Front Page Titles (by Subject) Theory of Dimensions of Economic Quantities. - The Theory of Political Economy
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Theory of Dimensions of Economic Quantities. - William Stanley Jevons, The Theory of Political Economy 
The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.
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Theory of Dimensions of Economic Quantities.
In the recent progress of physical science, it has been found requisite to use notation for the purpose of displaying clearly the natures and relations of the various kinds of quantities concerned. Each different sort of quantity is, of course, expressed in terms of its own appropriate unit—length in terms of yards, or metres; surface, or area, in terms of square yards or square metres; time in terms of seconds, days, or years; and so forth. But the more complicated quantities are evidently related to the simpler ones. Surface is measured by the square yard—that is to say, the unit of length is involved twice over, and if by L we denote one dimension of length, then the dimensions of surface are LL, or L2. The dimensions of cubic capacity are in like manner LLL, or L3.
In these cases the dimensions all enter positively, because the number of units in the cubical body, for instance, is found by multiplying the numbers of units in its length, breadth, and depth. In other cases a dimension enters negatively. Thus denoting time by T, it is easy to see that the dimensions of velocity will be L divided by T, or LT -1, because the number of units in the velocity of a body is found by dividing the units of length passed over by the units of time occupied in passing. In expressing the dimensions of thermal and electric quantities, fractional exponents often become necessary, and the subject assumes the form of a theory of considerable complexity. The reader to whom this branch of science is new will find a section briefly describing it in my Principles of Science, 3d ed., p. 325, or he may refer to the works there mentioned.1
Now, if such a theory of dimensions is requisite in dealing with the precise ideas of physical magnitudes, it seems to be still more desirable as regards the quantities with which we are concerned in Economics. One of the first and most difficult steps in a science is to conceive clearly the nature of the magnitudes about which we are arguing. Heat was long the subject of discussion and experiment before physicists formed any definite idea how its quantity could be measured and connected with other physical quantities. Yet, until that was done, it could not be considered the subject of an exact science. For one or two centuries economists have been wrangling about wealth, demand and supply, value, production, capital, interest, and the like; but hardly any one could say exactly what were the natures of the quantities in question. Believing that it is in forming these primary ideas that we require to exercise the greatest care, I have thought it well worth the trouble and space to enter fully into a discussion of the dimensions of economic quantities.
Beginning with the easiest and simplest ideas, the dimensions of commodity regarded merely as a physical quantity will be the dimensions of mass. It is true that commodities are measured in various ways,—thread by length, carpet by length, corn and liquids by cubic measure, eggs by number, metals and most other goods by weight. But it is obvious that, though the carpet be sold by length, the breadth and the weight of the cloth are equally taken into account in fixing the terms of sale. There will generally be a tacit reference to weight, and through weight to mass of materials in all measurement of commodity. Even if this be not always the case, we may, for the sake of simplifying our symbols in the first treatment of the subject, assume that it is so. We need hardly recede to any ultimate analysis of the physical conditions of the commodity, but may take it to be measured by mass, symbolised by M, the sign usually employed in physical science to denote this dimension.
A little consideration will show, however, that we have really little to do with absolute quantities of commodity. One hundred sacks of corn regarded merely by themselves can have no important meaning for the economist. Whether the quantity is large or small, enough or too much, depends in the first place upon the number of consumers for whom it is intended, and, in the second place, upon the time for which it is to last them. We may perhaps throw out of view the number of consumers in this theory, by supposing that we are always dealing with the single average individual, the unit of which population is made up. Still, we cannot similarly get rid of the element of time. Quantity of supply must necessarily be estimated by the number of units of commodity divided by the number of units in the time over which it is to be expended. Thus it will involve M positively and T negatively, and its dimensions will be represented by MT -1. Thus in reality supply should be taken to mean not supply absolutely, but rate of supply.
Consumption of commodity must have the same dimensions. For goods must be consumed in time; any action or effect endures a greater or less time, and commodity which will be abundant for a less time may be scanty for a greater time. To say that a town consumes fifty million gallons of water is unmeaning per se. Before we can form any judgment about the statement, we must know whether it is consumed in a day, or a week, or a month.
Following out this course of thought we shall arrive at the conclusion that time enters into all economic questions. We live in time, and think and act in time; we are in fact altogether the creatures of time. Accordingly it is rate of supply, rate of production, rate of consumption, per unit of time that we shall be really treating; but it does not follow that T -1 enters into all the dimensions with which we deal.
