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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.) + + + +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% Project Gutenberg's The Alphabet of Economic Science, by Philip H. 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+ +\pagestyle{empty} +\pagenumbering{alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +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 *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +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.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\pagenumbering{roman} +\pagestyle{empty} + +%% -----File: 001.png---Folio iv------- +% [Blank Page] +%% -----File: 002.png---Folio v------- + +\begin{center} +\setlength{\TmpLen}{0.2in}% +\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 +\pdfbookmark[-1]{Back Matter}{Back Matter} + +%%%% LICENSE %%%% +\pagenumbering{Alph} +\pagestyle{fancy} +\phantomsection +\pdfbookmark[0]{Project Gutenberg License}{License} +\fancyhf{} +\fancyhead[C]{\CtrHeading{Project Gutenberg License}} +\SetPageNumbers + +\begin{PGtext} +End of the Project Gutenberg EBook of The Alphabet of Economic Science, by +Philip H. 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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] + +Language: English + +Character set encoding: ISO-8859-1 + +*** 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.) + + + +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% Project Gutenberg's The Alphabet of Economic Science, by Philip H. 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+\newcommand{\ie}{\textit{i.e.}} +\newcommand{\Ie}{\textit{I.e.}} +\newcommand{\NB}{\textit{N.B.}} +\newcommand{\QED}{Q.E.D\@.} +\newcommand{\asterism}{${\smash[t]{{}^*}_*}^*$\quad} + +\newcommand{\Ditto}{\raisebox{1ex}{\textquotestraightdblbase}} +\newcommand{\Strut}{\rule{0pt}{14pt}} +\renewcommand{\arraystretch}{1.2} + +\newcommand{\Person}[1]{\textit{#1}} + +% For corrections. +\newcommand{\DPtypo}[2]{#2}% \DPtypo{txet}{text} +\newcommand{\DPnote}[1]{}% \DPnote{[** Text of note]} + +% For alignment; \PadTxt[a]{Size}{Text} sets Text in a box as +% wide as Size, with optional alignment (a = c, l, or r). +\newlength{\TmpLen} +\newcommand{\PadTxt}[3][c]{% + \settowidth{\TmpLen}{#2}% + \makebox[\TmpLen][#1]{#3}% +} +% Same, for math mode +\newcommand{\PadTo}[3][c]{% + \settowidth{\TmpLen}{$#2$}% + \makebox[\TmpLen][#1]{$#3$}% +} + +\newcommand{\Z}{\phantom{0}} + +\makeindex +\emergencystretch1em + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{alph} +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\Pagelabel{PGBoilerplate} +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +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] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE ALPHABET OF ECONOMIC SCIENCE *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +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.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter + +\pagenumbering{roman} +\pagestyle{empty} + +%% -----File: 001.png---Folio iv------- +% [Blank Page] +%% -----File: 002.png---Folio v------- + +\begin{center} +\setlength{\TmpLen}{0.2in}% +\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 +\pdfbookmark[-1]{Back Matter}{Back Matter} + +%%%% LICENSE %%%% +\pagenumbering{Alph} +\pagestyle{fancy} +\phantomsection +\pdfbookmark[0]{Project Gutenberg License}{License} +\fancyhf{} +\fancyhead[C]{\CtrHeading{Project Gutenberg License}} +\SetPageNumbers + +\begin{PGtext} +End of the Project Gutenberg EBook of The Alphabet of Economic Science, by +Philip H. 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