ and $y=0$.}
and if we desire to see how Robinson will
\begin{figure}[hbt]
\Pagelabel{60}%
  \begin{center}
    \Fig{14}
    \Input[4in]{081a}
  \end{center}
\end{figure}
apportion any quantity of time~$Oq_{3}$ between the two
\index{Time, distribution of}%
occupations we shall simply have to erect a perpendicular
at~$q_{3}$, and where it cuts the dotted curve draw a parallel
to the axis of~$x$, cutting the other curves at $p_{2}$~and~$p_{1}$.
We shall then have divided the whole time of~$Oq_{3}$ into
%% -----File: 082.png---Folio 61-------
two parts, $Oq_{1}$~and~$Oq_{2}$ ($=pp_{1}$~and~$pp_{2}$), such that if $Oq_{1}$
is devoted to the one occupation and $Oq_{2}$ to the other
the maximum satisfaction will be secured.

If we take $Oq_{3}=7$ we shall find we get $Oq_{1}=2$, $Oq_{2}=5$,
as above.\footnote
  {Note that when the hours of work have been distributed between
  the two occupations they pass into concrete results in the shape of
  commodity. Thus, strictly speaking, we measure \emph{hours} along the axis
  of~$x$ when dealing with the dotted curve, but \emph{hour-results} in commodity
  when we come to the other curves. If $Oq_{3}=7$, then, whereas
  $Oq_{3}=7 \text{\emph{ hours}}$, $Oq_{1}$~and~$Oq_{2}$ represent respectively $2$~and~$5$~\emph{units of
  commodity}, each unit being the result of an hour's work.}

\begin{Remark}
This is a principle of the utmost importance, applicable to
a great variety of problems, such as the most advantageous
distribution of a given quantity of any commodity between
two or more different uses. It is particularly important in
the pure theory of the currency. It need hardly be pointed
out that these diagrams do not pretend to assist any one in
practically determining how to divide his time. They are
merely intended to throw light on the process by which he
effects the distribution. In any concrete investigation we
should have direct access to the result but not to the conditions
of want and estimated satisfaction which determine
it; so that the actual distributions would be our data and
the preceding conditions of desire, etc.~our quæsita.\Pagelabel{61}%
\end{Remark}

We have now reached a stage of our investigations
at which it will be useful to recapitulate and expand
our conclusions as to the marginal usefulness of commodities.
In doing so we must bear in mind especially
what has been said as to the nature of our diagrammatic
curves (\Pageref{12}). The law of a curve is the law of the
connection between the corresponding pairs of values of
two varying quantities, one of which is a function of the
other. The curve on \Figref{7}, for instance, is not the
``curve of the heat produced by given quantities of
carbon in a furnace,'' nor yet the ``curve of the quantities
of carbon which effect given degrees of heat in a
furnace,'' but ``the curve of the connection between
varying quantities of carbon burned and varying degrees
%% -----File: 083.png---Folio 62-------
of heat produced,'' each of which magnitudes severally is
always measured by a vertical or horizontal straight line.

\Pagelabel{62}%
In like manner, the first curve in \Figref{13} is not ``the
curve of the varying marginal usefulness of esculent
roots to Robinson at given margins,'' nor ``the curve
of the varying quantities of esculent roots which
correspond to given marginal usefulnesses,'' but ``the
curve of the connection between the quantity of roots
Robinson possesses and the marginal usefulness of roots
to him.''

When this fact is fully grasped it will become obvious
that there are only two things which can conceivably
alter the marginal usefulness of a commodity to me:
either the quantity I possess must change, or the law
must change which connects that quantity and the
marginal usefulness of the commodity. If \emph{both} these
remain the same, obviously the marginal utility must
remain the same. Or, in symbols, if $y=f(x)$\footnote
  {Note that the symbol $f(x)$ is perfectly general, and signifies any
  kind of function of~$x$. It therefore includes and may properly represent
  the class of functions we have hitherto represented by letters with
  a dash, $f'(x)$, $\phi'(x)$, etc.}
the value
of~$y$ can only be altered by changing the value of~$x$, or
by changing the function signified by~$f$. The necessity
for insisting upon this axiomatic truth will become
evident as we proceed. Meanwhile,
\begin{center}
\begin{tabular}{l}
One charge, one sovereign charge I press,\\
And stamp it with reiterate stress,
\end{tabular}
\end{center}
viz.~to bear in mind, so as to recognise it under all disguises,
the fundamental and self-evident truth, that the
marginal usefulness of a commodity always depends
upon the quantity of the commodity possessed [$y=f(x)$],
and that if the \emph{nature of the dependence} [the form of
the function~$f$] and the quantity of the commodity
possessed [the value of~$x$] remain the same, then the
marginal usefulness of the commodity [the value of~$y$]
likewise remains unchanged. Whatever changes it must
%% -----File: 084.png---Folio 63-------
do so either by changing the nature of its dependence
upon the quantity possessed or by changing that quantity
itself; nothing which cannot change either of
these can change the marginal usefulness; and whatever
changes the marginal usefulness does so by means
of changing one of these. The length of the vertical
intercept cannot change unless \emph{either} the course of the
curve changes \emph{or} the position of the bearer is shifted.

These remarks, of course, apply to total utility as
well as to marginal usefulness.

Now, hitherto we have considered changes in the
quantity possessed only; and have supposed the nature
of the connection between the quantity and the total
utility or marginal usefulness to remain constant, \ie~we
have shifted our bearers, but have supposed our
curves to remain fixed in their forms. But obviously
\Pagelabel{63}%
in practical life it is quite as important to consider the
shifting of the curve as the shifting of the bearer and
the quantity-index. To revert to our first example.
The law that connects the quantity of coal I burn with
\index{Coal}%
the sum of advantages I derive from its consumption is
not the same in winter and in summer, or in the house
I now live in and the house I left ten years ago. And
in other cases, where there is a less obvious external
cause of change, a man's tastes and desires are nevertheless
perpetually varying. The state of his health, the
state of his affections, the nature of his studies, and a
thousand other causes change the amount of enjoyment
or advantage he can derive from a given quantity of a
given commodity; and if we wish to have an adequate
conception of the real economic conditions of life we
must not only imagine what we have called the ``bearer,''
that carries the vertical or quantity-index moving freely
along the axis of~$x$, but we must also imagine the form
of the curve to be perpetually flowing and changing.

\begin{Remark}
The obvious impossibility of adequately representing on
diagrams the flux and change of the curves presents a great
%% -----File: 085.png---Folio 64-------
difficulty to the demonstrator. Some attempt will here be
made to convey to the reader an elementary conception of
the nature of these changes.

We will take the simplest case, that of the straight line,
as an illustration. Suppose (a not very probable supposition)
that the quantity-and-marginal-usefulness curve of a certain
commodity for a certain man at a certain time is represented
by
\[
y=12-2x.
\]
By giving successive values to~$x$ we shall find the corresponding
\begin{figure}[hbt]
  \begin{center}
    \Fig{15}
    \Input{085a}
  \end{center}
\end{figure}
values of~$y$, and shall see that the curve is the
highest of the straight lines represented on \Figref{15}~(\textit{a}). Now
suppose that, owing to some cause or other, the man comes
to need the commodity less, so that its marginal utility,
while still decreasing by the same law as before, shall now
begin at ten instead of twelve. The formula of the curve
will then be $y=10-2x$, and the curve will be the second
straight line in \Figref{15}~(\textit{a}). By taking the formula, $y=8-2x$,
we may obtain yet another line, and so on indefinitely.
%% -----File: 086.png---Folio 65-------

What we have now been doing may be represented by the
formula
\[
y=f(z,x)=z-2x,
\]
where $y$ is a function of two variables, namely $z$~and~$x$, and
we proceed by giving $z$ successive values, and then for each
several value of~$z$ giving $x$ successive values. If instead of
taking the values $12$, $10$, $8$ for~$z$, we suppose it to pass continuously
through all values, it is obvious that we should
have a system of parallel straight lines, one of which would
pass through any given point on the axis of $x$ or~$y$.

But we have supposed the modifications in the position of
the line always to be of one perfectly simple character;
whereas it is easy to imagine that the man whose wants we
are considering might find that for some reason he needed a
smaller and smaller quantity of the commodity in question
completely to satisfy his wants, whereas his initial desire
remained as keen as ever. Such a case would be represented
by
\[
y=f(z,x)=12-zx,
\]
in which we may give $z$ the values of $2$, $3$, $4$, $6$ successively,
and then trace the lines in \Figref{15}~(\textit{b}) by making $x$~pass
through all values from $0$ to~$\dfrac{12}{z}$, after which the values of~$y$
would be negative.

But again we might suppose that while the quantity of
the commodity needed completely to sate a man remained
the same, the eagerness of his initial desire might abate.
This case might be represented by
\[
y=f(z,x)=z-\frac{z}{6}x,
\]
where by making $z$ successively equal to $12$, $10$, $8$, $6$,~etc.,
we shall get a system of lines such as those in \Figref{15}~(\textit{c}).

This is very far from exhausting the different modifications
our curve might undergo while still remaining a straight
line. For instance we might have a series of lines, one of
which should run from $12$ on the axis of~$y$ to $6$ on the axis
of~$x$, as before, while another ran from $8$ on the axis of~$y$
to $12$ on the axis of~$x$, and so on. This would indicate that
two independent causes were at work to modify the man's
want for the commodity.

Passing on to a case rather less simple, we may take the
first curve of \Figref{13}, which was drawn to the formula
\[
y=f(x)=\frac{24-3x}{4+x},
\]
%% -----File: 087.png---Folio 66-------
and confining ourselves to a single modification, may regard
it as
\[
y=f(z,x)=\frac{24-3x}{z+x},
\]
when, by making $z$ successively equal $4$, $6$, $8$, and $12$, we
shall get the four curves of \Figref{16}.

If we suppose that $z$~and~$x$ are both changing at the same
time, \ie~that the quantity of the commodity \emph{and} the nature
of the dependence of its marginal usefulness upon its quantity
are changing together, then the effect of the two changes
may be that each will intensify the other, or it may be that
\begin{figure}[hbt]
  \begin{center}
    \Fig{16}
    \Input[2.5in]{087a}
  \end{center}
\end{figure}
they will counteract each other.  Thus in $y=f(z,x)=
\dfrac{24-3x}{z+x}$, if $x$~is first~$5$ and then~$3$, while $z$ at the same time
passes from $4$ to~$12$, we shall have for the two values of~$y$
$\dfrac{24 - 3×5}{4+5}$ and $\dfrac{24 - 3×3}{12+3}$, and in either case $y=1$. This is
shown on the figure by the lines at $a$~and~$b$.

We must remember, then, that two things, and only two,
can alter the marginal usefulness of a commodity, viz.\ (i)~a
change in its quantity and (ii)~a change in the connection
between its quantity and its marginal usefulness. In the
diagrams these are represented by (i)~a movement of the
``bearer'' carrying the vertical to and fro on the base line,
and (ii)~a change in the form or position of the curve. In
%% -----File: 088.png---Folio 67-------
symbols they are represented (i)~by a change in the value of~$x$,
and (ii)~by a change in the meaning of~$f$. Anything that
changes the value of~$y$ must do so \emph{by} changing one of these.
Generally speaking the causes that affect the nature of the
function (\ie~the shape and position of the curve), so far as
they lend themselves to investigation, must be studied under
the ``theory of consumption;'' while an examination of the
causes which affect the magnitude of~$x$ (\ie~the position of the
``quantity-index'') will include, together with other things,
the ``theory of production.''
\Pagelabel{67}%
\end{Remark}
%% -----File: 089.png---Folio 68-------


\Chapter[II. Social]{II}

\Pagelabel{68}%
We have seen that the most varied and heterogeneous
wants and desires that exist \emph{in one mind} or ``subject''
may be reduced to a common measure and compared
one with another; but there is another truth which must
never be lost sight of on peril of a total misconception of
all the results we may arrive at in our investigations;
and that is, that by no possibility can desires or wants,
even for one and the same thing, which exist \emph{in different
minds}, be measured against one another or reduced to a
common measure. If $x$,~$y$, and~$z$ are all of them objects
\Pagelabel{69}% [** TN: Attempted to locate as closely as possible]
of desire to~\Person{A}, we can tell by his actions which of them
he desires most, but if \Person{A},~\Person{B}, and~\Person{C} all desire~$x$ no possible
process can determine which of them desires it
most. For any method of investigation is open to the
fatal objection that it must use as a standard of measurement
something that may not mean the same in
the different minds to be compared. Lady Jane Grey
\index{Lady@{\textsc{Lady Jane Grey}}}%
studies Plato while her companions ride in Bradgate
\index{Bradgate Park}%
\index{Plato}%
Park, whence we learn that an hour's study was more
than an equivalent to the ride to Lady Jane and less
than its equivalent to the others. But who is to tell
us whether Greek gave \emph{her} more pleasure than hunting
gave \emph{them}? Lady Jane fancied it did, but she may
have been mistaken. My account-book, intelligently
\index{Account-book@{\textsc{Account-book}}}%
studied, may tell you a good deal as to the equivalence
of various pleasures and comforts to me, but it can
establish no kind of equation between the amount of
pleasure which I derive from a certain article and the
%% -----File: 090.p n g----------
%[Blank Page]
%% -----File: 091.p n g----------
\begin{figure}[hbtp]
\Pagelabel{70}% [** TN: Attempted to locate as closely as possible]
  \begin{center}
    \Fig{17}
    \Input{091a}
  \end{center}
\end{figure}
% [To face page 69.]
%% -----File: 092.png---Folio 69-------
amount of pleasure you would derive from it. \Person{B}~wears his
black coats out to the bitter end and goes shabby three
\index{Coats}%
months in every year in order to get a few pounds
worth of books per annum. \Person{A}~would never think of
\index{Books}%
doing so---but whether because he values books less or
a genteel appearance more than~\Person{B} does not appear.
Nay, it is even possible he values books more, but
that his sensitiveness in the matter of clothing exceeds
\Person{B}'s in a still higher degree. \Person{C}~may be willing
to wait three hours at the door of a theatre to get a
place, whereas \Person{D} will not wait more than ten minutes;
but this does not show that \Person{C}~wants to witness the
representation more than \Person{D}~does; it may be that \Person{D} has
less physical endurance than~\Person{C}, and would suffer severely
from the exhaustion of long waiting; or it may be that
\index{Theatre, waiting}%
\index{Waiting@{Waiting (at theatre)}}%
\Person{C}~has nothing particular to do with his time and so
does not value it as much as \Person{D} does his.

Look at it how we will, then, it is impossible to
establish any scientific comparison between the wants
and desires of two or more separate individuals. Yet
it is obvious that almost the whole field of economic
investigation is concerned with collective wants and
desires; and we shall constantly have to speak of the
relative intensity of the demand for different articles or
commodities not on the part of this or that individual,
but on the part of society in general. In like manner
we shall speak of the marginal usefulness and utility
of such and such an article, not for the individual but
for the community at large. What right have we to use
such language, and what must we take it to mean?

To answer this question satisfactorily we must make
the relative intensity of the desires and wants of the
individual our starting-point. Let us suppose that \Person{A}
possesses stocks of $U$,~$V$, $W$,~$X$, $Y$,~$Z$, the marginal utility
to him of the customary unit (pound, yard, piece, bushel,
hundredweight, or whatever it may be) of each of
these articles being such that, calling a unit of~$U$, $u$,
a unit of~$V$, $v$,~etc., we shall have $3u$ or $10v$ or $4w$ or
%% -----File: 093.png---Folio 70-------
$\dfrac{x}{4}$ or~$\dfrac{3y}{2}$, applied at the margin, just equivalent to~$z$ (\ie~one
unit of~$Z$) at the margin. Portions of arbitrary
curves illustrating the supposed cases of $U$,~$X$, and~$Z$
are given in \Figref{17}~(\Person{A}). The curves represent the marginal
usefulness per unit of~$U$ as being one-third as great
as that of~$Z$. That is to say, if $u$ is but a very small
fraction of \Person{A}'s whole stock of~$U$, then, in the limit, $3u=z$.
In like manner $\dfrac{x}{4}=z$, in the limit. Now let us take
another man,~\Person{B}. We may find that he does not possess
(and possibly is not aware of definitely desiring) any $V$,~$W$,
or~$Y$ at all; but we will suppose that he possesses
stocks of $U$,~$X$, and~$Z$. In this case (neglecting the
practically very important element of friction) we shall
find that the units of $U$,~$X$, and~$Z$ stand in exactly the
same \emph{relative} positions for him as they do for~\Person{A}; that is
to say, we shall find that for~\Person{B}, as for~\Person{A}, $3u$ or~$\dfrac{x}{4}$ is exactly
equivalent to~$z$. For were it otherwise the conditions
for a mutually advantageous exchange would
obviously be present.

Suppose, for instance, we have
\[
\frac{x}{3} \text{ equivalent to~$2u$\qquad for~\Person{B}},
\]
as represented in Fig~17~(\Person{B}), while
\[
\frac{x}{4} \text{ is equivalent to~$3u$\qquad for~\Person{A}},
\]
as before. Then, reducing to more convenient forms,\footnote
  {This process is legitimate if $x$~and~$u$ are ``small'' units of $X$~and~$U$,
  so that the marginal usefulness of~$U$ remains sensibly constant
  throughout the consumption of $3u$,~etc.}
we shall have
\begin{align*}
 6u \text{ equivalent to~$x$} & \qquad \text{for~\Person{B}}, \\
12u \text{ equivalent to~$x$} & \qquad \text{for~\Person{A}}.
\end{align*}

\begin{Remark}
Observe that though we may suppose there will frequently
be some general similarity of form between the curves that
%% -----File: 094.png---Folio 71-------
connect the quantity of~$U$ with its marginal usefulness in
the cases of \Person{A}~and~\Person{B} respectively, yet we have no right
whatever to assume any close resemblance between these
curves.
\end{Remark}

Now since six units of~$U$ are equivalent to a unit of~$X$
for~\Person{B}, he will evidently be glad to receive anything
\emph{more than six} units of~$U$ in exchange for a unit of~$X$;
whereas \Person{A}~will be glad to give \emph{anything less than twelve}
units of~$U$ for a unit of~$X$. The precise terms on which
we may expect the exchange to take place will not be
investigated here, but it is obvious that there is a wide
margin for an arrangement by which \Person{A} can give~$U$ in
exchange for~$X$ from~\Person{B}, to the mutual advantage of the
two parties. The result of such an exchange will be to
change the quantities and make the quantity indices
move in the directions indicated by the arrow heads;
\Person{A}'s~stock of~$U$ decreasing and his stock of~$X$ increasing,
while \Person{B}'s~stock of~$U$ increases and his stock of~$X$
decreases. But this very process tends to bring the
ratio $\dfrac{\text{marginal usefulness of~$U$}}{\text{marginal usefulness of~$X$}}$ or $\dfrac{\text{marginal utility of~$u$}}{\text{marginal utility of~$x$}}$
nearer to unity (\ie~increase it) for~\Person{A}, for whom it
is now~$\frac{1}{12}$, and to remove it farther from unity
(\ie~decrease it) for~\Person{B}, to whom it is now~$\frac{1}{6}$. This
is obvious from a glance at the figures or a moment's
reflection on what they represent. Using $\dfrac{u}{x}$ as a
symbol of $\dfrac{\text{marginal utility of~$u$}}{\text{marginal utility of~$x$}}$ we may, therefore, say
that the ratio~$\dfrac{u}{x}$ will increase for~\Person{A}, to whom it is now
lowest, and decrease for~\Person{B}, to whom it is now highest.
If this movement continues long enough,\footnote
  {Compare below, \Pageref{73} and the note.}
there must
come a point at which $\dfrac{u}{x}$ will be the same for \Person{A}~and~\Person{B}.
Now until this point is reached the causes which produce
%% -----File: 095.png---Folio 72-------
the motion towards it continue to be operative, for it is
always possible to imagine a ratio of exchange~$\dfrac{u}{x}$ which
shall be greater than \Person{A}'s~$\dfrac{u}{x}$ and less than \Person{B}'s~$\dfrac{u}{x}$, and shall
therefore be advantageous to both. But when \Person{A}'s~$\dfrac{u}{x}$
and \Person{B}'s~$\dfrac{u}{x}$ have met there will be equilibrium. Hence
if the \emph{relative} worth, at the margin, of units of any two
commodities $U$~and~$X$ should not be identical for two
persons \Person{A}~and~\Person{B}, the conditions of a profitable exchange
between them exist, and continue to exist, until the
resultant changes have brought about a state of equilibrium,
in which the relative worths, at the margin, of
units of the two commodities are identical for the two
individuals.

