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Dimension of Interest. - William Stanley Jevons, The Theory of Political Economy 
The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.
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Dimension of Interest.
The formula which we obtained in the preceding section has been subjected to close criticism by an eminent mathematician, who proposed several alternative formulæ, but finally accepted my solution of the question as correct. As Professor Adamson, however, has also raised some objections to the formula, it seems desirable to explain its meaning and mode of derivation more fully than was done in the first edition.
In the first place, as regards the theory of dimensions the formula is clearly correct. The rate of interest expresses the ratio which the annual sum paid per annum for the loan of capital bears to the capital. The interest and the capital are quantities of the same nature, their ratio being an abstract number. Dividing by length of time rate of interest will have the dimension T -1.
Or we may put it in this way—Interest is paid per annum, or per month, or per other unit of time, and the less the magnitude of this unit, the less must be the numerical expression of the rate of interest. Simple interest at five per cent per annum is 0.416... per cent per month, and so on. Hence time enters negatively, and the dimension of the rate of interest will be T -1. Or, again, we may state it thus symbolically—The capital advanced may be taken as having the dimension M; the annual return has the dimensions MT. Dividing the former by the latter we obtain
Now the formula
clearly agrees with this result; for the denominator is a certain unknown function of the time of advance of the capital t. We may assume that it can be expressed in a finite series of the powers of t, and the numerator, being the differential coefficient of the same function, will be of one degree of power less than Ft. Hence the dimensions of the formula will be
It must be carefully remembered that it is the rate of interest which has the dimension T -1, not interest itself, which, being simply commodity of some kind, has the dimension of commodity, namely M, of the same nature, and having the same dimensions.
The function of capital is simply this, that labour which would produce certain commodity m1, if that commodity were needed immediately for the satisfaction of wants, is applied so as to produce m2 after the lapse of the time t. The reason for this deferment is that m2 usually exceeds m1, and the difference or interest m2 - m1 is commodity having the same dimensions as m1. Hence the rate of interest, apart from the question of time, would be m2 - m1 divided by m1, and the quantities being of the same nature, the ratio will be an abstract number devoid of dimensions. But the time for which the results of labour are foregone is as important a matter as the quantity of commodity. The amount of deferment is m1t, so that the rate of interest is m2 - m1 divided by m1t, which will have the dimension T -1.
Exactly the same result would be obtained, however, if we regarded the use of capital from a different point of view. Capital and deferment of consumption are not needed only in order to increase production, that is to say, the manufacture of goods; they are needed also to equalise consumption, and to allow commodity to be consumed when its utility is at the highest point. Now, when certain commodity is consumed within an interval of time, the utility produced will, as we have seen, possess the dimensions MUT -1 T, or MU. Suppose that instead of being consumed within that interval, the commodity is held in hand for a time before being consumed at all. Then the amount of deferment of utility will be proportional both to the interval of time over which it is deferred, and to the utility which is deferred. Thus the amount of deferment will have the dimensions MUT. The increase of utility due to deferment will clearly have the same dimensions as were previously determined, namely MU. Hence the ratio of this increase to the amount of deferment will have the dimensions or T -1, and this result corresponds with the dimension of the rate of interest as otherwise reached.