The Eight Tests for a Good Index Number
20th Century Irving Fisher EnglishHaving reviewed the headings of the vertical columns of the table, we have next to note the headings of the horizontal rows. These headings are the eight tests of index numbers. The first six tests are arranged in pairs, the odd being expressed in terms of prices and the even in terms of quantities.
The eight tests are intended to include all the tests which have been hitherto applied in the study of index numbers and some others. They are:
We shall first define each of these tests in general terms and then proceed to illustrate them by actual applications.
Test of proportionality as to prices. A formula for the price index should be such that the price index will agree with all individual price ratios when these all agree with each other. Thus, if in 1910 the price of everything is 10 per cent higher than in 1909, the index number should register 10 per cent higher.
Test of proportionality as to trade. Likewise the correlative formula for the trade index should be such that the trade index will agree with all individual trade ratios when these all agree with each other.
Test of determinateness as to prices. A price index should not be rendered zero, infinity, or indeterminate by an individual price becoming zero. Thus, if any commodity should in 1910 be a glut on the market, becoming a "free good," that fact ought not to render the index number for 1910 zero.
Test of determinateness as to trade. The correlative trade index should not be rendered zero, infinity, or indeterminate by an individual quantity becoming zero. Thus, if any commodity should go completely out of use in 1910 so that its quantity exchanged becomes zero, that fact ought not to render the trade index for 1910 indeterminate.
Test of withdrawal or entry as to prices. A price index should be unaffected by the withdrawal or entry of a price ratio agreeing with the index. Thus, if the price index of a certain number of goods, not including sugar, should be 105 in 1910 as compared with 1900, and the price of sugar itself should be 105 in 1910 as compared with 1900, then the inclusion of sugar in the calculation of the index number ought not to change the index from 105.
Test of withdrawal or entry as to trade. The correlative trade index should be unaffected by the withdrawal or entry of a quantity ratio agreeing with the index.
Test by changing base. The ratios between various price indexes (and therefore also, as we shall see, the ratios between the correlative trade indexes) should be unaffected by reversing or changing the base. Thus, if the index number for 1910 is twice that for 1900, when calculated on the basis of 1860, it should remain twice, when calculated on the basis of 1870.
Test by changing unit of measurement. The ratios between various price indexes (and therefore also, as we shall see, the ratios between the correlative trade indexes) should be unaffected by changing any unit of measurement. Thus, if the index number for 1910 is twice that for 1900 when coal is measured by the ton, it should remain twice, when coal is measured by the pound.
The statements of tests 7 and 8 are expressed in each case both as to prices and quantities; in these cases it was implied that what holds true of price indexes holds true also of trade indexes, and vice versa. To show this reciprocal relation for test 7 (base shifting), let the price index for year 1 in terms of year 0 be designated by P1, 0 instead of by P1 as heretofore, in order that the base year may be specifically designated, and let us compare years 1 and 2 by using first year 0 as a base and then (say) year 8. If the base-shifting test is fulfilled for the P's, i.e. if, we are to prove that the corresponding relation is also true for the T's, viz.
Divide (1) by (2) and (3) by (4). The quotients are
Comparing the right sides of these equations, we find that the "S" ratios are identical in the two cases, and we know that the P ratios are equal by hypothesis. Consequently the entire right sides of the two equations, and therefore the left sides, are also equal, and this is what was to be proved. The converse reasoning is also evident.
Like the base-shifting test, the unit-shifting test, No. 8, cannot apply to prices without applying also to quantities, and vice versa. To show this we employ the equation
T =.
Evidently the numerator of the right side of this equation is unaffected by a change of unit. For instance, if coal should be measured in ounces instead of tons, thus greatly increasing the number (say Q) representing its quantity, the value (pQ) will not be disturbed, since the number (p) representing the price will be correspondingly diminished. Consequently, if the denominator (P) meets the corresponding test, i.e. is likewise unaffected by a change in unit, the quotient (T) must be unaffected. That is, if the unit-shifting test is met for P, it must be met for T. As the converse reasoning also applies, the proposition is proved.
As will have been noted, the first six tests are expressed alternately in terms of prices and in terms of quantities. We now wish to point out that those expressed in terms of prices have a significance for quantities also, and that those expressed in terms of quantities have a significance for prices as well. That is, all the tests have significance both as to prices and as to quantities.
To emphasize this fact, which is important, let us note the price significance of each test. Since the price significance of tests 1, 3, 5, 7, 8 is evidently expressed in the statement of the test, we have left merely to express the price significance of tests 2, 4, 6.
Test 2 tells us that if all the trade ratios agree, their index should agree with them; that is,
The question now before us is, assuming this condition to hold as to the Q's, what condition holds true as to the p's. The answer evidently is:—
[obtained by substituting kQ2 for Q1, kQ'2 for Q1, etc., and kT2 for T1].