This proposition is of such crucial and fundamental
importance that we will repeat the demonstration with a
more sparing use of symbols, and without reference to
the figures.\Pagelabel{71}% [** TN: Attempted to locate as closely as possible.]

\Person{B}, who is glad to get anything more than~$6u$ for~$x$,
and \Person{A},~who is glad to give anything short of~$12u$ for~$x$,
exchange $U$~and~$X$ to their mutual advantage, \Person{B}~getting
$U$ and giving~$X$, while \Person{A}~gets $X$ and gives~$U$.

But by this very act of exchange \Person{B}'s~stock of~$X$ is
decreased and his stock of~$U$ increased, and thereby the
marginal usefulness of~$X$ is raised and that of~$U$ lowered,
so that \Person{B}~will now find $6u$~less than the equivalent
of~$x$; or in other words, the interval between the worth
of a unit of~$X$ and that of a unit of~$U$ is increasing,
and at the same time \Person{A}'s~stock of~$X$ is increasing and
his stock of~$U$ diminishing, whereby the marginal usefulness
of~$U$ increases and that of~$X$ diminishes, so that
now less than twelve units of~$U$ are needed to make an
equivalent to one unit of~$X$; or in other words, the
interval between the worths at the margin of a unit of~$U$
and a unit of~$X$ is diminishing. To begin with, then,
%% -----File: 096.png---Folio 73-------
$u$~and~$x$ differ less in worth, at the margin, to~\Person{B} than
they do to~\Person{A}, but the difference in worth to~\Person{B} is constantly
increasing and that to~\Person{A} constantly diminishing
as the exchange goes on. There must, therefore,
come a point at which the expanding smaller difference
and the contracting greater difference will coincide.\footnote
  {Unless, indeed, the whole stock of \Person{A}'s~$X$ or of \Person{B}'s~$U$ is exhausted
  before equilibrium is reached. See \Pageref{82}.}
The conditions for a profitable exchange will then cease
\Pagelabel{73}%
to exist; but at the same moment the marginal worths
of $u$~and~$x$ will come to stand in precisely the same ratio
for~\Person{A} and for~\Person{B}. Wherever, then, articles possessed in
common by \Person{A} and~\Person{B} differ in the ratio of their unitary
marginal utilities as estimated by \Person{A} and~\Person{B}, the conditions
of a profitable exchange exist, and this exchange itself
tends to remove the difference which gives rise to it.
We may take it, then, that in a state of equilibrium the
ratios of the unitary marginal utilities of any articles, $X$,~$Y$,
$Z$,~etc., possessed in common by \Person{A},~\Person{B}, \Person{C},~etc., taken
two by two, viz.\ $x : y$, $x : z$, $y : z$,~etc., \emph{are severally identical
for all the possessors}. Any departure from this state of
equilibrium tends to correct itself by giving rise to
exchanges that restore the equilibrium on the same or
another basis.

To give precision and firmness to this conception, we
may work it out a little farther. Let us call such a
table as the one given on pp.~\Pageref[]{69},~\Pageref[]{70} a ``scale of the relative
unitary marginal utilities to~\Person{A} of the commodities he
possesses,'' or briefly, ``\Person{A}'s~relative scale.'' How shall
we bring the relative scales of~\Person{B}, \Person{C},~etc.\ into the form
most convenient for comparison with~\Person{A}'s? In \Person{A}'s~relative
scale the unitary marginal utilities of all the articles,
that is to say, $u$,~$v$, $w$,~$x$, $y$,~$z$, were expressed in terms of
the unitary marginal utility of~$Z$, that is to say,~$z$. And
in like manner \Person{B}'s~relative scale expressed $u$~and~$x$ in terms
of~$z$. But now suppose \Person{C}~possesses $S$,~$T$, $V$,~$X$, and~$Y$,
but no $U$,~$W$, or~$Z$. It is obvious that, in so far as he
possesses the same commodities as \Person{A}~and~\Person{B}, his relative
%% -----File: 097.png---Folio 74-------
scale, when there is equilibrium, must coincide with
theirs. But when we attempt to draw out that scale by
direct reference to \Person{B}'s~wants, we find ourselves unable
to express the unitary marginal utilities of his commodities
in terms of the unitary marginal utility of~$Z$, for
since he has no~$Z$ (and perhaps does not want any) we
cannot ask him to estimate its marginal usefulness to
him.\footnote
  {We shall see presently (\Pageref{82}) that the estimate must positively
  be made in terms of a commodity possessed, and that even if \Person{B} wants~$Z$,
  and knows exactly how much he wants a first unit of it, that want
  will not serve as the standard unit of desire unless he actually possesses
  some quantity of~$Z$.}
But it is obvious that \Person{A}'s~scale fixes the relative
marginal utilities of the units $v$,~$x$, and~$y$ in terms
of each other as well as in terms of~$z$, and unless they
are the same to~\Person{C} that they are to~\Person{A} the conditions
of an advantageous exchange between \Person{A}~and~\Person{C} will
arise and will continue till $v$,~$x$,~$y$ coincide on the
two relative scales. In like manner \Person{B}'s~scale expresses
the marginal utilities of the units $s$~and~$t$ in terms
of each other, and \Person{C}'s~scale must, when there is
equilibrium, coincide with~\Person{B}'s in respect of these two
units. Now, even though \Person{C} not only possesses no~$Z$,
but does not even desire any, there is nothing to prevent
him, for convenience of transactions with \Person{A}~and~\Person{B},
from estimating $s$,~$t$, $v$,~$x$, and~$y$ not in terms of each
other, but in terms of~$z$, placing it hypothetically in his
own scale in the same place relatively to the other units
which it occupies for \Person{A}~and~\Person{B}. Thus he may express
his desire for the commodities he has or wants to have,
in terms of a desire to which he is himself a stranger,
but the relative strength of which in other men's minds
he has been able to ascertain.

Lastly, if \Person{C} knows that he can at any time get $S$~and~$T$
from~\Person{B}, and $V$,~$X$ and~$Y$ from~\Person{A}, in exchange for~$Z$,
on definite terms of exchange, then, although he may
not want~$Z$ for himself, and may have no possible use
for it, yet he will be glad to get it, though only as representing
the things he does want, and for which he
%% -----File: 098.png---Folio 75-------
will immediately exchange it, unless indeed he finds it
more convenient to keep a stock of~$Z$ on hand ready to
exchange for~$S$, $T$,~etc.\ as he wants them for actual
consumption than to keep those commodities themselves
in any large quantities.

All this is exactly what really takes place. Gold
(in England) is the~$Z$ adopted for purposes of reference
(and also, though less exclusively, as a vehicle of
exchange). Gold is valuable for many purposes in
the arts and sciences, and, therefore, there are always
a number of persons who want gold to use, and
will give other things in exchange for it. Most of
us possess, and use in a very direct manner, a small
quantity of gold which we could not dispense with
without great immediate suffering and the risk of serious
ultimate detriment to our health, viz., the gold stoppings
\index{Gold stoppings in teeth}%
of some of our teeth. There is a constant demand for
gold for this use. Lettering and ornamenting the backs
of books is another use of gold in which vast numbers
of persons have an immediate interest as consumers.
Plate and ornaments are a more obvious if not more
important means of employing gold for the direct
gratification of human desires or supply of human wants.
In short, there are a great number of well-known and
easily accessible persons who, for one purpose or another
of direct use or enjoyment, desire gold, and since these
persons desire many other things also, their wants
furnish a scale on which the unitary marginal utilities
of a great variety of articles are registered in terms of
the unitary marginal utility of gold, and if the relative
scales of any two of these gold-and-other-commodities-desiring
individuals differ, then exchanges will be made
until they coincide. Other persons who have no direct
desire or use for gold desire a number of the other commodities
which find a place in the scale of the gold-desiring
persons, and can, therefore, compare the
relative positions they occupy in their own scale of
desires with that which is assigned them in the scale of
%% -----File: 099.png---Folio 76-------
the gold-desiring people, and if these relative positions
vary exchanges may advantageously be made until they
coincide. Thus the non-gold-desiring people may find
it convenient to express their desires in terms of the
gold-desire to which they are themselves strangers, and
seeing that the gold-desiring people are accessible and
numerous, even those who have no real personal gold-desire
will always value gold, because they can always
get what they want in exchange for it from the gold-desiring
people. Indeed, as soon as this fact is generally
known and realised, people will generally find it convenient
to keep a certain portion of their possessions not
in the form of anything they really want, but in the
form of gold.

We may, therefore, measure all concrete utilities in
terms of gold, and so compare them one with another.
Only we must remember that by this means we reach
a purely objective and material scale of equivalence, and
that the fact that I can get a sovereign for either of
two articles does not prove, or in any way tend to prove,
that the two articles really confer equivalent benefits,
\emph{unless it is the same man who is willing to give a sovereign
for either}.

\Person{A}'s and \Person{B}'s desires for $U$~and~$W$, when measured in
their respective desires for~$Z$, are indeed equivalent;
but the \emph{measure itself} may mean to the two men things
severed by a hell-wide chasm; for \Person{A}'s desire for~$U$, $W$,
and~$Z$ alike may be satisfied almost to the point of
satiety, so that an extra unit of~$Z$ would hardly confer
any perceptible gratification upon him; whereas \Person{B} may
be in extreme need alike of~$U$, $W$, and~$Z$, so that an
extra unit of~$Z$ would minister to an almost unendurable
craving.

Or again, \Person{A} may possess certain commodities, $V$, $X$,
$Y$, which \Person{B} does not possess, and is not conscious of
wanting at all (say billiard tables, pictures by old
\index{Billiard-tables}%
\index{Pictures}%
masters, and fancy ball costumes), and in like manner
\index{Fancy ball costumes}%
\Person{B} may possess $W$~and~$T$ (say corduroy breeches and
\index{Corduroys}%
%% -----File: 100.png---Folio 77-------
tripe), which \Person{A} neither possesses nor desires. Now in
\index{Tripe}%
\Person{B}'s scale of marginal utilities we may find that $t=\dfrac{z}{80}$
(taking $t$ = one cut of tripe, and $z$ = the gold in a
sovereign),\footnote
  {These cannot be regarded as ``small'' units in the technical
  sense, in this case. We are speaking in this example strictly of the
  values of units at the margin, and they will not coincide even roughly
  with the ideal ``usefulness'' of the commodity at the margin.}
whereas in \Person{B}'s scale one $v=50z$. Then
taking one~$z$ as a purely objective standard, and neglecting
the difference of its meaning to the two men, and
regarding \Person{A}~and~\Person{B} as forming a ``community,'' we
might say that in that community $z=80t$ and $v=50z$,
or $v=4000t$, \ie~one~$v$ is worth $4000$~times as much as
one~$t$. By this we should mean that the man in the
community who wants~$Z$ will give $4000$~times as much
for a unit of it as you can get out of the man who
wants~$T$ in exchange for a unit of that. But this does
not even tend to show that a unit of~$V$ will give the
man who wants it $4000$~times the pleasure which the
other man would derive from a unit of~$T$. Nay, it is
quite possible that the latter satisfaction might be positively
the greater of the two.\Pagelabel{77}%

\begin{Remark}
Note, then, that the function of gold, or money, as a
standard, is to reduce all kinds of services and commodities
to an objective scale of equivalence; and this constitutes its
value in commercial affairs, and at the same time explains
the instinctive dislike of money dealings with friends which
many men experience. Money is the symbol of the exact
balancing and setting off one against the other of services
rendered or goods exchanged; and this balancing can only
be affected by absolutely renouncing all attempts to arrive at
a \emph{real} equivalence of effort or sacrifice, and adopting in its
place an external and mechanical equivalence which has no
tendency to conform to the real equivalence. It is the
systematising of the individualistic point of view which says,
``One unit of~$Z$ may be a very different thing for \Person{A}~or~\Person{B} to
\Pagelabel{78}%
\emph{give}, but it is exactly the same thing for me to \emph{get}, wherever
%% -----File: 101.png---Folio 78-------
it comes from; and, therefore, I regard it as the same thing
all the world over, and measure all that I get or give in
terms of it.'' Where the relations to be regulated are themselves
prevailingly external and objective, this plan works excellently.
But amongst friends, and wherever friendship or
any high degree of conscious and active goodwill enters into
the relations to be regulated, two things are felt. In the first
place we do not wish to keep an evenly balanced account, and
to set services, etc., against each other, but we wish to act on the
principle of the mutual gratuitousness of services; and in the
second place, so far as any idea of a rough equivalence enters
our minds at all, we are not satisfied with anything but a
real equivalence, an equivalence, that is, of sacrifice or effort;
and this may depart indefinitely from the objective equivalence
in gold. This also explains the dislike of money and money
dealings which characterises such saints as St.~Francis of
\index{Francis of Assisi}%
Assisi. Money is the incarnate negation of their principle of
mutual gratuitousness of service.

Under what circumstances the objective scale might be
supposed roughly, and taken over a wide area, to coincide
with the real scale, we shall ask presently. If such circumstances
were realised, and in as far as they actually are
realised, it is obvious that the objective scale has a social
and moral, as well as a commercial, value. (Compare \Pageref{86}.)
\end{Remark}

In future we may speak of a man's desire or want of
``gold'' without implying that he has any literal gold-desire
at all, but using the ``unitary marginal utility of
gold'' as the standard unit of desire, and expressing
the (objective) intensity of any man's want of anything
in terms of that unit. It is abundantly obvious from
what has gone before in what way we shall reduce to
this unit the wants of a man who has no real desire for
gold at all. When we use gold in this extended and representative
sense we shall indicate the fact by putting it in
quotation marks: ``gold.'' Thus any one who possesses
anything at all must to that extent possess ``gold,''
though he may be entirely without gold.

The result we have now reached is of the utmost
importance. We have shown that in any catallactic community,\footnote
  {I mean by a catallactic community one in which the individuals
  freely exchange commodities one with another, each with a view to
  making the enjoyment he derives from his possessions a maximum.}
%% -----File: 102.png---Folio 79-------
when in the state of equilibrium, the marginal
utilities of units of all the commodities that enter into the
circle of exchange will arrange themselves on a certain
relative scale or table in which any one of them can
be expressed in terms of any other, and that that scale
will be general; that is to say, it will accurately translate
or express, \emph{for each individual in the community}, the
worth at the margin of a unit of any of the commodities
he possesses, in terms of any other.

The scope and significance of this result will become
more and more apparent as we proceed; but we
can already see that the desiredness at the margin of a
unit of any commodity, expressed in terms of the desiredness
at the margin of a unit of any other commodity,
is the same thing as the \emph{value-in-exchange} (or exchange-value)
of the first commodity expressed in terms of the
second.

We have therefore established a precise relation between
value-in-use and value-in-exchange; for we have
discovered that the value-in-exchange of an article conforms
to the place it occupies on the (necessarily coincident)
relative scales of all the persons in the community
who possess it. Now to every man the
marginal utility of an article, that is to say of a unit of
any commodity, is determined by the average between
the marginal usefulness of the commodity at the beginning
and its marginal usefulness at the end of the
acquisition of that unit; and this marginal usefulness
itself is the first derived function, or the differential
coefficient, of the total utility of the stock of the commodity,
which the man possesses. Or briefly, \emph{the value-in-exchange
\Pagelabel{79}%
of a commodity is the differential coefficient of
the total \DPtypo{utilily}{utility}, to each member of the community, of the stock
of the commodity he possesses}.

``The things which have the greatest value-in-use
%% -----File: 103.png---Folio 80-------
have frequently little or no value-in-exchange; and, on
the contrary, those which have the greatest value-in-exchange
have frequently little or no value-in-use. Nothing
is more useful than water; but it will purchase scarce
\index{Water}%
anything; scarce anything can be had in exchange for
it'' (Adam Smith). Now that we know exchange-value
to be measured by marginal usefulness, we can well
understand this fact. For as the total value in use of a
thing approaches its maximum its exchange-value tends
to disappear. Were water less abundant its value-in-use
would be reduced, but its exchange-value would be
so much increased that there would be ``scarce anything
that could not be had in exchange for it.'' As it
is the total effect of water is so near its maximum that
its effectiveness at the margin is comparatively small.

\Pagelabel{80}%
Before proceeding farther we will look somewhat
more closely into this matter of the identity of the
exchange-value of a unit of any commodity and its
desiredness at the margin of the stocks of the persons
who possess it.

%[** TN: Kept pound signs upright on this page; italicized in original.]
In practical life, if I say that the exchange-value of a
horse is £31, I am either speaking from the point of view
\index{Horse}%
of a buyer, and mean that a horse of a certain quality could
be got in exchange for $8$~oz.~of gold;\footnote
  {About $7.97$~oz.~of gold is contained in £31.}
or I am speaking from
the point of view of a seller, and mean that a man could
get $8$~oz.~of gold for the horse; but I cannot mean both,
for notoriously (if all the conditions remain the same)
the buying and selling prices are never identical. What
then do I mean when, speaking as an economist, I suppose,
without further specification, that the exchange-value
of a horse in ounces of gold is~$8$? I mean that
the offer of anything \emph{more} than the $8$~oz.~of gold for
a horse of the quality specified will \emph{tend to induce} some
possessor of such a horse to part with him, and the offer
of such a horse for anything \emph{less} than $8$~oz.~of gold will
\emph{tend to induce} some possessor of gold to take the horse
in exchange for some of it; and if I reduce the friction
%% -----File: 104.png---Folio 81-------
of exchange (both physical and mental) towards the
vanishing point, I may say that every man who is
willing to give \emph{any} more than 8~oz.\ of gold for a horse
can get him, and every man who is willing to take \emph{any}
less than 8~oz.\ of gold for a horse can sell him.

The exchange-value of a horse, then, in ounces of gold,
represents a quantity of gold such that a man can get
anything short of it for a horse, and can get a horse for
anything above it. And obviously, if the conditions remain
the same, every exchange will tend to destroy the
conditions under which exchanges will take place, for
after each exchange the number of people who desire to
exchange on terms which will ``induce business'' tends to
be reduced by two.

Thus if the exchange value of a horse is 8~oz.\ of
gold, that means that the ratio ``1 horse to 8~oz.\ gold''
is a point \emph{on either side of which} exchanges will take
place, each exchange, however, tending to produce an
equilibrium on the attainment of which exchange will
cease.

Now we have shown in detail that the relative scale
of marginal utilities is a table of precisely such ratios,
between units of all commodities that enter into the
circle of exchange. Any departure in the relative scale
of any individual from these ratios will at once induce
exchanges that will tend to restore equilibrium. We
find, then, that the relative scale is, in point of fact, \emph{a
table of exchange values}, and that the exchange value of
an article is simply its marginal utility measured in the
marginal utility of the commodity selected as the standard
of value. And, after all, this is no more than the
simplest dictate of common sense and experience; for we
have seen that the conditions of exchange are that some
one should be willing, as a matter of business, to give more
(or take less) than 8~oz.\ of gold for a horse; but what could
induce that willingness except the fact that the marginal
utility of a horse is greater, to the man in question, than
the marginal utility of 8~oz.\ gold? And what should
%% -----File: 105.png---Folio 82-------
induce any other man to do business with him except
the fact that to that other man the marginal utility of a
horse is \emph{not} greater than that of 8~oz.\ of gold? In other
words, the conditions of exchange only exist when there
is a discrepancy in the relative scales of two individuals
who belong to the same community; and, as we have seen,
the exchange itself tends to remove this discrepancy.

\Pagelabel{82}%
Thus, \emph{the function of exchange is to bring the relative
scales of all the individuals of a catallactic community into
correspondence}, and the equilibrium-ratio of exchange
between any two commodities is the ratio which exists
between their unitary marginal utilities when this correspondence
has been established. Thus if the machinery
of exchange were absolutely perfect, then, \emph{given the
initial possessions of each individual in the community}, there
would be such a redistribution of them that no two men
who could derive mutual satisfaction from exchanges
would fail to find each other out; and so in a certain
sense the satisfactions of the community would be
maximised by the flow of all commodities from the
place in which they were relatively less to the place in
which they were relatively more valued. But the conformity
of the net result to any principle of justice or
of public good \emph{would depend entirely on initial conditions}
prior to all exchange.

