# Researches into the Mathematical Principles of the Theory of Wealth/Of Changes in Value, Absolute and Relative

CHAPTER II
OF CHANGES IN VALUE, ABSOLUTE AND RELATIVE
7. Whenever there is occasion to go back to the fundamental conceptions on which any science rests, and to formulate them with accuracy, we almost always encounter difficulties, which come, sometimes from the very nature of these conceptions, but more often from the imperfections of language. For instance, in the writings of economists, the definition of value, and the distinction between absolute and relative value, are rather obscure: a very simple and strikingly exact comparison will serve to throw light on this.
We conceive that a body moves when its situation changes with reference to other bodies which we look upon as fixed. If we observe a system of material points at two different times, and find that the respective situations of these points are not the same at both times, we necessarily conclude that some, if not all, of these points have moved; but if besides this we are unable to refer them to points of the fixity of which we can be sure, it is, in the first instance, impossible to draw any conclusions as to the motion or rest of each of the points in the system.
However, if all of the points in the system, except one, had preserved their relative situation, we should consider it very probable that this single point was the only one which ​had moved, unless, indeed, all the other points were so connected that the movement of one would involve the movement of all.
We have just pointed out an extreme case, viz., that in which all except one had kept their relative positions; but, without entering into details, it is easy to see that among all the possible ways of explaining the change in the state of the system there may be some much simpler than others, and which without hesitation we regard as much more probable.
If, without being limited to two distinct times, observation should follow the system through its successive states, there would be hypotheses as to the absolute movements of the different points of the system, which would be considered preferable for the explanation of their relative movements. Thus, without reference to the relative size of the heavenly bodies and to knowledge of the laws of gravitation, the hypothesis of Copernicus would explain the apparent motions of the planetary system more simply and plausibly than those of Ptolemy or Tycho.
In the preceding paragraph we have only looked on motion as a geometric relation, a change of position, without reference to any idea of cause or motive power or any knowledge of the laws which govern the movements of matter. From this new point of view other considerations of probability will arise. If, for instance, the mass of the body A is considerably greater than that of the body B we judge that the change in the relative situation of the bodies A and B is more probably due to the displacement of B than of A.
​Finally, there are some circumstances which may make it certain that relative or apparent movements come from the displacement of one body and not of another. Thus the appearance of an animal will show by unmistakable signs whether it is stopping or starting. Thus, to return to the preceding example, experiments with the pendulum, taken in connection with the known laws of mechanics, will prove the diurnal motion of the earth; the phenomenon of the aberration of light will prove its annual motion; and the hypothesis of Copernicus will take its place among established truths.
8. Let us now examine how some considerations perfectly analogous to those which we have just considered, spring from the idea of exchangeable values.
Just as we can only assign situation to a point by reference to other points, so we can only assign value to a commodity by reference to other commodities. In this sense there are only relative values. But when these relative values change, we perceive plainly that the reason of the variation may lie in the change of one term of the relation or of the other or of both at once; just as when the distance varies between two points, the reason for the change may lie in the displacement of one or the other or both of the ​two points. Thus again when two violin strings have had between them a definite musical interval, and when after a certain time they cease to give this interval, the question is whether one has gone up or the other gone down, or whether both of these effects have joined to cause the variation of the interval.
We can therefore readily distinguish the relative changes of value manifested by the changes of relative values from the absolute changes of value of one or another of the commodities between which commerce has established relations.
Just as it is possible to make an indefinite number of hypotheses as to the absolute motion which causes the observed relative motion in a system of points, so it is also possible to multiply indefinitely hypotheses as to the absolute variations which cause the relative variations observed in the values of a system of commodities.
However, if all but one of the commodities preserved the same relative values, we should consider by far the most probable hypothesis, the one which would assign the absolute change to this single article; unless there should be manifest such a connection between all the others, that one cannot vary without involving proportional variations in the values of those which depend on it.