The last form is derived from the next to the last by multiplying both numerator and denominator by k and then substituting Q1 for kQ2, Q'1 for kQ'2, etc.
The resulting two formulæ for P1/P2 express test No. 2 in terms of the conditions to which prices must conform. These formulæ will be recognized as those discussed in § 7 of the Appendix to Chapter II, the significance of which was there explained. It was there shown that a change in M, or a change in V, or a uniform change in all the Q's, or any combination of these changes will, through the equation of exchange, affect the price level in the manner expressed by the formula:—
Thus the equation of exchange itself prescribes test No. 2; for the fundamental theorems which the equation of exchange has taught us are that prices vary directly as M and as the V's and inversely as the Q's; and the only forms of index numbers which will faithfully reflect these changes, i.e. will vary directly with M and inversely with the Q's (assuming that all Q's vary in unison), are those forms of index numbers which conform to test No. 2. Any other form of index number, when M (and M') increased 50 per cent and there was no change in V's or Q's, might register a rise of 49 per cent or 51 per cent. That is, no other forms of index numbers will enable us to say that when the quantity of money changes, the velocity of circulation and the Q's remaining the same, the index number of prices will vary proportionately. No other forms will enable us to state the corresponding theorem as to the effect of a change in velocity or of a (uniform) change in the Q's. But these theorems are fundamental. The very concept of an index number is that it shall replace the divergent individual variations and enable us to state of its proportionate changes the same theorems which hold true when prices all change alike.
Test No. 2 is therefore of such fundamental importance that we may profitably pause a moment to restate it in words. To be concrete, let us suppose two years, 1900 and 1910. Let us assume that the quantity of every kind of goods sold in 1910 is (say) exactly double the quantity sold in 1900. Then the only proper index number showing the level of prices in 1910 (year 1) as compared with the level of prices in 1900 (year 0) is
the ratio of the total value of the goods sold in 1910 to what that value would have been at the prices of 1900; or, what amounts to the same thing, it is
the ratio of what the total value of the goods sold in 1900 would have been at the prices of 1910 to what it actually was at the prices of 1900.
Of the 44 formulæ in the table, only the following reduce to the required formula when the Q's change uniformly: (2) of Drobisch, (4), (6), (8), (10), (11), (28), (30), (34), (38), (40). All these are even-numbered except formula 11. Several others will reduce to the required formula, provided one of the years compared is the base year.
The formulæ of the tables which fail to meet test 2 at all would not even allow us to say of MV+M'V'=PT that if all the Q's remain the same, T will remain constant and P will vary as the other side of the equation. For these formulæ T fails as a true index of the Q's, and its error in one direction implies a corresponding error in P in the opposite direction.
Test 2 seems therefore in some respects the most important of all the eight tests for prices; although primarily it was not stated in terms of prices, but in terms of quantities. It is the only test which indicates the kind of weighting required. It completely prescribes the conditions which, while permitting any individual changes in prices, however divergent, enable us to say that a change in M or the two V's or in all the Q's in a given ratio will affect prices "on the average" in that same ratio (directly, of course, for the M's and V's and inversely, for the Q's).
Test 2 in fact points out the true form of the index number of prices as prescribed by the equation of exchange under all possible circumstances except when the Q's vary in unequal proportions. It also points to the proper weights required. These weights may be said to depend either on the Q1's or on the Q0's, interchangeably. The formula suggested by the Q1's is formula 11; that suggested by the Q0's is formula 12. Either will be perfectly satisfactory when the Q1's and Q0's are proportional, while when they are not, their discrepancy is negligible. When the Q's vary unequally, however, there seems to be no perfectly satisfactory formula. Under these circumstances the two systems of weights—one in terms of Q1's, the other in terms of Q0's—conflict with each other. But the conflict has been shown by Edgeworthto be slight. In fact, the weights are of much less importance in determining an index number of prices than the prices themselves.
The discussion of test 2 will be resumed later when in § 7 we come to compare the various forms of index numbers.
As to test 4, this states that if an individual quantity becomes zero this fact should not render the quantity or trade-index zero, infinity, or indeterminate. But according as the index does or does not become zero, infinity, or indeterminate, will the price index become or not become infinity, zero, or indeterminate respectively. This is clear from the relation.
Hence test 4 possesses a significance as to prices similar to that which it possesses as to quantities.
The price significance of test 6 is more complex and of no apparent importance. Its statement is included in the explanatory table on LF-BK133Page 407. In the preceding table of 44 index numbers the "score" for test 6 is bracketed to indicate that it has no important price significance, and is to be omitted in the totals.
Mutatis mutandis, each of the tests expressed in terms of prices (tests 1, 3, 5) has a significance as to quantities also.
The preceding explanatory table exhibits in algebraic terms both hypothesis and conclusion for each of the eight tests with respect both to prices and quantities.