It must never be forgotten that the coincident relative
scales of the individuals who make up a community
severally contain the things actually possessed (or commanded)
only, not all the things \emph{wanted} by the respective
individuals. If a man's \emph{initial} want of~$X$ relatively to
his (marginal) want of ``gold'' is not so great as the
\emph{marginal} want of~$X$ relatively to the (marginal) want of
gold experienced by the possessors of~$X$, then he will not
come into the possession of~$X$ at all, and all that we
shall learn from the fact of his having no~$X$, together
with an inspection of the position of~$X$ in the relative
scale of marginal utilities, is that he desires~$X$ with less
\emph{relative} intensity than its possessors do. But this does
%% -----File: 106.png---Folio 83-------
not by any means prove that his actual want of~$X$ is less
pressing than theirs. It may very well be that he wants
X far more than they do, but seeing that he has very
little of anything at all, his want of ``gold'' exceeds
theirs in a still higher degree. And, again, if one man
wants~$X$ but does not want~$Y$, and another wants~$Y$ but
does not want~$X$, and if the man who wants~$X$ wants it
more, relatively to ``gold,'' than the man who wants~$Y$,
it does not in the least follow that the one wants~$X$
absolutely more than the other wants~$Y$, for we have no
means of comparing the want of ``gold'' in the two
cases, so that we measure the want of~$X$ and the want
of~$Y$ in two units that have not been brought into
any relation with each other. All this is only to
say that because I cannot ``afford to buy'' a thing it
does not follow that I have less need of it or less desire
to have it than another man who can and does afford it.

Obvious as this is, it is constantly overlooked in
amateur attempts ``to apply the principles of political
economy to the practical problems of life.'' We are
told, for instance, that where there is no ``demand'' for
a thing it shows that no one really wants it. But before
we can assent to this proposition we must know what is
meant by ``demand.''

Now if I want a thing that I have not got, there are
many ways of ``demanding'' it. I may beg for it. I
may try to make people uncomfortable by forcing the
extremity of my want upon them. I may try to terrify
them into giving me what I want. I may attempt to
seize it. I may offer something for it which stands
lower than it on the relative scale of marginal utilities
in my community. I may offer to work for it. All
these forms of ``demand,'' and many more, the economists
have with fine, if unconscious, irony classed
together under one negative description. Not one of
them constitutes an ``effective'' demand. An ``effective''
demand (generally described, with the omission of the
adjective, as ``demand'' simply) is that demand, and
%% -----File: 107.png---Folio 84-------
that demand only, which expresses itself in the offer in
exchange for the thing demanded of something else that
stands at least as high as it does on the relative scale of
marginal utilities. No demand which expresses itself in
any language other than such an offer is recognised as a
demand at all---it is not ``effective.'' Now this phraseology
is convenient enough in economic treatises, but
unhappily the lay disciples of the economists have a
tendency to adopt their conclusions and then discard
their definitions. Thus they learn that it is waste of
effort to produce a commodity or render a service which
is less wanted than some other commodity or service
that would demand no greater expenditure (whether of
money, time, toil, or what not); they learn that what
men want most they will give most for; and the conclusion
which seems obvious is announced in such terms
as these: ``Political economy shows that it is a mistake
and a waste to produce or provide anything for people
which they are not willing to pay for at a fair remunerative
rate;'' or, ``It is false political economy to subsidise
anything, for if people won't pay for a thing it
shows they don't want it.'' Of course political economy
does not really teach any such thing, for if it did it
would teach that a poor man never ``wants'' food as
much as a rich one, that a poor man never ``wants'' a
holiday as much as a rich one; in a word, that a man who
\index{Holiday}%
has not much of anything at all has nearly as much of
everything as he wants---which is shown by his being
willing to give so very little for some more.

The fallacy, of course, lies in the use made of the
assertion that ``what men want most they will give most
for.'' This is true only if we are always speaking of the
\emph{same men}, or if we have found a measure which can
determine which of two different men is really giving
``most.'' Neither of these conditions is fulfilled in the
case we are dealing with. ``When two men give the
same thing, it is not the same thing they give,'' and if
$A$ spends £100 on a continental tour and $B$ half a crown
%% -----File: 108.png---Folio 85-------
on a day at the sea-side no one can say, or without
further examination can even guess, which of them has
given ``most'' for his holiday.
\index{Holiday}%

\begin{Remark}
Again, some confusion may be introduced into our
thoughts by the fact that desires not immediately backed by
any ``effective'' demand for gratification sometimes succeed in
getting themselves indirectly registered by means of secondary
desires which they beget in the minds of well-disposed
persons who are in a position to give ``effect'' to them.
Thus we may suppose that Sarah Bernhardt is charging three
\index{Sarah@{\textsc{Sarah Bernhardt}}}%
hundred guineas as her fee for reciting at an evening party,
and that the three hundred guineas would provide a weeks'
holiday in the country for six hundred London children. A
benevolent and fashionable gentleman is in doubt which of
these two methods of spending the sum in question he shall
adopt, and after much debate internal makes his selection.
What do we learn from his decision? We learn whether \emph{his}
desire to give his friends the treat of hearing the recitation or
to give the children the benefit of country air is the greater.
It tells us nothing whatever of the relative intensity of the
desire of the guests to hear the recitation and of the children
to breathe the purer air. The primary desires concerned have
not registered their relative intensities at all, it is only the
secondary desires which they beget in the benevolent host
that register themselves; and if the result proclaims the fact
that the marginal utility of a recitation from the tragic
actress is just six hundred times as great as the marginal
utility of a week in the country to a sick child, this does not
mean that the pleasure or advantage conferred on the company
by the recitation is (or is expected to be) six hundred
times as great as that conferred upon each child by the holiday;
nor does it mean that the company would have estimated
their pleasure in their own ``gold'' at the same sum
as that at which the six hundred children would have estimated
their pleasure in their ``gold,'' but that the host's
desire to give the pleasure to the company is as great as
his desire to give the pleasure to the six hundred children.
And since we have supposed the host's desires to be the
only ``effective'' ones, they alone are commercially significant.
No kind of equation---not even an objective one---is established
%% -----File: 109.png---Folio 86-------
between the primary desires in question, viz.\ those of
the guests and of the children respectively.\footnote
  {It is interesting to note that there are considerable manufactures
  of things the direct desire for which seldom or never asserts itself at
  all. There are immense masses of tracts and Bibles produced, for
\index{Bibles}%
\index{Tracts}%
\Pagelabel{86}%
  instance, which are paid for by persons who do not desire to use them
  but to give them away to other persons whose desire for them is not
  in any way an effective factor in the proceeding. And there are
  numbers of expensive things made expressly to be bought for ``presents,''
  \index{Presents}%
  and which no sane person is ever expected to buy for himself.}
\end{Remark}

The exchange value, then, of any commodity or service
indicates its position on \emph{its possessors'} relative scale
of unitary marginal utilities; and if expressed in ``gold''
it indicates the ratio between the unitary marginal
desiredness of the commodity and that of ``gold'' upon
all the (necessarily coincident) relative scales of \emph{all the
members of the community who possess it}.

\begin{Remark}
\index{Poor men's wares|(}%
\index{Rich men's wares|(}%
I have repeatedly insisted on the fact that we have no
common measure by which we can compare the necessities,
wants, or desires of one man with those of another. We
cannot even say that ``a shilling is worth more to a poor
man than to a rich one,'' if we mean to enunciate a rule that
can be safely applied to individual cases. The most we can
say is, that a shilling is worth more to a man \emph{when he is poor}
than (\textit{c{\oe}teris paribus}) to \emph{the same man} when he is rich.

But if we take into account the principle of averages, by
which any purely personal variations may be assumed to
neutralise each other over any considerable area, then we
may assert that shillings either are or ought to be worth
more to poor men than to rich. I say ``either are or ought
to be;'' for it is obvious that the rich man already has his
desires gratified to a greater extent than the poor man, and
if in spite of that they still remain as clamorous for one
shilling's worth more of satisfaction, it must be because his
tastes are so much more developed and his sensitiveness to
gratification has become so much finer that his organism even
when its most imperative claims are satisfied still remains
more sensitive to satisfactions of various kinds than the
other's. But if the poor man owes his comparative freedom
%% -----File: 110.png---Folio 87-------
from desires to a low development and blunted powers, then
the very fact that though he has so few shillings yet one in
addition would be worth no more to him than to his richer
neighbour is itself the indication of social pressure and
inequality. On the assumption, then, that the humanity of
\Pagelabel{87}%
all classes of society ought ideally to receive equal development,
we may say that shillings either are or ought to be
worth more to poor men than to rich. Thus, if \Person{A}~manufactures
articles which fetch 1s.~each in the open market and
are used principally by rich men, and if \Person{B}~produces articles
which fetch the same price but are principally consumed by
poor men, then the commercial equivalence of the two wares
does not indicate a social equivalence, \ie\ it does not indicate
that the two articles confer an equal benefit or pleasure on
the community. On the contrary, if the full humanity of
\Person{B}'s~customers has not been stunted, then his wares are of
higher social significance than~\Person{A}'s.

It is obvious, too, that if \Person{C}'s wares are such as rich and
poor consume alike, the different lots which he sells to his
different customers, though each commercially equivalent to
the others, perform different services to the opulent and the
needy respectively.

Now, anything which tends to the more equal distribution
of wealth tends to remove these discrepancies. Obviously if
all were equally rich the neutralising, over a wide area, of
individual variations would take full effect; and if a thousand
men were willing to give a shilling for \Person{A}'s~article and five
hundred to give a shilling for~\Person{B}'s, it would be a fair assumption
that though fewer men wanted \Person{B}'s~wares than~\Person{A}'s, yet
those who did want them wanted them (at the margin) as
much; nor would there be any reason to suppose that different
lots of the same ware ministered, as a rule, to widely
different intensities of marginal desire; the irreducible variations
of personal constitution and habit being the only
source of inequality left.

It is true that the desire for \Person{A}'s~and~\Person{B}'s wares might not
be equally legitimate, from a moral point of view. I may
``want'' a shameful and hurtful thing as much as I ``want''
a beautiful and useful one. The State usually steps in to
say that certain wants must not be provided for at all---in
England the ``want'' of gaming tables, for instance---and a
%% -----File: 111.png---Folio 88-------
man's own conscience may preclude him from supplying many
other wants. But on the supposition we are now making
equal intensity of commercial demand would at least represent
(what no one can be sure that it represents now) equal
intensity of desire on the part of the persons respectively
supplied. If wealth were more equally distributed, therefore,
it would be nearer the truth than it now is to say that
when we supply what will sell best we are supplying what is
wanted most.
\index{Rich men's wares|)}%
\index{Poor men's wares|)}%

These considerations are the more important because, in
general, this index of price is almost the only one we can
have to guide us as to what really is most wanted. When
we enter into any extensive relations with men of whom we
have little personal knowledge it is impossible that we should
form a satisfactory opinion as to the real ``equivalence'' of
services between ourselves and them, and it would be an
immense social and moral amelioration of our civilised life if
we could have some assurance that a moderate conformity
existed, over every considerable area, between the price a
thing would fetch and the intensity of the marginal want of
it. This would be an ``economic harmony'' of inestimable
importance. Within the narrower area of close and intimate
personal relations attempts would still be made, as now, to
get behind the mere ``averaging'' process and consider the
personal wants and capacities of the individuals, the ideal
being for each to ``contribute according to his powers and
receive according to his needs.'' Thus the different principles
of conducting the affairs of business and of home would
remain in force, but instead of their being, as they are now,
in many respects opposed to each other the principles of
business would be a first approximation---the closest admitted
by the nature of the case---to the principles on which
we deal with family and friends.

Now certain social reformers have imagined an economic
Utopia in which an equal distribution of wealth, such as we
have been contemplating, would be brought about as follows:---Certain
industrial, social and political forces are supposed to
be at work which will ultimately throw the opportunities of
acquiring manual and mental skill completely open; and
skill will then cease to be a monopoly. Seeing, then, that
there will only be a small number of persons incapable of
%% -----File: 112.png---Folio 89-------
doing anything but heaving, it will follow that the greater
part of the heaving work of the world will be done by persons
capable of doing skilled work. And hence again it will
follow that every skilled task may be estimated in the foot-tons,
which would be regarded by a heaver as its equivalent
in irksomeness. And if we ask ``What heaver?''\ the answer
will be ``The man at present engaged in heaving who estimates
the relative irksomeness of the skilled task most lightly,
and would therefore be most ready to take it up.'' Then the
reward, or wages, for doing the task in question will be the
same as for doing its equivalent (so defined) in foot-tons.
If more were offered some of the present heavers would
apply. If less were offered some of those now engaged in
the skilled work would do heaving instead. To me personally
heaving may be impossible or highly distasteful, but
as long as some of my colleagues in my task are capable of
heaving and some of the heavers capable of doing my task, a
scale of equivalence will be established at the margin between
them, and this will fix the scale of remuneration. Thus earnings
will tend to equality with efforts, estimated in foot-tons.

From this it would follow that inequalities of earnings
could not well be greater than the natural inequalities of
mere brute strength; for since foot-tons of labour-power are
the ultimate measure of all remunerated efforts, he who has
most foot-tons of labour-power at his disposal is potentially
the largest earner.

Again, the reformers who look forward to this state of
things hold that forces are already at work which will ultimately
dry up all sources of income except earnings, so that
we shall not only have earnings proportional to efforts, estimated
in foot-tons, but also incomes proportional to earnings.
Thus inequalities in the distribution of wealth will be restrained
within the limits of inequalities of original endowment
in strength.

The speculative weakness of this Utopia obviously lies in
its taking no sufficient account of differences of personal
ability. Throwing open opportunities might level the rank
and fill up all trades, including skilled craftsmen, artists, and
heavers; but it would hardly tend to diminish the distance,
for example, between the mere ``man who can paint'' and
the great artist.
%% -----File: 113.png---Folio 90-------

Nevertheless it is interesting to inquire how things would
go in such a Utopia. In the first place we are obviously as
far as ever from having established any common measure
between man and man or any abstract reign of justice; for a
foot-ton is not the same thing to~\Person{A} and to~\Person{B}, neither is there any
justice in a strong man having more comforts than a weak one.

Nevertheless there would be greater equality. For the
number of individual families whose ``means'' in foot-tons of
labour-power lie near about the average means, is much
greater than the number of families whose present means in
``gold'' lie near the average means. As this statement deals
with a subject on which there is a good deal of loose and inaccurate
thought, it may be well to expand the conception.

If $\dfrac{a+b+c+d+e}{5}$ remains the same, then the arithmetical
average of the five quantities remains the same. Suppose
that average is~$200$. Then we may have $a=b=c=d=e=200$,
or we may have $a=996$, $b=c=d=e=1$, or $a=394$,
$b=202$, $c=198$, $d=200$,~$e=6$. In all these cases the
average is~$200$, but in the second case not one of the several
quantities lies anywhere near the average. So again, if we
pass from the case $a=b=c=d=e=200$ to the case $a=997$,
$b=c=d=e=1$, we shall actually have raised the average,
but we shall have removed each quantity, severally, immensely
farther away from that average.

Now if we reflect that the average income of a family of
five in the United Kingdom is estimated at £175~per annum,
it is obvious that an enormous number of families have incomes
a long way below the average. It is held to be self-evident
that a smaller number of families fall conspicuously
short of the average means in labour-power.

Further, the extremes evidently lie within less distance of
the average in the case of labour-power than in the case of
``gold.'' There are, it is true, some families of extraordinary
\index{Athletes}%
athletic power, races of cricketers, oarsmen, runners, and so
forth, but if we imagine such a family, while still remaining
an industrial unit, to contain six or seven members each able
to do the work of a whole average family, we shall probably
have already exceeded the limit of legitimate speculation,
and this would give six or seven times the average as the
upper limit. Whereas the average ``gold'' income (as given
%% -----File: 114.p n g----------
%[Blank Page]
%% -----File: 115.p n g----------
\begin{figure}[p]
  \begin{center}
    \Fig{18}
    \Input[4.5in]{115a}
  \end{center}
\end{figure}
%[To face page 91.]
%% -----File: 116.png---Folio 91-------
above) being £175, we have only to think of the incomes of
our millionaires to see how much further above the average
the upper limit of ``gold'' incomes rises than it could possibly
do in the case of labour-power.

The lower limit being zero in both cases does not lend
itself to this comparison.

It may be urged, further, that there is no such broad
distinction between the goods required by the strong (?~skates,
\index{Skates}%
bicycles, etc.) and those required by the ``weak'' (?~respirators,
\index{Bicycles}%
\index{Respirators}%
reading-chairs, etc.) as there is between those demanded
\index{Reading-chairs}%
by the ``rich'' and those demanded by the ``poor.'' So
that the analogue of the cases mentioned on \Pageref{87} would
hardly occur; especially when we take into account the
balancing effect of the association of strong and weak in the
same family.

The whole of this inquiry may be epitomised and elucidated
by a diagramatic illustration.

The unitary marginal utilities of $U$~and~$V$ stand in the
ratio of~$3:4$ on the relative scale of the community in which
\Person{A}~and~\Person{B} live. \Person{A}~possesses a considerable supply both of $U$~and~$V$.
Parts of the curves are given in \Figref{18}~\Person{A}~(i), where
the ``gold'' standard is supposed to be adopted in measuring
marginal usefulness and utility. \Person{B}~possesses a little~$V$, but
no~$U$, and would be willing (as shown on the curves \Figref{18}~\Person{B}~(i\DPtypo{.}{}))
to give $\dfrac{v}{2}$ for~$u$ ($v$~and~$u$ being small units of $V$~and~$U$),
but since $u$ is only worth half as much as $v$ to him, he will
not buy it on higher terms than this. Now we have supposed
the ratio of utilities of $u$~and~$v$ on the relative scale to
be~$3:4$. That is to say, if $u$ contains three small units of
utility then $v$ contains four. Therefore $\dfrac{u}{3}$ has the same value-in-exchange
or marginal utility as $\dfrac{v}{4}$, and $\dfrac{3u}{3}$, or $u$ has the
same value-in-exchange as $\dfrac{3v}{4}$; therefore an offer of $\dfrac{3v}{4}$, but
nothing lower than this, constitutes an ``effective'' demand
for~$u$; whereas \Person{B} only offers $\dfrac{v}{2}$ or $\dfrac{2v}{4}$ for it. Measuring the
intensity of a want by the offer of ``gold'' it prompts, we
should say, that \Person{B} wants $v$ as much as \Person{A} does, but wants $u$
%% -----File: 117.png---Folio 92-------
less than \Person{A} does. This, however, is delusive, for we do not
know how much each of them wants the units of ``gold'' in
which all his other wants are estimated. Suppose we say,
``What a man wants he will work for,'' and ascertain that \Person{A}
would be willing to do half a foot-ton of work for a unit of
``gold,'' whereas \Person{B} would do one and a half foot-tons for it.
This would show that, measured in work, the standard unit
was worth three times as much to \Person{B} as to~\Person{A}. Reducing the
units on the axis of~$y$ to $\frac{1}{2}$ for~\Person{A}, and raising them to $\frac{3}{2}$ for~\Person{B},
we shall have the curves of \Figref{18}~\Person{A}~(ii) and \Person{B}~(ii) showing
the respective ``wants'' of \Person{A}~and~\Person{B} estimated in willingness
to do work. It will then appear that \Person{B} wants $v$ three
times as much and $u$ twice as much as \Person{A} does; but his
demand for~$u$ is still not effective, for he only offers $\dfrac{v}{2}$ or $\dfrac{2v}{4}$
for it, and its exchange-value is $\dfrac{3v}{4}$. There is only enough
$U$ to supply those who want a unit of it at least as much as
they want $\frac{3}{4}$ of a unit of $V$, and \Person{B} is not one of these.

Now if \Person{A} and \Person{B} had both been obliged to earn their
``gold'' by work, with equal opportunities, then obviously
the unitary marginal utility of ``gold,'' estimated in foot-tons,
must have been equally high for both of them, since each
would go on getting ``gold'' till at the margin it was just
worth the work it cost to get and no more. And therefore
the marginal utilities of $u$~and~$v$ (whether measured in foot-tons
or in ``gold'') must also have stood at the same height
for \Person{A}~and~\Person{B}. Hence \Person{B} could not have been wholly without
$U$ while \Person{A} possessed it, unless, measured in foot-tons, its
marginal usefulness was less to him than to~\Person{A}.

