§ 11 (to Chapter XII, § 7) Mutual Adjustments of Calculated Values of M, M', V, V', P, T

There are various methods of calculating the best adjustments, involving the theory of least squares. But the problem may be greatly simplified by dividing the process into a few separate steps. First, we ascertain the best adjustments of the calculated values of each side of the equation of exchange considered as a whole. We shall need to exercise judgment in deciding the relative errancy of the two sides, but the total adjustments are so small that differences in judgment could not make much difference in the results.

After a careful weighing of all the evidence, it is believed that the errors in the right side (PT) are liable to be about double those in the left (MV + M'V'). Accordingly, the discrepancy between the two sides is corrected by changing PT twice as much as MV + M'V'; that is, by applying to PT a correction equal to two thirds of the total discrepancy, and by applying the remaining one third to MV + M'V', the two corrections being, of course, opposite and such as to bring the two sides into agreement. Thus, for 1899, the total discrepancy is 5 per cent, of which we assign about a third, say 2 per cent, to MV + M'V', and the remaining 3 per cent to PT. That is, we propose to increase the calculated figures for MV + M'V', by 2 per cent and decrease those of PT by 3 per cent. The result will bring them nearly into agreement at 185 billions. Sometimes the results will not exactly agree, as this method of adding and subtracting percentage corrections is only approximately correct; but any remaining slight discrepancies are readily adjusted by slight empirical changes in the factors. The result is shown in the Figure 20, which gives MV + M'V' and PT (reduced by dividing by 1.11) as originally calculated, and a mean (dotted) curve which is the revised estimate of both MV + M'V' and PT.

The corrections which are thus made in MV + M'V' and PT, by which they are brought into mutual agreement, are small; but the corrections necessary in the individual factors, M, V, M', V', P, T, are smaller still. We assume, for simplicity, that the percentage corrections to be made in M and M' are equal to each other and also that the corrections to be made in V and V' are equal to each other. This is a reasonable assumption; but even if some other assumption were made, the final results would be scarcely changed.

A correction of 1 per cent simultaneously in M and M' will produce a correction of 1 per cent in MV + M'V'. Likewise a correction of 1 per cent simultaneously in V and V' will produce a correction of 1 per cent in MV + M'V'. We may then regard the correction of MV + M'V' as practically consisting of two parts: one, the correction of M and M' and the other, the correction of V and V'. As the M's are more accurately ascertained than the V's, their correction should be smaller. Thus, for 1897, the total correction assigned to MV + M'V' is 3 per cent, of which we assign 1 per cent to M and M', and the remaining 2 per cent to V and V'. That is, we increase the calculated values of M and M' by 1 per cent and those of V and V' by 2 per cent, thus effecting (approximately) the desired increase of 3 per cent in MV + M'V'. In like manner the total correction assigned to PT is distributed over P and T, assigning the major part to T. By thus distributing the corrections over (1) M and M', (2) V and V', (3) P, and (4) T, we find that only very slight individual corrections are needed, the maximum being only 5 per cent and the vast majority (50 out of 56 cases) not exceeding 2 per cent. In fact, a decided majority (35 out of 56 cases) are within 1 per cent. It is really astonishing to think that a correction of only 2 per cent or less is usually required in our calculated values of M, M', V, V', P, T, in order to make them conform perfectly to the equation of exchange. In fact, 2 per cent is less than what might naturally be considered the probable error in most of the figures as calculated. This fact justifies confidence in the general correctness of our results.

Having thus corrected, by mutual adjustment, all the factors in the equation of exchange, we are left with a figure for P which is not 100 per cent for any one year. As we prefer to call 1909 the unit year, the figures for P are adjusted on that basis and the figures for T accordingly. This change disturbs the system of corrections as measured relatively to the original figures. It reduces to zero the correction of P for 1909. In general, it makes smaller the corrections for P and T for years near 1909 and makes correspondingly larger those for years remote from 1909. But, even so, the corrections never exceed 10 per cent for T nor 6 per cent for P. As the entire scheme of corrections thus outlined is a matter of judgment and each figure was frankly "doctored" on its own individual merits in view of all the circumstances in the case, it seems inadvisable to burden these pages by any fuller statement of the voluminous details of the process. The results as shown in Figures 13, 14, 15, and 16, already given in the text, speak for themselves.