As was fully explained in Chapter II., the ultimate quantities which we treat in Economics are Pleasures and Pains, and our most difficult task will be to express their dimensions correctly. In the first place, pleasure and pain must be regarded as measured upon the same scale, and as having, therefore, the same dimensions, being quantities of the same kind, which can be added and subtracted; they differ only in sign or direction. Now, the only dimension belonging properly to feeling seems to be intensity, and this intensity must be independent both of time and of the quantity of commodity enjoyed. The intensity of feeling must mean, then, the instantaneous state produced by an elementary or infinitesimal quantity of commodity consumed.
Intensity of feeling, however, is only another name for degree of utility, which represents the favourable effect produced upon the human frame by the consumption of commodity, that is by an elementary or infinitesimal quantity of commodity. Putting U to indicate this dimension, we must remember that U will not represent even the full dimensions of the instantaneous state of pleasure or pain, much less the continued state which extends over a certain duration of time. The instantaneous state depends upon the sufficiency or insufficiency of supply of commodity. To enjoy a highly pleasurable condition, a person must want a good deal of commodity, and must be well supplied with it. Now, this supply is, as already explained, rate of supply, so that we must multiply U by MT -1 in order to arrive at the real instantaneous state of feeling. The kind of quantity thus symbolised by MUT -1 must be interpreted as meaning so much commodity producing a certain amount of pleasurable effect per unit of time. But this quantity will not be quantity of utility itself. It will only be that quantity which, when multiplied by time, will produce quantity of utility. Pleasure, as was stated at the outset, has the dimensions of intensity and duration. It is then this intensity which is symbolised by MUT -1, and we must multiply this last symbol by T in order to obtain the dimensions of utility or quantity of pleasure produced. But in making this multiplication, MUT -1 T reduces to MU, which must therefore be taken to denote the dimensions of quantity of utility.
We here meet with an explanation of the fact, so long perplexing to me, that the element of time does not appear throughout the diagrams and problems of this theory relating to utility and exchange. All goes on in time, and time is a necessary element of the question; yet it does not explicitly appear. Recurring to our diagrams, that for instance on p. 46, it is obvious that the dimension U, or degree of utility, is measured upon the perpendicular axis oy. The horizontal axis must, therefore, be that upon which rate of supply of commodity or MT -1 is measured, strictly speaking. If now we introduce the duration of the utility, we should apparently need a third axis, perpendicular to the plane of the page, upon which to denote it. But were we to introduce this third dimension, we should obtain a solid figure, representing a quantity truly of three dimensions. This would be erroneous, because the third dimension T enters negatively into the quantity represented by the horizontal axis. Thus time eliminates itself, and we arrive at a quantity of two dimensions correctly represented by a curvilinear area, one dimension of which corresponds to each of the factors in MU.
This result is at first sight paradoxical; but the difficulty is exactly analogous to that which occurs in the question of interest, and which led so profound a mathematician as Dean Peacock into a blunder, as will be shown in the Chapter upon Capital. Interest of money is proportional to the length of time for which the principal is lent, and also to the amount of money lent and the rate of interest. But this rate of interest involves time negatively, so that time is ultimately eliminated, and interest emerges with the same dimensions as the principal sum. In the case of utility we begin with a certain absolute stock of commodity, M. In expending it we must spread it over more or less time, so that it is really rate of supply which is to be considered; but it is this rate MT -1, not simply M, which influences the final degree of utility, U, at which it is consumed. If the same commodity be made to last a longer time, the degree of utility will be higher, because the necessity of the consumer will be less satisfied. Thus the absolute amount of utility produced will, as a general rule, be greater as the time of expenditure is greater: but this will also be the case with the quantity symbolised by MU, because the quantity U will under those circumstances be greater, while M remains constant.
To clear up the matter still further if possible, I will recapitulate the results we have arrived at.
M means absolute amount of commodity.
[]J.D. Everett's Illustrations of the Centimetre-gramme-second System of Units, 1875; Fleeming Jenkin's Text-Book of Electricity and Magnetism, 1873; Clerk Maxwell's Theory of Heat, or the commencement of his great Treatise on Electricity, vol. i. p. 2.