It would remain possible that a foot-ton might represent
widely different things to the two men; but the contention is
that this is less probable, and possible only within narrower
limits, than in the corresponding case of ``gold'' under our
present system. I need hardly remind the reader that the
assumptions of \Figref{18} are arbitrary, and might have been
so made as to yield any result desired. The figure illustrates
a perhaps rational supposition, and throws light on the
nature and effects of a change of the standard unit of utility.
It does not prove anything as to the actual result which
would follow upon any specified change of the standard.
%% -----File: 118.png---Folio 93-------

The whole of this note must be regarded as a purely speculative
examination of the conditions (whether possible of
approximate realisation or not) under which it might be
roughly true that ``what men want most they will pay most
for.''
\end{Remark}

\Pagelabel{93}%
We have now gained a distinct conception of what
is meant by the exchange-value of a commodity. It is
identical with the marginal utility which a unit of the
commodity has to every member of the community
who possesses it, expressed in terms of the marginal
utility of some concrete unit conventionally agreed
upon. There is no assignable limit to the divergence
that may exist in the \emph{absolute} utility of the standard
unit at the margin to different members of the community,
but the \emph{relative} marginal utilities of the standard
unit and a unit of any other article must be identical to
every member of the community who possesses them, on
the supposition of perfectly developed frictionless exchange,
and ``small'' units.

We may now proceed to show the principle on which
to construct collective or social curves of quantity-possessed-and-marginal-usefulness
without danger of
being misled by the equivocal nature of the standard,
or measure, of usefulness which we shall be obliged to
employ.

In approaching this problem let us take an artificially
simple case, deliberately setting aside all the secondary
considerations and complications that would rise in
practice.

We will suppose, then, that a man has absolute control
\index{Mineral spring}%
of a medicinal spring of unique properties, and that
its existence and virtues are generally known to the
medical faculty. We will further suppose that the
owner is actuated by no consideration except the desire
to make as much as he can out of his property, without
exerting himself to conduct the business of bottling and
disposing of the waters. He determines, therefore, to
allow people to take the water on whatever terms
%% -----File: 119.png---Folio 94-------
prove most profitable to himself, and to concern himself
no further in the matter.

Now there are from time to time men of enormous
wealth who would like to try the water, and would give
many pounds for permission to draw a quart of it, but
these extreme cases fall under no law. One year the
owner might have the offer of £50 for a quart, and for
the next ten years he might never have an offer of more
than £5, and in neither case would there be any regular
flow of demand at these fancy prices. He finds that in
order to strike a broad enough stratum of consumers to
give him a basis for averaging his sales even over a series
of years he must let people draw the water at not more
than ten shillings a quart, at which price he has a small
but appreciable and tolerably steady demand, which he
can average with fair certainty at so much a year. This
means that there is no steady flow of patients to whom
the marginal utility of a quart of the water is greater
than that of ten shillings. In other words, the initial
utility of the water to the community is ten shillings a
quart. Clearly, then, the curve of quantity-and-marginal-usefulness
of the water cuts the axis of~$y$ (that is to say,
begins to exist for our purposes) at a value representing
ten shillings a quart. If we were to take our unit on $x$ to
represent a quart and our unit on~$y$ to represent a shilling,
then we should have the corresponding values $x=0$, $y=10$.
But since we shall have to deal with large quantities of
the water, it will be convenient to have a larger unit for
diagramatic purposes; and since the rate of 10s.~per
quart is also the rate of £5000 per $10,000$ quarts, we
may keep our corresponding values $x=0$, $y=10$, while
interpreting our unit on~$x$ as $10,000$ quarts and our unit
on~$y$ as £500 ($= 10,000$ shillings). The curve, then,
cuts the axis of~$y$ at the height~$10$; which is to say that
the initial \emph{usefulness} of the water to the community is
£500 per $10,000$ quarts, or ten shillings a quart, which
latter estimate being made in ``small'' units may be
converted into the statement that the initial \emph{utility} of a
%% -----File: 120.png---Folio 95-------
quart of the water is equal to that of ten shillings, of
two quarts twenty shillings, etc.\footnote
  {Whereas it cannot be said that the initial utility of $10,000$ quarts
  is £500, for the initial usefulness is not sustained throughout
  the consumption of $10,000$ quarts.}

But at this price customers are few, and the owner
makes only a few pounds a year. He finds that if he
lowers the price the increased consumption more than
compensates him, and as he gradually and experimentally
lowers the price he finds his revenue steadily rising.
Even a reduction to nine shillings enables him to sell
\begin{figure}[hbt]
\Pagelabel{96}%
  \begin{center}
    \Fig{19}
    \Input{120a}
  \end{center}
\end{figure}
about $1000$ quarts a year, and so to derive a not inconsiderable
income (£450) from his property. A further
reduction of a shilling about doubles his sale, and he
sells $2000$ quarts a year at eight shillings, making £800
income. When he lowers the price still further to six
shillings, he sells between $5000$ and $6000$ quarts a year,
and his income rises to £1500.

Before following him farther we will look at the problem
%% -----File: 121.png---Folio 96-------
from the other side. At first no one could get a
quart of the water unless its marginal utility to him
was as great as that of ten shillings. Now the issue
just suffices to supply every one whose marginal want of
a quart is as high as six shillings. These and these only
possess the water, and on their relative scales it stands
as having a marginal utility of six shillings a quart.
This, then, may be called the marginal utility of the
water \emph{to the community}; only we must bear in mind that
we have no reason to suppose that the marginal wants
of the possessors are \emph{in themselves} either all equal to
each other or all more urgent than those of the yet unsupplied;
but relatively to ``gold'' they will be so.

We will now suppose that the owner tries the effect
of lowering the price further still, and finds that when
he has come down to four shillings a quart he sells
$11,000$ quarts a year, so that his revenue is still increasing,
being now more than £2200 per annum. This means
that over $11,000$ quarts are needed to supply all those
members of the community to whom the marginal utility
of a quart is as great as the marginal utility of four
shillings. Still the owner lowers the price, and discovers
at every stage \emph{what quantity of the water it is that has the
unitary marginal utility to the community corresponding to
the price he has fixed}. By this means he is tracing the
curve of price-and-quantity-demanded, and he is doing so
by giving successive values to~$y$ and ascertaining the
values of~$x$ that severally correspond to them. \Figref{19}
shows the supposed result of his experiments, which,
however, he will not himself carry on much beyond
$y=1$, which gives $x=10$,\footnote
  {In the diagram $y=\dfrac{120-x}{10x+10} - \dfrac{x^2-20x+100}{50}$.}
and represents an income of
ten units of area, each unit representing £500, or £5000
in all. The price is now at the rate of £500 per $10,000$
quarts, or one shilling per quart, and the annual sales
amount to $100,000$ quarts. Up to this point we have
supposed that every reduction of the price has increased
%% -----File: 122.png---Folio 97-------
the total pecuniary yield to the owner. But this cannot
go on for ever, inasmuch as the owner is seeking to
increase the value of $x × y$ by diminishing $y$ and increasing
$x$, and since in the nature of the case $x$ cannot be
indefinitely extended (there being a limit to the quantity
of the water wanted by the public at all) it follows that
as $y$ diminishes a point must come at which the increase
of~$x$ will fail to compensate for the decrease of~$y$, and $xy$
will become smaller as $y$ decreases. This is obvious from
the figure. We suppose, then, that when the owner has
already reduced his price to one shilling a quart he finds
that further reductions fail to bring in a sufficient increase
of custom to make up for the decline in price. To make
the public take $160,000$ quarts a year he would not only
have to give it away, but would have to pay something
for having it removed.

We have supposed the owner to fix the price and to
let the quantity sold fix itself to correspond. That is,
we have supposed him to say: Any one on whose relative
scale of marginal utilities a quart of this water
stands as high as $y$~shillings may have it, and I will see
how many quarts per annum it will take to meet
the ``demand'' of all such. Hence he is constructing
a curve in which the price is the variable and the
quantity demanded at that price is the function. This
is a curve of price-and-quantity-demanded. It is usual
to call it a ``curve of demand'' simply, but this is
an elliptical, ambiguous, and misleading phrase, which
should be strictly excluded from elementary treatises.
We have seen (\Pageref{12}) that a curve is never a curve
of height, time, quantity, utility, or any other \emph{one} thing,
but always a curve of connection between some \emph{two}
things. The amounts of the things themselves are always
represented by straight lines, and it is the connection of
the corresponding pairs of these lines that is depicted on
the curve. If we not only always bear this in mind,
but always express it, it will be an inestimable safeguard
against confusion and ambiguity, and we may
%% -----File: 123.png---Folio 98-------
make it a convention always to put the magnitude
which we regard as the variable first. Thus the curve
we have just traced is a curve of price-and-quantity-demanded.

But it would have been just as easy to suppose our
owner to fix the quantity issued, and then let the price
fix itself. The curve itself would, of course, be the
same (compare pp.~\Pageref[]{3},~\Pageref[]{13}), but we should now regard it as
a curve of quantity-issued-and-intensity-of-demand. The
price obtainable always indicating the intensity of the
demand for more when just so much is issued. From
this point of view also it might be called a ``curve of
demand,'' but ``demand'' would then mean intensity of
demand (the quantity issued being given), and would
be measured by the price or~$y$. In the other case ``demand''
would mean quantity demanded (at a given
price), and would be measured by~$x$.

Now this curve of quantity-issued-and-intensity-of-demand
is the same thing as the curve of quantity-possessed-(by
the community)-and-marginal-usefulness,
or briefly quantity-and-price. Thus if we call the curve
a curve of price-and-quantity we indicate that we are
supposing the owner to fix the price and let the
quantity sold fix itself, whereas if we call it the curve
of quantity-and-price we are supposing the owner to fix
the amount he will issue and let the price fix itself. In
either case we put the variable first, and call it the
curve of the variable-and-function.

Regarding the curve as one of quantity-and-price
then, we suppose the owner to say: I will draw $x$~times
$10,000$ quarts (of course $x$ may be a fraction) from my
spring every year, and will see how urgent in comparison
with the want of ``gold'' the want that the last quart
meets turns out to be. In this case it is obvious that
as the owner increases the issue the new wants satisfied
by the larger supply will be less urgent, relatively to
``gold,'' than the wants supplied before, but still the
marginal utility of a quart relatively to ``gold'' will be
%% -----File: 124.png---Folio 99-------
the same to all the purchasers, and will be greater to
them than to any of those who do not yet take any.
Thus as the issue increases the marginal utility to the
community of a quart steadily sinks on the relative scale
of the community, and shows itself, as in the case of the
individual, to be a decreasing function of the quantity
possessed, each fresh increment meeting a less urgent
want than the last. But meanwhile the \emph{total} service
done to the community by the water is increased by
every additional quart. The man who bought one
quart a year for ten shillings, and who buys two quarts
a year when it comes down to eight shillings, and ten
quarts a year when it is only a shilling, would still be
willing to give ten shillings for a single quart if he could
not get it cheaper, and the second and following quarts,
though not ministering to so urgent a want as the first,
yet in no way interfere with or lessen the advantage it has
already conferred, while they add a further advantage of
their own. Thus from his first quart the man now gets
for a shilling the full advantage which he estimated at ten
shillings, and from the second quart the advantage he
estimated at eight shillings, and so on. It is only the last
quart from which he derives an advantage no more than
equivalent to what he gives for it. We may, therefore,
still preserving the ``gold'' standard, say that the total
utility of the $q$~quarts which \Person{A} consumes in the year is
made up of the whole sum he would have given for
one quart rather than have none, \emph{plus} the whole quantity
he would have given for a second quart sooner than
have only one $+ \ldots +$ the whole sum he gives for the
$q$th~quart sooner than be satisfied with $(q-1)$. In like
manner the successive quarts, up to~$p$, which \Person{B} adds to
his yearly consumption as the price comes down, each
confers a fresh benefit, while leaving the benefits already
conferred by the others as great as ever. Thus we
should construct for \Person{A},~\Person{B}, \Person{C}, etc., severally, curves of
quantity-and-total-utility of the water, on which we
could read the total benefit derived from any given
%% -----File: 125.png---Folio 100-------
quantity of the water by each individual measured in
terms of the marginal utility to him of the unit of gold.
And regarding the total utility as a function of the
quantity possessed, we shall, of course, find that each
consumer goes on possessing himself of more till the
first derived function (rate at which more is adding to
his satisfaction) coincides with the price at which he can
purchase the water.

In like manner we may, if we choose, add up all the
utilities of the successive quarts to \Person{A},~\Person{B}, \Person{C}, etc.,
measured in ``gold,'' as they accrue (neglecting the fact
that they are not subjectively but only objectively
commensurate with each other), and may make a curve
showing the grand total of the utility to the community
of the whole quantity of water consumed. And this
curve would of course continue to rise (though at a
decreasing rate) as long as any one who had anything to
give in exchange wanted a quart more of the water than
he had.

Thus we have seen that as the issue increases the
utility of a quart at the margin to each individual and
to the whole community continuously falls on the relative
scale, the exchange value of course (recorded in the
price) steadily accompanying it; while at the same time
each extra quart confers a fresh advantage on the
community without in any way interfering with or
lessening the advantages already conferred; that is to
say, the total advantage to the community increases as
the issue increases, whereas the marginal usefulness constantly
decreases. The maximum total utility would
be realised when the issue became free, and every one
was allowed as much of the water as he wanted, and
then the marginal utility would sink to nothing, that is
to say, no one would attach any value to more than he
already had. This is in precise accordance with the
results already obtained with reference to a single individual.
The total effect is at its maximum when the
marginal effectiveness is zero.
%% -----File: 126.png---Folio 101-------

But now returning to the owner of the spring, we
note that his attention is fixed neither upon the total
nor the marginal utility of the water, but on the total
price he receives, and we note that that price is represented
in the diagram by a rectangle, the base of
which is~$x$, or the quantity sold measured in the unit
agreed upon, and the height~$y$, the price or rate per unit
(determined by its marginal usefulness) at which when
issued in that quantity the commodity sells. The area,
therefore, is~$xy$. And this brings us to the important
principle involved in what is known as the ``law of indifference.''
By this law the owner finds himself obliged to
sell \emph{all} his wares at the price which \emph{the least urgently needed}
will fetch, for he cannot as a rule make a separate bargain
with each customer for each unit, making each pay as
much for each successive unit as that unit is worth to him;
since, unless he sold the same quantity at the same price
to all his customers, those whom he charged high would
deal with those whom he charged low, instead of directly
with him. ``There cannot be two prices for the same
article in the same market.'' Thus we see again, and
see with ever increasing distinctness, that the exchange
value of a commodity is regulated by its marginal
utility, and is independent of the service which that
particular specimen happens to render to the particular
individual who purchases it.

Thus (if we bear in mind the purely relative and
therefore socially equivocal nature of our standard of
utility) we may now generalise the conclusions we
reached in the first instance with exclusive reference to
the individual. From the collective as from the individual
point of view the marginal utility of a commodity
is a function of the quantity of it possessed or commanded.
If the quantity changes, the communal marginal
utility and therefore the exchange-value changes
with it; and this altogether irrespective of the nature
of the causes which produce the change in quantity.
Whether it is that nature provides so much and no
%% -----File: 127.png---Folio 102-------
more, or that some one who has power to control the
supply chooses, for whatever reason, to issue just so
much and no more, or that producers think it worth
while to produce so much and no more---all this, though
of the utmost consequence in determining whether and
how the supply can be further changed, is absolutely immaterial
in the primary determination of the marginal
utility, and therefore of the exchange-value, so long as
just so much and no more \emph{is} issued. This amount is
the variable, and, given a relation between the variable
and the function (\ie~given the curve), then, when the
variable is determined, no matter how, why, or by
whom, the function is thereby determined also (compare
\Pageref{62}).

\emph{Exchange value, then, is relative marginal value-in-use,
and is a function of quantity possessed.}\Pagelabel{102}%

\begin{Remark}
The ``Law of Indifference'' is of fundamental importance
in economics. Its full significance and bearing cannot be
grasped till the whole field of economics has been traversed;
but we may derive both amusement and instruction, at the
stage we have now reached, from the consideration of the
various attempts which are made to evade it, and from the light
which a reference to it throws upon the real nature of many
familiar transactions.

In the first place, then, sale by auction is often an attempt
\index{Auction}%
to escape the law of indifference. The auctioneer has, say,
ten pictures by a certain master whose work does not often
come into the market, and his skill consists in getting the
man who is most keen for a specimen to give his full price
for the first sold. Then he has to let the second go cheaper,
because the keenest bidder is no longer competing; but he
tries to make the next man give \emph{his} outside price; and so on.
The bidders, on the other hand, if cool enough, try to form a
rough estimate of the \emph{marginal} utility of the pictures, that is
to say, of the price which the tenth man will give for a
picture when the nine keenest bidders are disposed of, and
they know that if they steadily refuse to go above this point
there will be one for each of them at the price. When the
%% -----File: 128.p n g----------
%[Blank Page]
%% -----File: 129.p n g----------
\begin{figure}[p]
  \begin{center}
    \Fig{20}
    \Input[4.5in]{129a}
  \end{center}
\end{figure}
%[To face page 103.]
%% -----File: 130.png---Folio 103-------
things on sale are such as can be readily got elsewhere, the
auctioneer is powerless to evade the law of indifference.

Another instance constantly occurs in the stock markets.
\index{Stock-broking}%
A broker wishes to dispose of a large amount of a certain
stock, which is being taken, say, at~$95$. But he knows that
only a little can be sold at that price, because a few thousands
would be enough to meet all demands of the urgency represented
by that figure. In fact, the stock he has to part with
would suffice to meet all the wants represented by $93$~and
upwards, and accordingly the law of indifference would compel
him to part with the first thousand at that rate just as
much as the last if he were to offer all he means to sell
at once. This, in fact, will be the selling price of the
whole when he has completed his operations. But meanwhile
he endeavours to hold the law of indifference at bay by
producing only a small part of his stock and doing business
at~$95$ till there are no more demands urgent enough to prompt
an offer of more than~$94\frac{7}{8}$. He then proceeds cautiously to
meet these wants likewise, obtaining in each case the maximum
that the other party is willing to give; and so on, till,
if completely successful, he has let the stock down~$\frac{1}{8}$ at a
time from $95$ to~$93$. By this time, of course, not only his own
last batch, but all the others that he has sold, are down at~$93$.
The law of indifference has been defeated only so far as he is
concerned, and not in its general operation on the market.