For instance, an observer who should see by inspection of a table of statistics of values from century to century, that the value of money fell about four-fifths towards the end of the sixteenth century, while other commodities preserved practically the same relative values, would consider it very probable that an absolute change had taken place in the value of money, even if he were ignorant of the ​discovery of mines in America. On the other hand, if he should see the price of wheat double from one year to the next without any remarkable variation in the price of most other articles or in their relative values, he would attribute it to an absolute change in the value of wheat, even if he did not know that a bad grain harvest had preceded the high price.
Without reference to this extreme case, where the disturbance of the system of relative values is explained by the movement of a single article, it is evident that among all the possible hypotheses on absolute variations some explain the relative variations more simply and more probably than others.
If, without being limited to consideration of the system of relative values at two distinct periods, observation follows it through its intermediate states, a new set of data will be provided to determine the most probable law of absolute variations, from all possibilities for satisfying the observed law of relative variations.
9. Let
$p_{1},p_{2},p_{3},{\mbox{ etc.}},$
be the values of certain articles, with reference to a gram of silver; if the standard of value is changed and a myriagram of wheat is substituted for the gram of silver, the values of the same articles will be given by the expressions
${\frac {1}{a}}p_{1},{\frac {1}{a}}p_{2},{\frac {1}{a}}p_{3},{\mbox{ etc.}},$
a being the price of the myriagram of wheat, or its value with reference to a gram of silver. In general, whenever it ​is desired to change the standard of value, it will suffice to multiply the numerical expressions of individual values by a constant factor, greater or less than unity; just as with a system of points conditioned to remain in a straight line, it would suffice to know the distances from these points to any one of their number, to determine by the addition of a constant number, positive or negative, their distances referred to another point of the system, taken as the new origin.
From this there results a very simple method of expressing by a mathematical illustration the variations which occur in the relative values of a system of articles. It is sufficient to conceive of a system composed of as many points arranged in a straight line as there are articles to be compared, so that the distances from one of these points to all the others constantly remain proportional to the logarithms of the numbers which measure the values of all these articles with reference to one of their number. All the changes of distance which occur by means of addition and subtraction, from the relative and absolute motions of such a system of movable points, will correspond perfectly to the changes by means of multiplication and division in the system of values which is being compared: from which it follows that the calculations for determining the most probable hypothesis as to the absolute movements of a system of points, can be applied, by going from logarithms back to numbers, to the determination of the most probable hypothesis for the absolute variations of a system of values.
But, in general, such calculations of probability, in view of the absolute ignorance in which we would be of the causes of variation of values, would be of very slight interest. ​What is really important is to know the laws which govern the variation of values, or, in other words, the theory of wealth. This theory alone can make it possible to prove to what absolute variations are due the relative variations which come into the field of observation; in the same manner (if it is permissible to compare the most exact of sciences with the one nearest its cradle) as the theory of the laws of motion, begun by Galileo and completed by Newton, alone makes it possible to prove to what real and absolute motions are due the relative and apparent motions of the solar system.
10. To sum up, there are only relative values; to seek for others is to fall into a contradiction with the very idea of value in exchange which necessarily implies the idea of a ratio between two terms. But also an accomplished change in this ratio is a relative effect, which can and should be explained by absolute changes in the terms of the ratio. There are no absolute values, but there are movements of absolute rise and fall in values.
Among the possible hypotheses on the absolute changes which produce the observed relative changes, there are some which the general laws of probability indicate as most probable. Only knowledge of the special laws of the matter in question can lead to the substitution of an assured decision for an opinion as to probability.
11. If theory should indicate one article incapable of absolute variation in its value, and should refer to it all others, it would be possible to immediately deduce their absolute variations from their relative variations; but very slight attention is sufficient to prove that such a fixed term ​does not exist, although certain articles approach much more nearly than others to the necessary conditions for the existence of such a term.
The monetary metals are among the things which, under ordinary circumstances and provided that too long a period is not considered, only experience slight absolute variations in their value. If it were not so, all transactions would be disturbed, as they are by paper money subject to sudden depreciation.