The general principle involved is illustrated, without
special reference to the cases cited, in \Figref{20}. The law of
indifference dictates that if the quantity~$Oq_4$ is to be sold,
then $Oq$, $qq_1$, $q_1q_2$, $q_2q_3$, $q_3q_4$ must all be treated indifferently,
and therefore sold at the price measured by $Op_4$~($=q_4m_4$).
This would realise an amount represented by the area~$p_4q_4$.
But the seller endeavours to mask the fact that $Oq_4$ is to be
sold, and by issuing separate instalments tries to secure the
successive areas $pq+s_1q_1+s_2q_2+s_3q_3+s_4q_4$. Obviously the
``limit'' of this process, under the most favourable possible
circumstances, is the securing of the whole area bounded by
the curve, the axes, and the line~$q_nm_n$ (where $q_n$~stands for the
last of the series $q$,~$q_1$,~etc.)\footnote
  {If $Op$ or~$q^m$ is~$f(Oq)$, \ie~if $y$ is~$f(x)$, then the area in question
  will be $\int_0^xf(x)\,dx$ (see pp.~\Pageref[]{23},~\Pageref[]{31}). The meaning of this symbol may
  now be explained. The sum of all the rectangular areas is $pq+s_1 q_1
  +s_2 q_2+ \text{etc.}$, or $qm\centerdot Oq+q_1 m_1\centerdot qq_1+q_2 m_2\centerdot q_1q_2+ \text{etc.}$, but $qm$ is
  $f(Oq)$, $q_1m_1$ is $f(Oq_1)$, $q_2 m_2$ is $f(Oq_2)$, etc. Therefore the sum of the
  areas is
  \[
  f(Oq)\centerdot Oq+f(Oq_1)\centerdot qq_1+f(Oq_2)\centerdot q_1q_2+ \text{etc.}
  \]
  But $Oq=qq_1=q_1q_2= \text{etc.}$ We may call this quantity ``the increment
  of $x$,'' and may write it $\Delta x$. The sum of the rectangular areas will then
  be
  \begin{gather*}
    \{f(Oq)+f(Oq_1)+f(Oq_2) + \text{etc.}\} \Delta x,\\
    \text{or}\ \operatorname{sum} \{f(Oq)\} \Delta x,\ \text{or}\ \textstyle\sum \{f(Oq)\} \Delta x.
  \end{gather*}
  When we wish to indicate the limit of any expression into which
  $\Delta x$, \ie~an increment of~$x$, enters, as the increment becomes smaller
  and smaller, it is usual to say that $\Delta x$becomes~$dx$. In the
  limit then $\sum \{f(Oq)\}\Delta x$ becomes $\int f(Oq)dx$, where $\int$ is simply the
  letter~\emph{s}, the abbreviation of ``sum.'' The symbol then means, the
  limit of the sum of the areas of the rectangles as the bases become
  smaller and the number of the rectangles greater. But we have further
  to indicate the limits within which we are to perform this summing of
  the rectangles. If we wished to express the area $q_1m_1m_3q_3$ the limits
  would be $Oq_1$~and~$Oq_3$. We should wish to sum all the rectangles
  bounded by~$f(Oq_1)$, \ie~$q_1m_1$, and~$f(Oq_3)$, \ie~$q_3m_3$.
  This we should
  indicate thus---
  \[
  \int^{O_{q_3}}_{O_{q_1}}f(O_q)\centerdot dx
  \]
  And the area~$OPm_nq_n$ will be
  \[
  \int_0^{Oq_n}f(Oq)\centerdot dx
  \]
  This means that the values successively assumed by~$Oq$ in the expression,
  $\operatorname{sum} (Oq\centerdot dx)$ are, respectively, all the values between $Oq_1$~and~$Oq_3$,
  or all the values between $O$~and~$Oq_n$. Finally, since the successive
  values of~$Oq$ are the successive values of~$x$, and since $Oq_n$ is the
  last value of~$x$ we are to consider, we may write the expression for
  $OPm_nq_n$
  \[
  \int_0^xf(x)\centerdot dx
  \]
  or the expression for $q_1m_1m_nq_n$
  \[
  \int_{q_1m_1}^x f(x)\centerdot dx
  \]
  remembering the $x$ in~$f(x)$ stands for all the successive values of the
  variable,~$x$, whereas in, $\int_0^x$ or $\int_{q_1m_1}^x$ or generally $\int_{\text{constant}}^x$ $x$ stands
  only for the \emph{last} of the values of the variable considered.}
If the law of indifference takes
%% -----File: 131.png---Folio 104-------
full effect the seller is apt to regard the area~$Pp_n m_n$ as a
territory to be reclaimed. The public, he thinks, has got it
without paying for it. If the law of indifference is completely
evaded, the public, in its turn, is apt to think that it
has been cheated to the extent of this area.

We may now consider some more special cases of attempts
to escape the action of the law of indifference. The system
of  ``two prices'' in retail dealing is a good instance. It is an
attempt to isolate two classes of customers and to confine the
action of the law of indifference to equalising the prices within
these classes, taken severally. In fact, the principle of ``fixed
prices in retail trade'' is strictly involved in the frank acceptance
of the law of indifference; and all evasions or modifications
of that principle are attempts to escape the action of
the law. The extent to which ``double prices'' prevail in
London is perhaps not generally realised. A differential
charge of a halfpenny or penny a quart on milk, for instance,
\index{Milkman@{Milkman's prices}}%
according to the average status (estimated by house rent) of
%% -----File: 132.png---Folio 105-------
the inhabitants of each street or neighbourhood, seems to be
common.

It is clear, too, that when he has established a system of
differential charges, the tradesman can, if he likes, sell to the
low-priced customer at a price which would not pay him\footnote
  {This phrase is used in anticipation, but is perhaps sufficiently
  clear (see below).}
if
charged all round; for the small profit he would make on each
transaction would not enable him to meet his standing expenses.
Having met them, however, from the profits of his high-priced
business, he may now put down any balance of receipts over
expenses out of pocket on the other business as pure gain. If in
\Figref{20} the rectangles represent not the actual receipts for the
respective sales, but the balance of receipts over expenses out of
pocket on each several transaction, we may suppose that the
dealer requires to realise an area of~$20$ in order to meet his
standing expenses and make a living. He can do business
to the extent of~$Oq_4$ at the (gross)\footnote
  {\textit{I.e.}~surplus of receipts over expenses out of pocket \emph{on that transaction},
  all standing expenses being already incurred.}
rate of profit~$Op_4$, which gives
him his area of~$20$, \ie~$p_4q_4$. If he did business to the extent
of~$Oq_n$ at a uniform (gross) profit of~$Op_n$, he would only
secure an area of~$18$, \ie~$p_nq_n$, and so could not carry on business
at all. But if he can keep $Oq_4$ at the profit~$Op_4$, and
%% -----File: 133.png---Folio 106-------
then without detriment to the other add $q_4q_n$ at a profit
$Op_n$, he secures $20+8$, \ie~$p_4q_4+s_nq_n$.  Nay, it is conceivable
enough that he could not carry on business at all except on
the principle of double prices. Suppose, in the case illustrated
by the figure, that he must realise an area of~$25$ in
order to go on. It will be found that no rectangle containing
so large an area can be drawn in the curve. The maximum
rectangle will be found to correspond to the value of
nearly $4.5$ for~$x$, which will give an area of only a little more
than $20$. If the law of indifference, then, takes full effect,
our tradesman cannot do business at all; but if he can deal
with $Oq_4$ and $q_4q_n$ separately, he may do very well.

In this case the ``double price'' system is the only possible
one; and the high-priced customers are not really paying an
unnaturally high price. For unless \emph{some one} pays as high as
that the ware cannot be brought into the market at all. But
it would be easy so to modify our supposition as to make the
tradesman a kind of commercial Robin Hood, forcing up the
price for one class of customers above the level at which they
would naturally be able to obtain their goods, and then
lowering it for others below the paying line.

The differential charges of railway companies illustrate
\index{Railway@{\textsc{Railway} charges, differential}}%
this. A company finds that certain goods~$Oq$ must necessarily
be sent on their line, whereas $qq_4$ may be equally well
sent by another line. An average surplus of receipts
over expenses out of pocket represented by an area of four
units per unit of~$x$ will pay the company; \ie~$Op_4$ per
unit, giving $p_4q_4$ or $20$ on the carriage of $Oq_4$ would pay.
On $Oq$ the company puts a charge which will yield gross
profits at the rate of~$Op$, and thus secure $pq=14$. They
then underbid the other company for the carriage of~$qq_4$. $Op_4$
being the minimum average gross profit that will pay (in
view of standing expenses), they offer to carry at a gross
profit of~$Op_n$, for their standing expenses are already incurred,
and they thus secure an extra gross profit of $qs_n$ ($=8$) which,
together with the $pq$ ($=14$) they have already secured, gives
them a total of~$22$, or $2$~more than if they had run at
uniform prices. Of the ten extra units of area which they
extracted from the consigners of~$Oq$, they have given eight to
the consigners of~$qq_4$ in the shape of a deduction from the
legitimate charge.
%% -----File: 134.p n g----------
%[Blank Page]
%% -----File: 135.p n g----------
%[** TN: Labels have been transcribed faithfully from the original.]
\begin{figure}[p]
  \begin{center}
    \Fig{21}
    \Input{135a}
  \end{center}
\end{figure}
% [To face page 107.]
%% -----File: 136.png---Folio 107-------

Another interesting case is that of a theatre. Here the
\index{Theatre, pit and stalls}%
``two (or more) price'' system is disguised by withholding
from the low-price customers certain conveniences which practically
cost nothing, but which serve as a badge of distinction
and enable the high-price customers to pay for the privilege
of being separated from the rest without offensively parading
before them that this separation is in fact the privilege for
which they are paying 8s.~each. The accommodation is
limited, and the nature of the demand varies according to the
popularity of the piece. Except under quite exceptional circumstances
custom fixes the charges for stalls and pit, to which we
will confine ourselves; and though the manager would rather
fill his floor with stalls than with benches, yet he is glad of all
the half-crowns which do not displace half-guineas, since his
expenses out of pocket for each additional pittite are trivial or
non-existent. Neglecting the difference of space assigned to
a sitter in a stall and on a bench, let us suppose the whole
floor to hold $800$~seats, $400$~of which are made into stalls.
Representing a hundred theatre-goers by a unit on~$x$, and the
rate of 1s.~a head, or £5 a 100 by the unit on~$y$, and so
making each unit of area represent £5 receipts, we may
read the two curves $a$~and~$a'$ in \Figref{21} thus. There is a
nightly supply of four hundred theatre-goers who value the
entertainment, accompanied by the dignity and comfort of a
stall at not less than 10s.~6d.\ a seat (rate of £52:10s.\ per
hundred seats.) There are also five hundred more who value
it, with the discomforts of the pit, at 2s.~6d.\ a seat (rate of
£12:10s.\ per hundred). There is not accommodation for all
the latter, since there are but four hundred pit seats, and
custom prevents the manager from filling his pit at a little
over 3s.~a place as he might do. So he lets his customers fight
it out at the door and takes in four hundred at 2s.~6d.\ each
(area~$p'a'$). His takings are $(10.5× 4+2.5× 4) \text{ times £5}=\text{£260}$,
since each unit of area represents~£5. The areas
are $pa$~and~$p'a'$. The former $pa$ is as great as the marginal
utility of the article offered admits of, but the latter
$p'a'$ is limited horizontally by the space available and vertically
by custom.

As the public gets tired of the play the curves $a$~and~$a'$ are
replaced by $b$~and~$b'$. The manager might fill his stalls by
going down to 8s., and might almost fill his pit at~2s. But
%% -----File: 137.png---Folio 108-------
custom forbids this. His prices are fixed and his issue of tickets
fixes itself. He has 200~stalls and 300~places in the pit
taken every night. Area $=pb+p'b'$. Receipts $(10.5× 2+2.5× 3)$
times £5 = £142:10s.

When the manager puts on a new piece the curves $c$~and~$c'$
\index{Theatre, waiting}%
\index{Waiting@{Waiting (at theatre)}}%
replace $b$~and~$b'$; and finding that he can issue six
hundred stall tickets per night at 10s.~6d., the manager
pushes his stalls back and cuts down the pit to two
hundred places, for which six or seven hundred theatre-goers
fight; several hundred more, who would gladly have
paid 2s.~6d.\ each for places, retreating when they find
that they must wait a few hours and fight with wild
beasts for ten minutes in addition to paying their half-crowns.
When the two hundred successful competitors find
that the manager has not sacrificed £80 a night for the
sake of keeping the four hundred seats they consider due to
them and their order, they try to convince him that a pittite
and peace therewith is better than a stalled ox and contention
with it. It would be interesting to know in what terms they
would state their case; but evidently the merely commercial
principles of ``business'' do not command their loyal assent.
The areas $pc+p'c'$ are $(10.5× 6+2.5× 2) \text{ times £5}=\text{£340}$.

The case of ``reduced terms'' at boarding schools is very
\index{Reduced terms at school}%
like the cases of the railway and the theatre. The reader
may work it out in detail. As long as the school is not full,
the ``reduced'' pupils do something towards helping things
along, if they pay anything more than they actually eat and
break. At the same time it would be impossible to meet the
standing expenses and carry on the school if the terms were
reduced all round. If pupils are taken at reduced terms
when their places could be filled by paying ones, then the
master is sacrificing the full amount of the reduction.

These instances, which might be increased almost
indefinitely, will serve to illustrate the importance of the law
of indifference and the attempts to escape its action.\Pagelabel{108}%
\end{Remark}

Having now a sufficiently clear and precise conception
of the marginal utilities of various commodities \emph{to the
community}, we may take up again from the general
point of view the investigation which we have already
%% -----File: 138.png---Folio 109-------
entered upon (on \Pageref{58}) with reference to the individual,
and may inquire what principles will regulate the direction
taken in an industrial community by the labour
(and other efforts or sacrifices, if there are any others)
needful to production.

Strictly speaking, this does not come within the
scope of our present inquiry. We have already seen
that the exchange value of an article is a function of the
quantity possessed, completely independent of the way
in which that quantity comes to be possessed; and
any inquiries as to the circumstances that determine, in
particular cases, the actual quantity produced and therefore
possessed, fall into the domain of the ``theory of
production'' or ``making'' rather than into that of the
``theory of value'' or ``worth.'' But the two subjects
have been so much confounded, and the connection
between them is in reality so intimate and so important,
that even an elementary treatment of the subject of
``value'' would be incomplete unless it included an
examination of the simplest case of connection between
value and what is called cost of production. The consideration
of any case except the simplest would be out
of place here.

Suppose \Person{A} can command the efforts and sacrifices
needed to produce either $U$~or~$V$, and suppose the production
of either will require the same application of
these productive agents per unit produced. Obviously~\Person{A},
if he approaches his problem from the purely mercantile
side, has simply to ask, ``Which of the two, when
produced, will be worth most in `gold' to the community?''\
\ie, he must inquire which of the two has the
highest relative marginal utility, or stands highest on the
relative scale. Suppose a unit~$u$ has, at the margin,
twice the relative utility of the unit~$v$; \Person{A}~will then
devote himself to the production of~$U$, for by so doing
he will create a thing having twice the exchange value,
and will therefore obtain twice as much in exchange, as
if he took the other course. He will therefore produce
%% -----File: 139.png---Folio 110-------
$u$ simply because, when produced, it will exchange for
more ``gold'' than~$v$. \Person{A}~will not be alone in this preference.
Other producers, whose productive forces are
freely disposable, will likewise produce~$U$ in preference
to~$V$, and the result will be a continual increase in the
quantity of~$U$. Now we have seen that an increased
quantity of~$U$ means a decreased marginal usefulness of~$U$
measured in ``gold,'' so that the production of~$U$ in
greater and greater quantities means the gradual declension
on the relative scale of its unitary marginal utility,
and its gradual approximation to that of~$V$, which will
cause the exchange values of $u$~and~$v$ to become more
and more nearly equal. But as long as the marginal
utility of~$u$ stands at all above that of~$v$ on the relative
scale, the producers will still devote themselves by preference
to the production of~$U$, and consequently its
marginal usefulness will continue to fall on the
scale until at last it comes down to that of~$V$\@. Then
the marginal utilities and exchange values of $u$~and~$v$
will be equal, and as the expenditure of productive
forces necessary to make them is by hypothesis equal
also, there will be no reason why producers should
prefer the one to the other. There will now be equilibrium,
and if more of \emph{either} is produced, then more of
\emph{both} will be produced in such proportions as to preserve
the equilibrium now established. In fact the diagram
(\Figref{14}, \Pageref{60}) by which we illustrated the principle upon
which a wise man would distribute his own personal
labour between two methods of directly supplying his
own wants, will apply without modification to the
principles upon which purely mercantile considerations
tend to distribute the productive forces in a mercantile
society. But though the diagram is the same there is a
momentous difference in its signification, for in the one
case it represents a genuine balancing of desire against
desire in one and the same mind or ``subject,'' where
the several desires have a real common measure; in the
other case it represents a mere mechanical and external
%% -----File: 140.png---Folio 111-------
equivalence in the desires gratified arrived at by
measuring each of them in the corresponding desires for
``gold'' existing respectively in \emph{different} ``\emph{subjects}.''

It only remains to generalise our conclusions. No
new principle is introduced by supposing an indefinite
number of alternatives, instead of only two, to lie before
the wielders of productive forces. There will always be
a tendency to turn all freely disposable productive forces
towards those branches of production in which the
smallest sum of labour and other necessaries will produce
a given utility; that is to say, to the production of
those commodities which have the highest marginal
utility in proportion to the labour, etc., required to produce
them; and this rush of productive forces into these
particular channels will increase the amount of the
respective commodities, and so reduce their marginal
usefulness till units of them are no longer of more value
at the margin than units of other things that can be
made by the same expenditure of productive forces.
There will then no longer be any special reason for
further increasing the supply of them.

The productive forces of the community then, like
the labour of a self-sufficing industrial unit, will tend to
distribute themselves in such a way that a given sum of
productive force will produce equal utilities at the
margin (measured externally by equivalents in ``gold'')
wherever applied.

To make this still clearer, we may take a single case
in detail, and supposing general equilibrium to exist
amongst the industries, may ask what will regulate the
extent to which a newly developed industry will be
taken up? But as a preliminary to this inquiry we
must define more closely our idea of a general equilibrium
amongst the industries. On \Pageref{73}~\textit{sqq} we established
the principle that if commodities $A$~and~$B$ are
freely exchanged, and commodities $B$~and~$C$ are freely
exchanged also, then the unitary marginal utilities, and
thus the exchange values of $a$~and~$c$, may be expressed
%% -----File: 141.png---Folio 112-------
each in terms of the other, even though it should happen
that no owners of~$A$ want~$C$, and no owners of~$C$ want~$A$,
and in consequence there is no direct exchange between
them. In like manner the principle of the distribution
of efforts and sacrifices just established enables us to
select a single industry as a standard and bring all the
others into comparison with it. It will be convenient,
as we took gold for our standard commodity, so to take
gold-digging as our standard industry; and as we have
\index{Gold-digging}%
written ``gold'' as a short expression for ``gold and all the
commodities in the circle of exchange, expressed in terms
of gold,'' so we may write ``gold-digging'' as a short expression
for ``gold-digging and all the industries open to
producers, in equilibrium with gold-digging,'' and we
shall mean by one industry being in equilibrium with
another that the conditions are such that a unit of
effort-and-sacrifice applied at the margin of either
industry will produce an equivalent utility.\footnote
  {To speak of the ``margin'' of an industry again involves an
  anticipation of matters not dealt with in this volume, but I trust it
  will create no confusion. It must be taken here simply to mean ``a
  unit of productive force added to those already employed in a certain
  industry,'' and the assumption is that all units are employed at the
  same advantage, the difference in the utility of their yields being due
  simply to the decreasing marginal utility of the same unit of the commodity
  as the quantity of the commodity progressively increases.}
If, then,
a sufficient number of persons have a practical option
between gold-digging~($\alpha$) and cattle-breeding~($\beta$), this
\index{Cattle-breeding}%
will establish equilibrium between these two occupations
$\alpha$~and~$\beta$ in accordance with the principle just laid
down; and if a sufficient number of other persons to
whom gold-digging is impossible have a practical option
between cattle-breeding~($\beta$) and corn-growing~($\gamma$), then
\index{Corn-growing}%
that will establish equilibrium between $\beta$ and~$\gamma$. But
since there will always be equilibrium between $\alpha$ and~$\beta$
as long as sufficient persons have the option between
them, and since that equilibrium will be restored, whenever
disturbed, by the forces that first established it, it
follows that if there is equilibrium between $\beta$ and~$\gamma$
%% -----File: 142.p n g----------
%[Blank Page]
%% -----File: 143.p n g----------
\begin{figure}[p]
  \begin{center}
    \Fig{22}
    \Input{143a}
  \end{center}
\end{figure}
% [To face page 113.]
%% -----File: 144.png---Folio 113-------
there will be equilibrium between $\alpha$ and~$\gamma$ also. We
may therefore conveniently select $\alpha$~or gold-digging as
the industry of general reference, and may say that a
man will prefer $\gamma$~or corn-growing to ``gold-digging'' as
long as the yield is higher in the former industry,
although as a matter of fact it is not the yield in gold-digging
but the yield in cattle-breeding (itself equilibrated
with gold-digging) with which he directly compares
his results in corn growing. Industries in equilibrium
with the same are in equilibrium with each
other.

We assume, then, that there is a point of equilibrium
about which all the industries, librated with each other
directly and indirectly, oscillate; and, neglecting the
oscillations, we use the yield to a given application of
productive forces in gold-digging as the representative
of the equivalent yield in all the other industries in
equilibrium with it.