On the other hand, articles such as wheat, which form the basis of the food supply, are subject to violent disturbances; but, if a sufficient period is considered, these disturbances balance each other, and the average value approaches fixed conditions, perhaps even more closely than the monetary metals. This will not make it impossible for the value so determined to vary, nor prevent it from actually experiencing absolute variations on a still greater scale of time. Here, as in astronomy, it is necessary to recognize secular variations, which are independent of periodic variations.
Even the wages of that lowest grade of labour, which is only considered as a kind of mechanical agent, the element often proposed as the standard of value, is subject like wheat to periodic as well as secular variations; and, if the periodic oscillations of this element have generally been less ​wide than those of wheat, on the other hand we may suspect that in future the progressive changes in the social status will cause it to suffer much more rapid secular variations.
But if no article exists having the necessary conditions for perfect fixity, we can and ought to imagine one, which, to be sure, will only have an abstract existence. It will only appear as an auxiliary term of comparison to facilitate conception of the theory, and will disappear in the final applications.
In like manner, astronomers imagine a mean sun endowed with a uniform motion, and to this imaginary star they refer, as well the real sun as the other heavenly bodies, and so finally determine the actual situation of these stars with reference to the real sun.
12. It would perhaps seem proper to first investigate the causes which produce absolute variations in the value of the monetary metals, and, when these are accounted for, to reduce to the corrected value of money the variations which occur in the value of other articles. This corrected money would be the equivalent of the mean sun of astronomers.
But, on one hand, one of the most delicate points in the theory of wealth is just this analysis of the causes of variation of the value of the monetary metals used as means of circulation, and on the other hand it is legitimate to admit, as has been already said, that the monetary metals do not suffer notable variations in their values except as we compare very distant periods, or else in case of sudden revolutions, now very improbable, which would be caused by the ​discovery of new metallurgical processes, or of new mineral deposits. It is, to be sure, a common saying, that the price of money is steadily diminishing, and fast enough for the depreciation of value of coin to be very perceptible in the course of a generation; but by going back to the cause of this phenomenon, as we have shown how to do in this chapter, it is plain that the relative change is chiefly due to an absolute upward movement of the prices of most of the articles which go directly for the needs or pleasures of mankind, an ascending movement produced by the increase in population and by the progressive developments of industry and labour. Sufficient explanations on this doctrinal point can be found in the writings of most modern economists.
Finally, in what follows, it will be the more legitimate to neglect the absolute variations which affect the value of the monetary metals, as we do not have numerical applications directly in view. If the theory were sufficiently developed, and the data sufficiently accurate, it would be easy to go from the value of an article in terms of a fictitious and invariable modulus, to its monetary value. If the value of an article, in terms of this fictitious modulus, was $p$, at a time when that of the monetary metal was $\pi$, and if at another time these quantities had taken other values, $p^{\prime }$ and $\pi ^{\prime }$, it is evident that the monetary value of the article would have varied in the ratio of
${\frac {p}{\pi }}{\mbox{ to }}{\frac {p^{\prime }}{\pi ^{\prime }}}.$
If the absolute value of the monetary metals during long periods only suffers slow variations, which are hardly ​perceptbile throughout the commercial world, the relative values of these very metals suffer slight variations from one commercial centre to another, which constitute what is known as the rate of exchange and of which the mathematical formula is very simple, as will be seen in the next chapter.
* ↑ See Newton, Principia, Book I, at the end of the preliminary definitions.
* ↑ It is almost needless to observe that for conciseness the word commodity is used in its most general sense, and that it includes the rendering of valuable services, which can be exchanged either for other services or for commodities proper, and which, like such commodities, have a definite price or a value in exchange. We shall not repeat this remark in the future, as it can easily be supplied firom the context.
* ↑ What characterizes a contract of sale, and distinguishes it essentially from a contract of exchange, is the invariability of the absolute value of the monetary metals, at least for the lapse of time covered by an ordinary business transaction. In a country where the absolute value of the monetary tokens is perceptibly variable, there are, properly speaking, no contracts of sale. This distinction should affect some legal questions.
* ↑ Montesquieu, Esprit des Lois, Book XXII, Chap. 8.