Now we imagine a new industry to be proposed, and
producers who command freely disposable efforts and
sacrifices to turn their attention to it. Their option is
between the new industry and ``gold-digging,'' in the
extended sense just explained. We are justified in
assuming, for the sake of simplicity, that the whole sum
of the productive forces under consideration would not
sensibly affect the marginal usefulness of ``gold'' (in the
extended sense, observe) if applied to ``gold-digging;''
that is to say, we assume that in no case will the new
industry draw to itself so great a volume of effort-and-sacrifice
as to starve the other industries of the world,
taken collectively, and make the general want of the things
they yield perceptibly more keen. Therefore, in examining
the alternative of ``gold-digging,'' we assume that the
whole volume of labour and other requisites of production,
or effort-and-sacrifice, which is in question might
be applied to ``gold-digging'' without reducing the marginal
usefulness of ``gold,'' or might be withdrawn from
it without increasing that usefulness.  The yield in
%% -----File: 145.png---Folio 114-------
``gold'' of any quantity of labour and other requisites,
then, would be exactly proportional to that quantity.

Fixing on any arbitrary unit of effort-and-sacrifice
(say $100,000$ foot-tons), and taking as our standard unit
of utility the gold that it would produce (say $30$~ounces),
we may represent the ``gold'' yield of any given amount
of labour and other requisites by the aid of a straight
line, drawn parallel to the abscissa at a distance of unity
from it (\Figref{22}). Thus if $Oq$~effort and sacrifice were
devoted to ``gold-digging,'' the area~$Gq$ would represent
the exchange value of the result. Now let the upper
curve on the figure be the curve of quantity-and-marginal-usefulness
of the new product, the unit of quantity
being that amount which the unit of labour and other
requisites ($100,000$ foot-tons) will produce. And here
we must make a simplification which would be violent
if we were studying the theory of production, but which
is perfectly legitimate for our present purpose. We
must suppose, namely, that however much or little of
the new product is secured it is always got under the
same conditions, so that the yield per unit effort-and-sacrifice
is the same at every stage of the process. But
though the \emph{quantity} produced by a unit of productive
force is always the same its marginal usefulness and
exchange value will of course descend, according to the
universal law, as the total quantity of the ware increases.
In the first instance, then, the commercial mind has
simply to ask, ``Are there persons to whom such an
amount of this article as I can produce by applying the
unit of productive force will be worth more than the
`gold' I could produce by the same application of force?''
In other words, ``Will the unit of productive force applied
to this industry produce more than the unit of utility?''
Under the conditions represented in the figure the
answer will be a decisive affirmative, and the producer
will turn his disposable forces of production into the new
channel. But as soon as he does so the most importunate
demands for the new article will be satisfied, and if any
%% -----File: 146.png---Folio 115-------
further production is carried on it must be to meet a
demand of decreasing importunacy, \ie~the marginal
utility of the article is decreasing, and the exchange
value of the yield of the unit of productive force in
the new industry is falling. Production will continue,
however, as long as there is any advantage in the new
industry over gold-production, \ie~till the yield of unit
productive force in the new industry has sunk to unit
utility.

Thus, if $Oq_1$~effort and sacrifice is devoted to the
new industry, the marginal usefulness of the product will
be measured by~$q_1f_1$, and the exchange value of the
whole output by the rectangle bounded by the dotted
line and $q_1f_1$,~etc. This is much more than $Gq_1$ the
alternative ``gold'' yield to the same productive force.
But there is still an advantage in devoting productive
forces to the new industry, since $q_1f_1$ is greater than~$q_1g_1$,
and even if the present producers are unable to
devote more work to it, or unwilling to do so, because
it would diminish the area of the rectangle (\Pageref{96}), yet
there will be others anxious to get a return to their
work at the rate of~$q_1f_1$ instead of~$q_1g_1$. Obviously,
then, the new commodity will be produced to the extent
of~$Oq$ where $qf=qg$, \ie,~the point at which the curve
cuts the straight line~$Gg$, which is the alternative ``gold''
curve. If production be carried farther it will be carried
on at a disadvantage. At~$q_2$, for instance, $q_2f_2$~is less
than~$q_2g_2$, that is to say, if the supply is already~$Oq_2$,
then a further supply will meet a demand the importunity
of which is less than that of the demand for the
``gold'' which the same productive force would yield.
This will beget a tendency to desert the industry, and
will reduce the quantity towards~$Oq$.

We have supposed our units of ``gold'' and the new
commodity so selected that it requires equal applications
of productive agencies to secure either, but in practice
we usually estimate commodities in customary units that
have no reference to any such equivalence. This of
%% -----File: 147.png---Folio 116-------
course does not affect our reasoning. If the unit of~$F$ is
such that our unit of labour and other necessaries yields
a hundred units of~$F$ and only one unit of~$G$, then,
obviously, we shall go on producing~$F$ until, but only
until, the exchange value of a hundred units of~$F$ (the
product of unit of labour, etc., in~$F$) becomes equal to the
exchange value of one unit of~$G$ (the product of unit of
labour, etc., in~$G$). Or, generally, if it needs $x$~times as
much effort and sacrifice to produce one unit~$A$ as it
takes to produce one unit~$B$, then it takes as much to
produce $x$~units $B$ as to produce one unit~$A$, and there
will always be an advantage either in producing~$xb$ or
in producing one~$a$, by preference, unless the exchange
value of both is the same; that is to say, unless the
marginal value of~$a$ equals $x$~times that of~$b$. Thus, \emph{if $a$~contains
$x$~times as much work as~$b$, then there will not be
equilibrium until $A$ and~$B$ are produced in such amounts as
to make the exchange value of~$a$ just $x$~times the exchange
value of~$b$}.

This, then, is the connection between the exchange
value of an article (that can be produced freely and in
indefinite quantities) and the amount of work it contains.
Here as everywhere the quantity possessed
determines the marginal utility, and with it the exchange
value; and if the curve is given us we have only
to look at the quantity-index in order to read the exchange
value of the commodity (see pp.~\Pageref[]{62},~\Pageref[]{67}). But in
the practically and theoretically very important case of
commodities freely producible in indefinite quantities
we may now note this further fact as to the principle
by which the position of the quantity-index is in its turn
fixed---that fluid labour-and-sacrifice tends so to distribute
itself and so to shift the quantity-indexes as to
make \emph{the unitary marginal utility of every commodity
directly proportional to the amount of work it contains}.

\begin{Remark}
This fact, that the effort-and-sacrifice needed to produce
two articles is, in a large class of cases (those, namely, in
%% -----File: 148.png---Folio 117-------
which production is free and capable of indefinite extension),
proportional to the exchange values of the articles themselves,
has led to a strange and persistent delusion not only amongst
the thoughtless and ignorant but amongst many patient and
earnest thinkers, who have not realised that the exchange
value of a commodity is a function of the quantity possessed,
and may be made to vary indefinitely by regulating
that quantity. The delusion to which I refer is that it is the
amount of effort-and-sacrifice or ``labour'' needed to produce
a commodity which \emph{gives that commodity its value in exchange}.
A glance at \Figref{22} will remind the reader of the magnitude
and scope of the error involved in this idea. The commodity,
on our hypothesis, always contains the same amount
of effort-and-sacrifice per unit, whether much or little is produced,
but the fact that only the unit of ``labour'' has been
put into it does not prevent its exchange value being more
than unity all the time till it exists in the quantity~$Oq$, nor
does the fact of its containing a full unit of labour keep its
exchange value up to unity as soon as it exists in excess of
the quantity~$Oq$. What gives the commodity its value in
exchange is the quantity in which it exists and the nature of
the curve connecting quantity and marginal usefulness; and
it is no more true and no more sensible to say that the
quantity of ``labour'' contained in an article determines its
value than it would be to say that it is the amount of money
which I give for a thing that makes it useful or beautiful.
The fact is, of course, precisely the other way. I give so
much money for the thing because I expect to find it useful
or think it beautiful; and the producer puts so much
``labour'' into the making of a thing because when made he
expects it to have such and such an exchange value. Thus
one thing is not worth twice as much as another because it
has twice as much ``labour'' in it, but producers have been
willing to put twice as much ``labour'' into it because they
know that when produced it will be worth twice as much,
because it will be twice as ``useful'' or twice as much
desired.

This is so obvious that serious thinkers could not have
fallen into and persisted in the error, and would not be
perpetually liable to relapse into it, were it not for certain
considerations which must now be noticed.
%% -----File: 149.png---Folio 118-------

In the first place, if we have not fully realised and completely
assimilated the fact that exchange value is a function
of the quantity possessed, and changes as the quantity-index
shifts, it seems reasonable to say, ``It is all very well to
say that because people want~$a$ twice as much as~$b$ they
will be \emph{willing to do} twice as much to get~$a$ as they will to
get~$b$, but how does it follow that they will be \emph{able to get} the
article~$a$ by devoting just twice as much labour to it as to~$b$?
Surely you cannot maintain that it \emph{always happens} that
the thing people want twice as much needs exactly twice as
much ``labour'' to produce as the other? And yet you
admit yourself that the thing which has twice the exchange
value always does contain twice the ``labour.'' If it is not
a chance, then, what is it?'' The answer is obvious, and the
reader is recommended to write it out for himself as clearly
and concisely as possible, and then to compare it with the
following statement: If people want~$a$ just twice as much
as~$b$, and no more, it does not follow that a producer will
find $a$ just twice as hard to get, but it does follow that if he
finds~$a$ is \emph{more} than twice as hard to get (say $x$~times as hard)
he will not get it at all, but will devote his productive
energies to making~$b$. Confining ourselves, for the sake
of simplicity, to these two commodities, we note that other
producers will, for the like reason, also produce~$B$ in preference
to~$A$. The result will be an increased supply of~$B$,
and, therefore, a decreased intensity of the want of it;
whereas the want of~$A$ remaining the same as it was, the
utility of~$a$ is now more than twice as great as the (diminished)
utility of~$b$; and as soon as the want of~$b$ relatively to the
want of~$a$ has sunk to~$\dfrac{1}{x}$, then one~$a$ is worth $x$~$b$'s, and as it
needs just $x$~times the effort-and-sacrifice to produce~$a$, there
is now equilibrium, and $A$ and~$B$ will \emph{both} be made in such
quantities as to preserve the equilibrium henceforth; but the
proportion of one utility to the other, and the proportion
of the ``labour'' contained in one commodity to that
contained in the other, do not ``happen'' to coincide; they
have been \emph{made} to coincide by a suitable adjustment of efforts
so as to secure the maximum satisfaction.

Another source of confusion lurks in the ambiguous use
of the word ``because''; and behind that in a loose conception
%% -----File: 150.png---Folio 119-------
of what is implied and what is involved in one thing being
the ``cause'' of another.

Thus we sometimes say ``$x$~is true because $y$ is true,''
when we mean not that $y$ being true is the \emph{cause}, but that it
is the \emph{evidence} of $x$ being true. For instance, we might say
``prime beef is less esteemed by the public than prime
mutton, because the latter sells at~$1$d.\ or~$\frac{1}{2}$d.\ more per pound
than the former.'' By this we should mean to indicate the
higher price given for mutton not as the cause of its being
more esteemed, but as the evidence that it is so.\footnote
  {Such psychological reactions as the desire to put one dish on the
  table in preference to another, simply because it is known to be more
  expensive, do not fall within the scope of this inquiry.}
So again,
``Is the House sitting?''---``Yes! because the light on the clock-tower
\index{House of Commons sitting}%
is shining.'' This does not mean that the light shining
causes the House to sit, but that it shows us it is sitting.

In like manner a man may say, ``If I want to know how
much the exchange value of~$a$ exceeds that of~$b$, I shall look
into the cost of producing them, and if I find four times as
much `labour' put into~$a$, I shall say $a$~is worth four times~$b$,
because I find that producers have put four times the
`labour' into it;'' and if he means by this that he knows
the respective values in exchange of $a$~and~$b$ on the evidence
of the amount of effort-and-sacrifice which he finds producers
willing to put into them respectively, then we have no fault
to find with his economics, though he is using language
dangerously liable to misconception. But if he means that
it is the effort-and-sacrifice, or ``labour,'' contained in them
which \emph{gives} them their value in exchange, he is entirely
wrong. As a matter of fact, the defenders of the erroneous
theory sometimes make the assertion in the erroneous sense,
victoriously defend it, when pressed, in the true sense, and
then retain and apply it in the erroneous sense.

Again, though it is never true that the quantity of
``labour'' contained in an article \emph{gives} it its value-in-exchange,
yet it may be and often is true, in a certain sense, that the
quantity of ``labour'' it contains is the \emph{cause} of its having
such and such a value in exchange. But if ever we allow
ourselves to use such language we must exercise ceaseless
vigilance to prevent its misleading ourselves and others.
%% -----File: 151.png---Folio 120-------
For what does it mean? The quantity-index and the curve
fix the value-in-exchange. But the quantity-index may run
the whole gamut of the curve, and we have seen that what
determines the direction of its movement and the point at
which it rests is, in the case of freely producible articles,
precisely the quantity of ``labour'' contained in the article.
This quantity of ``labour'' contained, then, determines the
amount of the commodity produced, and this again determines
the value-in-exchange. In this sense the amount of
``labour'' contained in an article is the cause of its exchange
value. But this is only in the same sense in which the
approach of a storm may be called the cause of the storm-signal
\index{Storm-signal}%
rising. The approach of the storm causes an intelligent
agent to pull a string, and the tension on the string causes
the signal to rise. In this sense the storm is the cause of
the signal rising. But it would be a woful\DPnote{** [sic] legitimate variant} mistake, which
might have disastrous consequences, to suppose that there is
any immediate causal nexus between the brewing of the
storm and the rising of the ball. And if our mechanics
were based on the principle that a certain state of the atmosphere
``gives an upward movement to a storm-signal,'' the
science would stand in urgent need of revision. So in our
case: Relative ease of production makes intelligent agents
produce largely if they can; increasing production results in
falling marginal utilities and exchange-values; therefore, in a
certain sense, ease of production causes low marginal utilities
and exchange-values. But there is no immediate causal
nexus between ease of production and low exchange-values.
Exchange values, high and low, are found in things which
cannot be produced at all; and if (owing to monopolies,
artificial or natural) the intelligent agents who observe how
easily a thing is produced are not in a position to produce it
abundantly, or have reasons for not doing so, the ease of
production may coexist with a very high marginal utility,
and consequently with a very high exchange value. In such
a case the amount of ``labour'' contained in the article will
be small out of all proportion to its exchange-value; and the
quantity produced may be regulated by natural causes that
have no connection with effort and sacrifice, or by the desire
on the part of a monopolist to secure the maximum gains.

Finally, there are certain phenomena, of not rare occurrence
%% -----File: 152.png---Folio 121-------
in the industrial world, which really seem at first
sight to give countenance to the idea that the exchange-value
of a commodity is determined, not by its marginal
desiredness, but by the quantity of ``labour'' it contains.
These phenomena are for the most part explained by the
principle of ``discounting,'' or treating as present, a state of
things which is foreseen as certain to be realised in a near
future. For instance, suppose a new application of science to
industry, or the rise into favour of a new sport or game, suddenly
\index{Games@{\textsc{Games}}}%
creates a demand for special apparatus, and suppose one
or two manufacturers are at once prepared to meet it. They
may, and often do, take advantage of the urgency of the want
of those who are keenest for the new apparatus, and sell it at its
full initial exchange-value, only reducing their price as it becomes
necessary to strike a lower level of desire, and thus
travelling step by step all down the curve of quantity-and-value-in-exchange
till the point of equilibrium is at last reached, and
every one can buy the new apparatus who desires it as much
as the ``gold'' that the same effort-and-sacrifice would produce.
But it may also happen that the manufacturers who are
already on the field foresee that others will very soon be
ready to compete with them, and that it will require a comparatively
small quantity of the new apparatus to bring it
down to its point of equilibrium, inasmuch as it cannot,
in the nature of the case, be very extensively used. They
feel, therefore, that they have not much to gain by securing
high prices for the first specimens, and on the other hand, if
they ``discount'' or anticipate the fall to the point of equilibrium,
and at once offer the apparatus on such terms as will
secure all the orders, they will prevent its being worth while
for any other manufacturers to enter upon the new industry,
and will secure the whole of the permanent trade to themselves.

Any intermediate course between these two may likewise
be adopted; but the discounting or anticipation of the foreseen
event only disguises and does not change the nature of
the forces in action.

A more complicated case occurs when a man wants a
single article made for his special use which will be useless
to any one else. Let us say he wants a machine to do certain
work and to fit into a certain place in his shop. The importance
%% -----File: 153.png---Folio 122-------
to him of having such a machine is great enough
to make him willing to give £100 for it sooner than go
without it. But the ``labour'' (including the skill of the
designer) needed to produce it would, if applied to making
other machines, or generally to ``gold-digging,'' only produce
an article of the exchange-value of £50. ``In this case,'' it
will be said, ``the marginal utility of the machine is measured
by £100, yet the manufacturer (if his skill is not a monopoly)
can only get £50 for making it, because it only contains
labour and other requisites to production represented by that
sum. Does not this show conclusively that it is the ``labour''
contained in an article, not its final utility, which determines
its exchange-value?'' To judge of the validity of this objection,
let us begin by asking exactly what our theory would
lead us to anticipate, and then let us compare it with the
alleged facts. We have seen that in equilibrium the marginal
utility of the unit of a commodity must occupy the same
place on the relative scales of all those who possess it;
and further, that if ever that marginal utility should be
higher on \Person{A}'s relative scale than on \Person{B}'s, then (if \Person{B} possesses
any of the commodity) the conditions for a mutually profitable
exchange exist, though on what terms that exchange
will be made remains, as far as our investigations have taken
us, indeterminate, within certain assignable limits. Now if
we suppose the machine to be actually made we shall have
this situation: \Person{A}, on whose relative scale the marginal
utility of the machine stands at £100 has not got it. \Person{B},
on whose relative scale it stands at zero, possesses it. The
conditions of a mutually advantageous exchange therefore
exist. But the terms on which that exchange will take place
are indeterminate between 0~and~£100. When a single
exchange has been made, on whatever terms, then the
article will stand at zero on every relative scale except
that of its possessor, and no further exchange will be
made. \emph{If the machine exists}, therefore, its exchange-value
will be indeterminate between zero and £100. Now if
we consistently carry out our system of graphic representation
this position will be reproduced with faultless accuracy.
The curve of quantity-possessed-and-marginal usefulness with
reference to the community being drawn out, the vertical
intercept on the quantity-index indicates the exchange-value
%% -----File: 154.png---Folio 123-------
of the commodity. Now in this instance the curve in question
consists of the rectangle in \Figref{23}~(\textit{a}), where the unit on
the axis of~$y$ is £100~per machine, and the unit on the axis
of~$x$ is one machine. For the usefulness of the first machine
to the community is at the rate of £100~per machine, and
the usefulness of all other machines at the rate of $0$~per
machine. Therefore the curve falls abruptly from $1$ to $0$ \emph{at}
the value $x=1$. But the quantity possessed by the community
is one machine. Therefore the quantity index is at
\begin{figure}[hbtp]
  \begin{center}
    \Fig{23}
    \Input[3.5in]{154a}
  \end{center}
\end{figure}
the distance unity from the origin, \Figref{23}~(\textit{b}). What is the
length of the intercept? Obviously it is indeterminate between
$0$ and $1$. This is exactly in accordance with the facts.
Supposing the machine actually to exist, then, our theory
vindicates itself entirely. But if the machine does not yet
exist, what does our theory tell us of the prospect of its being
made? We have seen that a thing will be made if there is a
prospect of its exchange-value, when made, being at least as
great as that of anything else that could be made by the same
effort-and-sacrifice. Now the exchange-value is determined
by the intercept on the quantity-index. Before the machine
is made that intercept is $1$ ($=\text{£100}$), but that does not concern
the maker, for he wants to know what it \emph{will be} when
the machine is made, not what it is before. But it will be
indeterminate, as we have seen, and therefore there is no
security in making the machine. In order to get the
machine made, therefore, the man who wants it must remove
the indeterminateness of the problem by stipulating in
advance that he will give not less than £50 for it. But
what he is now doing is not getting the machine (which does
not exist) in exchange for ``gold.'' It is getting control or
%% -----File: 155.png---Folio 124-------
direction of a given application of labour, etc. in exchange
for ``gold,'' and this being so, it is not to be wondered at
that the price he pays for this ``labour'' should be proportionate
to the quantity of it he gets.

This is the general principle of ``tenders'' for specific
work.
\end{Remark}

\Pagelabel{124}%
We may appropriately close our study of exchange
value by a few reflections and applications suggested
by the ordinary expenditure of private income, and
especially shopping and housekeeping.

On \Pageref{58} we considered what would be the most
sensible way of distributing labour amongst the various
occupations which might claim it on a desert island.
There labour was the purchasing power, and the question
was in what proportions it would be best to exchange it
for the various things it could secure. We were not
then able to extend the principle to the more familiar
case of money as a purchasing power, because we had
not investigated the phenomena of exchange value and
price. We may now return to the problem under this
aspect. The principle obviously remains the same.
Robinson Crusoe, when industrial equilibrium is established
\index{Robinson Crusoe}%
in his island, so distributes his labour that the
last hour's work devoted to each several task results in
an equivalent mass or body of satisfaction in every case.
If the last hour devoted to securing \Person{A} produced less
satisfaction than the last hour devoted to securing \Person{B},
Robinson would reduce the former application of labour
till, his stock of \Person{A} falling and its marginal usefulness
rising, the last hour devoted to securing it produced a
satisfaction as great as it could secure if applied otherwise.
He would then keep his supply at this level, or
advance the supply of \Person{A} and \Person{B} together in such proportions
as to maintain this relation. If he lets his stock
of \Person{A} sink lower he incurs a privation which could be
removed at the expense of another privation not so
great; if he makes it greater he gets a smaller gratification
at a cost which would have secured a greater
%% -----File: 156.png---Folio 125-------
one if applied elsewhere. In equilibrium, then, the last
hour's work applied to each task produces an equal
gratification, removes an equal discomfort, or gratifies
an equal volume of desire; which is to say, that Robinson's
supply of all desired things is kept at such a
level that the unitary marginal utilities of them all
are directly proportional to the labour it takes to secure
them.

In like manner the householder or housewife must
\index{Housekeeper}%
\Pagelabel{125}%
aim at making the last penny (shilling, pound, or whatever,
in the particular case, is the \textit{minimum sensibile}\footnotemark)
\footnotetext{\Ie, the smallest thing he can ``feel.'' The importance of this
  qualification will become apparent presently (see \Pageref{129}).}
expended on every commodity produce the same gratification.
If this result is not attained then the money
is not spent to the best advantage. But how is it to be
attained? Obviously by so regulating the supplies of
the several commodities that the marginal utilities of a
pennyworth of each shall be equal. We take it that the
demand of the purchaser in question is so small a part
of the total demand for each commodity as not sensibly
to affect the position of its quantity-index on the national
register, and we therefore take the price of each commodity
as being determined, independently of his
demand, on the principles already laid down. There is
enough lump sugar available of a given quality to supply
\index{Sugar}%
all people to whom it is worth 3d.\ a pound. Our housewife
therefore gets lump sugar until the marginal utility
of one pound is reduced to the level represented by 3d.
Perhaps this point will be reached when she buys six
pounds a week. The difference between six pounds and
seven pounds a week is not worth threepence to her.
The difference between five pounds and six is. Sooner
than go without any loaf sugar at all she would perhaps
pay a shilling a week for one pound. That pound
secured, a second pound a week would be only worth,
say, eightpence. Possibly the whole six pounds may
represent a total utility that would be measured by
%% -----File: 157.png---Folio 126-------
$(12\text{d.} + 8\text{d.} + 5\tfrac{1}{2}\text{d.} + 4\text{d.} + 3\tfrac{1}{2}\text{d.}+ 3\text{d.})$ three shillings, or
an average of sixpence a pound, but the unitary marginal
utility of a pound is represented by threepence.
Another housekeeper might be willing to give one and
sixpence a week for a pound of sugar sooner than go
without altogether, and to give a shilling a week for
a second pound, but her demand, though more keen, may
be also more limited than her neighbour's. She gets a
third pound a week, worth, say, sevenpence to her, and
a fourth worth threepence, and there she stops, because
a fifth pound would be worth less than threepence to
her, and there is only enough for those who think it
worth 3d.\ a pound or more. She has purchased for a
shilling sugar the total utility of which is represented
by $(18\text{d.} + 12\text{d.} + 7\text{d.}+ 3\text{d.} =)$ 3s.~4d., but the unitary
marginal utility of a pound is 3d., as in the other case.

So with all other commodities. Each should be purchased
in such quantities that the marginal utility of one
pennyworth of it exactly balances the marginal utility of
one pennyworth of any of the rest; the absolute marginal
utility of the penny itself changing, of course, with
circumstances of income, family, and so forth, but the
relative utilities of pennyworths at the margin always
being kept equal to each other. The clever housekeeper
has a delicate sense for marginal utilities, and can
balance them with great nicety. She is always on the
alert and free from the slavery of tradition. She follows
changes of condition closely and quickly, and keeps
her system of expenditure fluid, so to speak, always
ready to rise or fall in any one of the innumerable and ever
shifting, expanding and contracting channels through
which it is distributed, and so always keeping or
recovering the same level everywhere. She keeps her
marginal utilities balanced, and never spends a penny on
A when it would be more effective if spent on B; and
combines the maximum of comfort and economy with
the minimum of ``pinching.''

The clumsy housekeeper spends a great deal too much
%% -----File: 158.png---Folio 127-------
on one commodity and a great deal too little on another.
She does not realise or follow the constant changes of
condition fast enough to overtake them, and buys
according to custom and tradition. Her system of
expenditure is viscous, and cannot change its levels
so fast as the channels change their bore. She can
never get her marginal utilities balanced, and therefore,
though she drives as hard bargains as any one,
and always seems to ``get her money's worth'' in
the abstract, yet in comfort and pleasure she does
not make it go as far as her neighbour does, and
never has ``a penny in her pocket to give to a boy,''\footnote
  {The absence of which was lamented by an old Yorkshire woman
  as the greatest trial incident to poverty and dependence.}
\index{Penny@{\textsc{Penny} ``to give to a boy''}}%
a
fact that she can never clearly understand because she
has not learned the meaning of the formula, ``My coefficient
of viscosity is abnormally high.''

\begin{Remark}
It is rather unfortunate for the advance of economic
science that the class of persons who study it do not as a rule
belong to the class in whose daily experience its elementary
principles receive the sharpest and most emphatic illustrations.
For example, few students of economics are obliged to
realise from day to day that a night's lodging, and a supper,
possess utilities that fluctuate with extraordinary rapidity;
and the tramps who, towards nightfall, in the possession of
twopence each, make a rush on suppers, and sleep out, if the
thermometer is at~$45°$, and make a rush on the beds and go
\index{Thermometer}%
supperless if it is at~$30°$, have paid little attention to the
economic theories which their experience illustrates. As a
rule it seems easier to train the intellect than to cultivate the
imagination, and while it is incredibly difficult to make the
well-to-do householder realise that there are people to whom
the problem of the marginal utilities of a bed and a bowl of
\index{Bed@{Bed \textit{versus} supper}}%
stew is a reality, on the contrary, it is quite easy to demonstrate
the general theory of value to any housekeeper who
has been accustomed to keep an eye on the crusts, even
though she may never have had any economic training. For
the great practical difficulty in the way of gaining acceptance
for the true theory is the impression on the part of all but
%% -----File: 159.png---Folio 128-------
the very poor or the very careful that it is contradicted by
experience. In truth our theory demands that no want
should be completely satisfied as long as the commodity that
satisfies it costs anything at all; for in equilibrium the
unitary marginal utilities are all to be proportional to the
prices, and if any want is completely satisfied then the
unitary marginal utility of the corresponding commodity
must be zero, and this cannot be proportional to the price
unless that is zero too. Again, since all the unitary marginal
utilities are kept proportional to the prices, it follows
that though none of them can \emph{reach} zero while the corresponding
commodity has any price, they must all \emph{approach} zero
together. Now all this, it is said, is contrary to experience.
In the first place, we all of us have as much bread and meat
and potatoes as we want, though they all cost something;
and in the next place, whereas the marginal utility of these
things has actually reached zero, the marginal utility of pictures,
horses, and turtle soup has not even approached it, for
\index{Turtle soup}%
we should like much more than we get of them all.

We have only to run this objection down in order to see
how completely our theory can justify itself; but we must
begin by reminding ourselves---first, that real commodities
are not infinitely divisible, and that we are obliged to choose
between buying a \emph{definite quantity} more or no more at all;
and second, that our mental and bodily organs are only capable
of discerning certain definite intervals. There may be
two tones, not in absolute unison, which no human ear could
distinguish; two degrees of heat, not absolutely identical, which
the most highly trained expert could not arrange in their
order of intensity. With this proviso as to the \emph{minimum
venale}\footnote
  {The reply, ``We don't make up ha'poths,'' which damps the
  purchasing ardour of the youth of Northern England, is constantly
  made by nature and by man to the economist who tries to apply the
  doctrine of continuity to the case of individuals.}
and the \textit{minimum sensibile}, let us examine the supposed
case in detail. A gentleman has as much bread but not as
much turtle soup as he would like. This is bad husbandry, for
he ought to stop short of the complete gratification of his desire
for bread at the point represented by a usefulness of sixteen-pence
a quartern (for we assume that he takes the best quality),
and the surplus which he now wastefully expends on reducing
%% -----File: 160.png---Folio 129-------
that usefulness to absolute zero might have been spent on
turtle soup. But let us see how this would work. We must
not allow him to adopt the royal precept of eating cake when
he has no bread, but must suppose him \textit{bona fide} to save on
his consumption of bread in order to increase his expenditure
on turtle and on nothing else. Probably he already
resembles Falstaff in incurring relatively small charges on
\index{Falstaff}%
account of bread---say his bill is 3d.~a~day. He has as much
\Pagelabel{129}%
as he wants, and therefore the marginal utility is zero, but the
curve descends rapidly, and if we reduce his allowance by
one-sixth, and his toast at breakfast, his roll at dinner and
lunch, and his thin bread-and-butter at tea, or with his white-bait,
are all of them a little less than he wants, he will find
that the marginal utility of bread has risen far above 1s.~4d.\
a quartern, and is more like a shilling an ounce. Taking
the unit of~$x$ as $1$~ounce, and the unit of~$y$ as 1d., it is a
delicate operation to arrest the curve for some value between
$x=2\tfrac{1}{2}$, $y=12$, and $x=3$, $y=0$. But let us suppose
our householder equal to it. He finds that $x=2\tfrac{3}{4}$ gives
$y=1$, and accordingly determines to dock himself of $\tfrac{1}{12}$
of his supply and save $\tfrac{1}{4}$d.~a~day on bread. But now
arises another difficulty. He wants always to have his bread
fresh, and the $\tfrac{1}{4}$d.~worth he saves to-day is not suitable
for his consumption to-morrow. The whole machinery
of the baking trade and of his establishment is too
rough to follow his nice discrimination. Its utmost delicacy
cannot get beyond discerning between $2\tfrac{1}{2}$d.~and~3d., and he
finds that to be sure of not letting the marginal utility of
bread down to zero he must generally keep it up immensely
above 1d.~per ounce. Suppose this difficulty also overcome.
Then our economist saves $\tfrac{1}{4}$d.~a~day on bread or 6d.\ in twenty-four
days. In one year and 139~days he has saved enough to
get an extra pint of turtle soup, which (if it does not reduce its
marginal utility below 10s.~6d.)\ fully compensates him for
his loss of bread---but not for the mental wear and tear and the
unpleasantness in the servants' hall which have accompanied
his fine distribution of his means amongst the objects of
his appetite. This is in fact only an elaboration of the principle
laid down on \Pageref{125}.

As a rule, however, it is by no means true that we all
have as much bread, meat, and potatoes as we want. Omitting
%% -----File: 161.png---Folio 130-------
all consideration of the great numbers who are habitually
hungry, and confining our attention to the comfortable classes
who always have enough to eat in a general way, we shall
nevertheless find that the bread-bill is very carefully watched,
and that a sensible fall in the price of bread would immediately
cause a sensible increase in the amount taken.
For instance, if bread were much cheaper, or if the housekeeping
\index{Resurrection pudding}%
allowance were much raised, many a crust would be
allowed to rest in peace which now reappears in the ``resurrection
pudding,'' familiar rather than dear to the schoolboy,
who has given it its name; but also known in villadom,
where his sister uncomplainingly swallows it without vilifying
it by theological epithets.

The assertion which for a moment seems to be true of
bread, though it is not, is obviously false when made concerning
milk, meat, potatoes, etc. The people who have ``as
much as they want'' of these things are few; and if in most
cases a more or less inflexible tradition in our expenditure
prevents us from quite realising that we save out of potatoes
to spend on literature or fashion, it is none the less true that
we do so. Indeed, there are probably many houses in which
sixpence a week is consciously saved out of bread, milk,
cheese, etc., for the daily paper during the session, when its
\index{Daily@{\textsc{Daily Paper}}}%
marginal utility is relatively high, to be restored to material
purposes when Parliament adjourns.

Before leaving the subject of domestic expenditure, I
would again emphasise the important part which tradition
and viscosity play in it. This is so great that sometimes a
loss of fortune, which makes it absolutely necessary to break
\index{Fortune, loss of}%
up the established system and begin again with the results of
past experience, but free from enslaving tradition, has been
found to result in a positive increase of material comfort and
enjoyment.

One of the benefits of accurate account-keeping consists in
\index{Account-keeping}%
the help it is found to give in keeping the distribution of
funds fluid, and preventing an undue sum being spent on any
one thing without the administrator realising what he is
doing.\Pagelabel{130}%
\end{Remark}

A few miscellaneous notes may be added, in conclusion,
on points for which no suitable place has been
%% -----File: 162.png---Folio 131-------
found in the course of our investigation, but which cannot
be passed over altogether.

\begin{Remark}
The reader may have observed a frequent oscillation
between the conceptions of ``so much a year, a month, a day,
etc.,'' and ``so much'' absolutely. If a man has one watch,
he will want a second watch less. But we cannot say that
if he has one loaf of bread he will want a second loaf less.
We can only say if he has one loaf of bread \emph{a week} (or a day,
or some other period) he will want a second less. Our
curves then do not always mean the same thing. Generally
the length on the abscissa indicates the breadth of a
stream of supply which must be regarded as continuously
flowing, for most of our wants are of such a nature as to
destroy the things that supply them and to need a perpetual
renewal of the stores provided to meet them. And in the
same way the area of the curve of quantity-and-marginal-usefulness
or the height of the curve of quantity-and-total-utility
does not indicate an absolute sum of gratification or
relief from pain, but a rate of enjoyment or relief per week,
month, year, etc. Thus, strictly speaking, the value of~$y$ in
one of our quantity-and-marginal-usefulness curves measures
the rate at which increments in the \emph{rate of supply} are increasing
the \emph{rate of enjoyment}; but we may, when there is no
danger of misconception, cancel the two last ``rates'' against
each other, and speak of the rate at which increments in the
\emph{supply} increase the \emph{gratification}; for the gratification (area)
and the supply (base), though rates absolutely, are not rates
with reference to each other, but the ratio of the increase of
the one to the increase of the other is a rate with reference
to the quantities themselves.

We must remember, then, that, as a matter of fact, it is
generally rates of supply and consumption, not absolute
quantities possessed, of which we are speaking; and especially
when we are considering the conditions of the maintenance
of equilibrium. It will repay us to look into this conception
more closely than we have hitherto done; and as the problem
becomes extremely complex, unless we confine ourselves
to the simplest cases, we will suppose only two persons, \Person{A}~and~\Person{B},
to constitute the community, and only two articles,
$V$~and~$W$, to be made and exchanged by them, $V$~being made
%% -----File: 163.png---Folio 132-------
exclusively by~\Person{A}, and $W$~exclusively by~\Person{B}. Let the curves on
\Figref{24} represent \Person{A}'s and \Person{B}'s curves of quantity-and-marginal-utility
of $V$~and~$W$; and let \Person{A} consume~$V$ at the rate of $q_{av}$~per
day (or month or other unit of time) and $W$~at the rate of~$q_{aw}$,
while \Person{B} consumes~$V$ at the rate of~$q_{bv}$, and $W$~at the rate of~$q_{bw}$.
And let the position of the amount indices in the figure
represent a position of equilibrium. Let us first inquire how
many of the data in the figures are arbitrary, and then ask
what inferences we can draw as to the conditions for maintaining
equilibrium and the effects of failure to comply with
those conditions.

Since the quantities $q_{av}$, $q_{aw}$, etc. represent rates of consumption,
it is evident that if equilibrium is to be preserved
the rate of production must exactly balance them. Now the
total rate of consumption, and therefore of production, of~$V$
is $q_{av}+q_{bv}$, and that of~$W$ is $q_{aw}+q_{bw}$, calling these respectively
$q_v$ and $q_w$, we have
\begin{align*}
\text{(i)\ \ } q_{av} &+ q_{bv} = q_v, \\
\text{(ii) }  q_{aw} &+ q_{bw} = q_w.
\end{align*}

If we call the ratio of the marginal utility of~$w$ to that of~$v$
on \Person{A}'s relative scale~$r$, then we shall know, by the general
law, that in equilibrium the respective marginal utilities
must bear the same ratio on the relative scale of~\Person{B}; and if \Person{A}'s
curve of quantity-and-marginal-usefulness in~$V$ be $y=\phi_a(x)$,
and if $y=\psi_a(x)$, $y=\phi_b(x)$, $y=\psi_b(x)$ be the other three curves
then we shall have
\[
\text{(iii)\ (iv) } \frac{\psi_a(q_{aw})}{\phi_a(q_{av})}=\frac{\psi_b(q_{bw})}{\phi_b(q_{bv})}=r,
\]
where $\phi_a(q_{av})$ etc.\ are the vertical intercepts on the figures,
and where each of the ratios indicated is the ratio of the
marginal utility of~$w$ to that of~$v$ on the relative scale. And,
finally, since \Person{B} gets all his~$V$ by giving $W$ in exchange for
it, getting $r$~units $v$ in exchange for one unit~$w$, and since the
rate at which he gets it is, on the hypothesis of equilibrium,
the rate at which he consumes it ($q_{bv}$), and the rate at which
he gives $W$ is the rate at which \Person{A}~consumes it~($q_{aw}$), we have
\[
\text{(v) } q_{bv}=rq_{aw},
\]
and we suppose, throughout, that the consumption and production
%% -----File: 164.p n g----------
%[Blank Page]
%% -----File: 165.p n g----------
\begin{figure}[p]
  \begin{center}
    \Fig{24}
%    \Input{165a}
  \end{center}
\end{figure}
%[To face page 133.]
%% -----File: 166.png---Folio 133-------
go on continuously, that is to say, not by jerks, so
that the conditions established are never disturbed.

Here, then, we have eleven quantities,
\[
q_v, q_w, q_{av}, q_{aw}, q_{bv}, q_{bw},
\phi_a(q_{av}), \psi_a(q_{aw}), \phi_b(q_{bv}), \psi_b(q_{bw}), r,
\]
and we have five relations between them. It follows that
we may arbitrarily fix any six of the eleven quantities. Our
five relations will then determine the other five.

Thus, if in the figures we assume that the four curves are
known, that is equivalent to assuming that $\phi(q_{av})$, etc. are
given in terms of $q_{av}$, etc., which reduces the number of our
unknown quantities to seven, between which we have five
relations. We may therefore arbitrarily fix two of them.
Say $q_v=13$, $q_w=7$. We shall then have
\begin{gather*}
\text{(i)\ \ }q_{av}+q_{bv}=13, \\
\text{(ii) }q_{bw}+q_{aw}=7, \\
\text{(iii)\ (iv) }\frac{\psi_a(q_{aw})}{\phi_a(q_{av})}=\frac{\psi_b(q_{bw})}{\phi_b(q_{bv})}=r, \\
\text{(v) }q_{bv}=rq_{aw},
\end{gather*}
which, if the meaning of $\phi_a(x)$ etc.\ be known, as we have
supposed, gives us five equations by which to determine five
unknown quantities. If $\phi_a(x)$ etc.\ were interpreted in accordance
with the formulæ of the curves in the figure, these
equations would yield the answers
\begin{align*}
q_{av} & = 5, \\
q_{aw} & = 4, \\
q_{bv} & = 8, \\
q_{bw} & = 3, \\
     r & = 2.
\end{align*}

I do not give the formulæ, and work out the calculation,
since such artificial precision tends to withdraw the attention
from the real importance of the diagrammatic method, which
consists in the light it throws on the nature of processes, not
in any power it can have of theoretically anticipating concrete
industrial phenomena.

Now suppose \Person{A} ceases, for any reason, to produce at the
rate of~$13$, and henceforth only produces at the rate of~$10$.
The equilibrium will then be disturbed and must be re-established
under the changed conditions. We shall have the
same five equations from which to determine the distribution
%% -----File: 167.png---Folio 134-------
of $V$ and~$W$, and the equilibrium exchange value between
them except that (i)~will be replaced by
\[
q_{av}+q_{bv}=10.
\]

If we wrote out $\phi_a(q_{av})$, etc., in terms of $(q_{av}$,~etc., according
to the formulæ of the curves, we might obtain definite
answers giving the values of $(q_{av}$,~etc., and $r$~for equilibrium
under the new conditions; but without doing so we can
determine by inspection the general character of the change
which will take place.

If \Person{A} continues, as before, to consume~$W$ at the rate of~$4$,
giving $V$ for it at the rate of~$8$, he will only be able to consume~$V$
at the rate of~$2$ himself, and the marginal utility of~$v$
will rise to more than half that of~$w$. He will therefore
find that he is buying his last increments of~$W$ at too high
a price, and will contract his expenditure on it, \ie,~the quantity
index of~$(q_{aw}$, will move in the direction indicated by the
arrow-head. But again, if \Person{A}~continues to consume~$V$ at the full
present rate of~$5$, he will only be able to use it for purchasing~$W$
at the rate of (the remaining)~$5$, instead of~$8$ as now, and he
will therefore get less than~$(q_{aw}$. The marginal utility of~$w$
will therefore be more than twice that of~$v$, and \Person{A}~will find
that he is enjoying his last increments of~$V$ at too great a
sacrifice of~$W$. He will therefore consume less~$V$, and the
quantity index will move in the direction indicated by the
arrow-head, \ie, \Person{A}~will consume less~$V$ and less~$W$, and the
unitary marginal values of both of them will rise.

But since we have seen that \Person{A}~gives less~$V$ to~\Person{B} (and
receives less~$W$ from him), it follows that~\Person{B}, who cannot
produce~$V$ himself, must consume it at a slower rate than
before. This is again indicated by the direction of the
arrow-head on the quantity-index of~$q_{bv}$. Lastly, since \Person{A}~now
receives less~$W$ than before there is more left for~\Person{B}, who
now consumes it at an increased rate; as is again indicated
by the arrow-head of the quantity-index of~$q_{bw}$.

Now since \Person{B}'s~quantity-indexes are moving in opposite
directions, and the one is registering a higher and the other
a lower marginal usefulness, it follows that the new value of~$r$
will be lower than the old one.  \Person{A}'s~quantity-indexes, then,
must move in such a way that the length intercepted on that
of~$q_{av}$ shall increase more than the length intercepted on that
%% -----File: 168.png---Folio 135-------
of~$q_{aw}$. Whether this will involve the former index actually
moving farther than the latter depends on the character of
the curves.

The net result is that though the rate of exchange has
altered in favour of~\Person{A}, yet he loses part of his enjoyment of
$V$~and of~$W$ alike, while \Person{B}~loses some of his enjoyment of~$V$,
but is partly (not wholly) compensated by an increased enjoyment
of~$W$.

If we begin by representing the marginal usefulness of $V$
and~$W$ as being not only relatively but absolutely equal for
\Person{A}~and~\Person{B}, then the deterioration in \Person{A}'s~position relatively to
\Person{B}'s after the change will be indicated by the final usefulness
of both articles coming to rest at a higher value for him than
for~\Person{B}.

The only assumption made in the foregoing argument is
that all the curves decline as they recede from the origin.

It should be noted---first, that we have investigated the
conditions with which the new equilibrium must comply
when reached, and the general character of the forces that
will lead towards it, but not the precise quantitative relations
of the actual steps by which it will be reached; and second,
that since the equations (iii)~and~(iv) involve quadratics (if
not equations of yet higher order), it must be left undetermined
in this treatise whether or not there can theoretically
be two or more points of equilibrium.

The investigation of the same problem with any number
of producers and articles is similar in character. But if we
discuss the conditions and motives that determine the amounts
of each commodity produced by \Person{A},~\Person{B},~etc.\ respectively, we shall
be trespassing on the theory of production or ``making.''

Now, if we turn from the problem of rates of consumption
and attempt to deal with \emph{quantities possessed}, in the strict
sense, without reference to the wearing down or renewal of
the stocks, we shall find the problem takes the following
form. Given \Person{A}'s~stock of~$V$, an imperishable article which
both he and~\Person{B} desire; given \Person{B}'s~stock of~$W$, a similar
article; and given \Person{A}'s and~\Person{B}'s curves of quantity-and-marginal-desiredness
for $V$ and~$W$ alike; on what principles and
in what ratio will \Person{A}~and~\Person{B} exchange parts of their stocks?
The problem appears to be the same as before, but on closer
inspection it is found that equation~(v) does not hold; for we
%% -----File: 169.png---Folio 136-------
cannot be sure that $V$ and~$W$ will be exchanged at a uniform
rate up to a certain point, and then not exchanged any more.
Therefore we cannot say
\[
q_{bv}= rq_{aw},
\]
for in the case of \emph{rates} of production, of exchange, and of consumption,
every tentative step is reversible at the next moment.
By the flow of the commodities the conditions assumed as
data are being perpetually renewed; and if either of the
exchangers finds that he can do better than he has done as
yet, he can try again with his next batch with exactly the same
advantages as originally, since at every moment he starts fresh
with his new product; and if the stream of this new product
flows into channels regulated in any other way than that
demanded by the conditions of equilibrium we have investigated,
then ever renewed forces will ceaselessly tend with
unimpaired vigour to bring it into conformity with those
conditions, so long as the curves and the quantities produced
remain constant. But when the stocks are absolute, and
cannot be replaced, then every partial or tentative exchange
\emph{alters the conditions}, and is so far irreversible; nor is there
any recuperative principle at work to restore the former conditions.
The problem, therefore, is indeterminate, since we
have not enough equations to find our unknown quantities
by. The limits within which it is indeterminate cannot be
examined in an elementary treatise. The student will find
them discussed in F.~Y. Edgeworth's \textit{Mathematical Psychics}
(London,~1881).

This problem of absolute quantities possessed is not only
of much greater difficulty but also of much less importance
than the problem of \emph{rates} of consumption. For when we
are considering the economic aspect of such a manufacture
as that of watches, for instance, though the wares are, relatively
\index{Watches}%
speaking, permanent, and we do not talk of the ``rate
of a man's consumption'' of watches, as we do in the case of
bread---or umbrellas,---yet the \emph{manufacturer} has to consider the
rate of consumption of watches per~annum, etc., regarded as a
stream, not the absolute demand for them considered as a volume.
Hence the cases are very few in which we have to deal
with absolute quantities possessed, from the point of view of
the community and of exchange values.  But this does not
%% -----File: 170.png---Folio 137-------
absolve us from the necessity of investigating the problem
with reference to the individual, for he possesses some things
and consumes others, and has to make equations not only
between possession and possession, and again between consumption
and consumption, but also between possession and
consumption. That is to say, he must ask not only, ``Do I
prefer to possess a book of Darwin's or a Waterbury watch?''
\index{Darwin's Works}%
\index{Watches}%
and, ``Do I prefer having fish for dinner or having a cigar
\index{Cigar}%
\index{Fish for dinner}%
with my coffee?'' but he must also ask, ``Do I prefer to
\emph{possess} a valuable picture or to \emph{consume} so much a year in
\index{Pictures}%
places at the opera?'' or, in earlier life, ``Is it worth while
\index{Opera@{\textsc{Opera}}}%
to give up \emph{consuming} ices till I have saved enough to \emph{possess}
\index{Ices@{\textsc{Ices}}}%
a knife?'' But these problems generally resolve themselves
\index{Knife}%
into the others. The picture is regarded as yielding a
revenue of enjoyment, so to speak, and so its possession
becomes a rate of consumption comparable with another rate
of consumption; and the abstinence from ices is of definite
duration and the total enjoyment sacrificed is estimated and
balanced against the total enjoyment anticipated from the
possession of the knife. If, however, the enjoyment of the
knife is regarded as a permanent revenue (subject to risks of
loss) it becomes difficult to analyse the process of balancing
which goes on in the boy's mind, for he seems to be comparing
a \emph{volume} of sacrifice and a \emph{stream} of enjoyment, and
the stream is to flow for an indefinite period. Mathematically
the problem must be regarded as the summing of a
convergent series; but if we are to keep within the
limits of an elementary treatise, we can only fall back
upon the fact that, however he arrives at it, the boy
``wants'' the knife enough to make him incur the privations
of ``saving up'' for the necessary period. He is balancing
``desires,'' and whether or not we can get behind them and
justify their volumes or weights it is clear that, as a matter
of fact, he can and does equate them.

This will serve as a wholesome reminder that we have
throughout been dealing with the balancing of \emph{desires} of
equal weight or volume. I have spoken indifferently of
``gratification,'' ``relief,'' ``enjoyment,'' ``privation,'' and so
forth, but since it is only with the \emph{estimated} volumes of all these
that we have to do the only things really compared are the
\emph{desires} founded on those estimates. And so too the ``sense
%% -----File: 171.png---Folio 138-------
of duty,'' ``love,'' ``integrity,'' and other spiritual motives all
\index{Duty, sense of}%
inspire desires which may be greater or less than others, but
are certainly commensurate with them. This thought, when
pursued to its consequences, so far from degrading life, will
help us to clear our minds of a great deal of cant, and to
substitute true sentiment for empty sentimentality. When
inclined to say, ``I have a great affection for him, and would
do anything I could for him, but I cannot give money for I
have not got it,'' we shall do well to translate the idea into
the terms, ``My marginal desire to help him is great, but
relatively to my marginal desire for potatoes, hansom cabs,
\index{Hansom@{\textsc{Hansom Cabs}}}%
books, and everything on which I spend my money, it is not
high enough to establish an `effective' demand for gratification.''
It may be perfectly right that it should be so; but
then it is not because ``affection cannot be estimated in
potatoes;'' it is because the gratification of this particular
affection, beyond the point to which it is now satisfied, is
(perhaps rightly) esteemed by us as not worth the potatoes
it would cost. Rightly looked upon, this sense of the
unity and continuity of life, by heightening our feelings of
responsibility in dealing with material things, and showing
that they are subjectively commensurable with immaterial
things, will not lower our estimate of affection, but will
increase our respect for potatoes and for the now no longer
``dismal'' science that teaches us to understand them in their
social, and therefore human and spiritual, significance.
\end{Remark}
%% -----File: 172.png---Folio 139-------


\Chapter[Summary---Definitions and Propositions]{%
Summary of Important Definitions and Propositions Contained in this Book.}
\Pagelabel{139}

\hspace*{\parindent}I\@.  One quantity is a function of another when any change in the
latter produces a definite corresponding change in the former (\Pagerange{1}{6}).

II\@.  The total utility resulting from the consumption or possession
of any commodity is a function of the quantity of the commodity
consumed or possessed (\Pagerange{6}{8}).

III\@. The connection between the quantity of any commodity
possessed and the resulting total utility to the possessor is theoretically
capable of being represented by a curve (\Pagerange{8}{15}).

IV\@. Such a curve would, as a rule, attain a maximum height,
after which it would decline; and in any case it would \emph{tend} to reach
a maximum height (\Pagerange{15}{19}).

V\@.  If such a curve were drawn, it would be possible to derive from
it a second curve, showing the connection between the quantity of
the commodity already possessed and the rate at which further increments
of it add to the total utility derived from its consumption or
possession; and the height of this derived curve at any point would
be the differential coefficient of the height of the original curve at
the same point (\Pagerange{19}{39}).

VI\@.  The differential coefficient of the total effect or value-in-use
of a commodity is its marginal effectiveness or degree of final
utility; as a rule marginal effectiveness is at its maximum when
total effect is zero, and marginal effectiveness is zero when total
effect is at its maximum (\Pagerange{39}{41}).

VII\@.  For small increments of commodity marginal \emph{effect} varies,
in the limit, as marginal effectiveness (\Pagerange{41}{46}).

VIII\@.  In practical life we oftener consider marginal effects than
total effects (\Pagerange{46}{48}).

IX\@.  In considering marginal effects we compare, and reduce to a
common measure, heterogeneous desires and satisfactions (\Pagerange{48}{52}).

X\@.  A unit of utility, to which economic curves may be drawn, is
conceivable (\Pagerange{52}{55}).

XI\@.  On such curves we might read the parity or disparity of
worth of stated increments of different commodities, the proper distribution
of labour between two or more objects, and all other
phenomena depending on ratios of equivalence between different
commodities (\Pagerange{55}{61}).

XII\@.  In practice the curves themselves will be in a constant
state of change and flux, and these changes, together with the
changes in the quantity of the respective commodities possessed,
%% -----File: 173.png---Folio 140-------
exhaust the possible causes of change in marginal effectiveness (\Pagerange{61}{67}).

XIII\@. The absolute intensities of two desires existing in two
different ``subjects'' cannot be compared with each other; but the
ratio of \Person{A}'s~desire for~$u$ to \Person{A}'s~desire for~$w$ may be compared with
the ratio of \Person{B}'s~desire for~$u$ or for~$v$ to \Person{B}'s~desire for~$w$ (\Pagerange{68}{71}).

XIV\@. Thus, though there can be no real subjective common
measure between the desires of different subjects, yet we may have
a conventional, objective, standard unit of desire by reference to
which the desires of different subjects may be reduced to an objective
common measure (\Pagerange{73}{77}).

XV\@.  In a catallactic community there cannot be equilibrium as
long as any two individuals, \Person{A}~and~\Person{B}, possessing stocks of the same
two commodities $U$~and~$W$ respectively, desire or esteem $u$~and~$w$
(at the margin) with unlike relative intensity (\Pagerange{71}{73}).

XVI\@.  The function of exchange is to bring about a state of
equilibrium in which no such divergencies exist in the relative intensity
with which diverse possessors of commodities severally
desire or esteem (small) units of them at the margin (\Pagerange{80}{82}).

XVII\@.  The relative intensity of desire for a unit of any given
commodity on the part of one who does \emph{not} possess a stock of it,
may fall indefinitely below that with which one or more of its possessors
desire it at the margin without disturbing equilibrium (\Pagerange{82}{86}).

XVIII\@.  Hence in every catallactic community there is a general
relative scale of marginal utilities on which all the commodities in
the circle of exchange are registered; and if any member of the
community constructs for himself a relative scale of the marginal
utilities, to him, of all the articles he possesses, this scale will (on
the hypothesis of frictionless equilibrium) coincide absolutely, as
far as it goes, with the corresponding selection of entries on the
general scale; whereas, if he inserts on his relative scale any article
he does \emph{not} possess, the entry will rank somewhere below (and may
rank \emph{anywhere} below) the position that would be assigned to it in
conformity with the general scale.

And this general relative scale is a table of \emph{exchange values}.

Thus the exchange value of a small unit of commodity is, in the
limit, measured by the differential coefficient of the total utility, to
any one member of the community, of the quantity of the commodity
he possesses; and this measure necessarily yields the same result
whatever member of the community be selected (\Pagerange{71}{86}).

XIX\@.  As a rule exchange value is at its maximum when value-in-use
is zero, and exchange value is zero when value-in-use is at its
maximum (pp.~\Pageref[]{79},~\Pageref[]{80}, \Pagerange{93}{102}).

XX\@.  If we can indefinitely increase or decrease our supplies of two
commodities, then we may indefinitely change the ratio between
the marginal effects to us, of their respective units (\Pagerange{108}{124}).

XXI\@.  Labour, money, or other purchasing power, secures the
maximum of satisfaction to the purchaser when so distributed that
equal outlays secure equal satisfactions to whichever of the alternative
margins they are applied (\Pagerange{124}{130}).
\Pagelabel{140}
%% -----File: 174.png---Folio 141-------

% INDEX OF ILLUSTRATIONS
\cleardoublepage%
\phantomsection\pdfbookmark[0]{Index of Illustrations}{Index}%
\label{indexpage}%
\printindex

\iffalse
Account-book@{\textsc{Account-book}}#Account-book 68

Account-keeping 130

Air, fresh#Air 52

Athletes 90

Auction 102

Bath-room@{\textsc{Bath-room}}#Bath-room 47

Bed@{Bed \textit{versus} supper}#Bed 127

Beer 8

Bibles 86

Bicycles 91

Billiard-tables 76

Blankets 6

Body, falling 2

Books 52, 69

Bradgate Park 68

Carbon@{\textsc{Carbon Furnace}}#Carbon 37

Cattle-breeding 112

China 50, 56

Cigar 137

Cloth, price of#Cloth 1

Coal 39, 47, 53, 63

Coats 69

Cooling iron 2

Corduroys 76

Corn-growing 112

Daily@{\textsc{Daily Paper}}#Daily 130

Darwin's Works 137

Duty, sense of#Duty 138

Eggs@{\textsc{Eggs}, fresh}#Eggs 52

Examination papers 53

Falling@{\textsc{Falling body}}#Falling 2

Falstaff 129

Fancy ball costumes 76

Fire@{Fire in ``practising'' room}#Fire 47

Fish for dinner 137

Foot-tons 53, 54

Fortune, loss of#Fortune 130

Francis of Assisi 78

Friendship 52

Games@{\textsc{Games}}#Games 121

Garden-hose 47

Gimlet 8

Gold-digging 112

Gold stoppings in teeth 75

Hansom@{\textsc{Hansom Cabs}}#Hansom 138

Holiday 84, 85

Horse 80

House of Commons sitting 119

Housekeeper 49, 125

Ices@{\textsc{Ices}}#Ices 137

Iron, cooling#Iron 2

Kitchen@{\textsc{Kitchen Fire}}#Kitchen 47

Knife 137

Lady@{\textsc{Lady Jane Grey}}#Lady 68

Linen 48, 54, 56

Meat@{\textsc{Meat}, butcher's}#Meat 16

Milkman@{Milkman's prices}#Milkman 104

Mineral spring 93

Museum, British 52

Opera@{\textsc{Opera}}#Opera 137

Penny@{\textsc{Penny} ``to give to a boy''}#Penny 127
%% -----File: 175.png---Folio 142-------

Pictures 76, 137

Plato 68

Poor men's wares 86, 87

Presents 86

Projectile 5, 8, 19, 32

Railway@{\textsc{Railway} charges, differential}#Railway 106

Rainfall 18

Reading-chairs 91

Reduced terms at school 108

Respirators 91

Resurrection pudding 130

Rich men's wares 86, 87

Robinson Crusoe 58, 124

Root-digging 58

Rossetti's Works 47

Rush-gathering 58

Sarah@{\textsc{Sarah Bernhardt}}#Bernhardt 85

Skates 91

Stock-broking 103

Storm-signal 120

Sugar 125

Testing@{\textsc{Testing Machine}}#Testing 13

Theatre, pit and stalls#Theatre 107

Theatre, waiting 69, 108

Thermometer 15, 127

Time, distribution of#Time 60

Tracts 86

Tripe 77

Turkish bath 14

Turtle soup 128

Waistcoat@{\textsc{Waistcoat}}#Waistcoat 47

Waiting@{Waiting (at theatre)}#Waiting 69, 108

Watches 7, 137, 136

Water 47, 80

Wheat 44

Wine 8

THE END
\fi
%% -----File: 176.png---Folio 143-------

%[Blank Page]

\backmatter
\phantomsection
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% End of the Project Gutenberg EBook of The Alphabet of Economic Science, by 
% Philip H. Wicksteed                                                     %
%                                                                         %
% *** END OF THIS PROJECT GUTENBERG EBOOK THE ALPHABET OF ECONOMIC SCIENCE ***
%                                                                         %
% ***** This file should be named 32497-t.tex or 32497-t.zip *****        %
% This and all associated files of various formats will be found in:      %
%         http://www.gutenberg.org/3/2/4/9/32497/                         %
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