§ 6. The Interior of the Table; Column 11 in Particular

We have reviewed briefly the headings of the table, including both those of the vertical columns and those of the horizontal rows. Their relations to each other are contained in the interior of the table. The object of the table is to show the degree of conformity of the 44 various formulæ for P (and their correlates for T) to the eight tests. In spite of all the mathematical ingenuity spent by many writers in devising index numbers, no known formula and apparently no possible formula will meet all eight of the tests.

For each test we note three possible degrees of conformity. It may be fulfilled by any particular formula in three degrees of conformity or nonconformity: (1) completely, (2) partially, or (3) not at all. These three degrees are indicated in the table which follows, by the numbers 1, ½, 0 respectively. A test is completely fulfilled by index numbers for any two years whatever (as year 1 and year 2). A test is partially fulfilled if it is fulfilled by index numbers for two years, one of which is the base year (year 0). Thus the former relates to the general case, the latter to a particular case. Since the general includes the special, if the test is fulfilled in general it is fulfilled in particular; the converse is not necessarily true. But if the test is not fulfilled in the particular case, it is not fulfilled in general; the converse of this proposition is not necessarily true. In short, an affirmative answer to the question of complete conformity carries with it an affirmative answer to the question of partial conformity; and a negative answer to the question of partial conformity carries with it a negative answer to the question of complete conformity. These two rules save much labor in working out the figures in the table.

Our next task is to illustrate the eight tests by applying them to a particular formula for P1 and the correlative formula for T1. We select for this illustration the pair of formulæ numbered (11) in the table, namely

P1 = and T1 = Sp0Q1, and we seek to know how far this pair of formulæ conform to the eight tests.

TEST 1. Proportionality as to prices.—We shall begin with "the particular case" where one of the two years compared is the base year. In concrete terms, this test means that, if all prices for the year 1 are any given number of times (k times) the prices for the base year 0, then the index number for the year 1 (in terms of the year 0) should be the same number k.

The test is best expressed in algebraic language as follows:—

the given constant price ratio, then also should

It is easy apply this test to our given pair of formulæ.

Therefore test 1 is fulfilled for the particular case when one of the years is the base year.

But, as we have noted, it does not follow that the test is fulfilled in general. For "the general case" of any two years the test may be thus stated: If the price of each good for the year 1 is k times the price of the same good for the year 2, then the index number for the year 1 (in terms of the year 0) should also be k times the index number for the year 2 (in terms of the year 0). That this may be true, the test requires, for the general case, that if

This general test, however, will not be fulfilled, as may be seen from the following:—

In order that this last expression should reduce to k, it is evident that

would have to be equal to.

But this cannot be assumed to be always true. If this equality should happen to hold true for any particular value of (say) Q2, it would evidently be disturbed by the slightest deviation from that value. If, for instance, Q2 should vary, the left side, of the supposed equality would be unaffected, but the first term in the numerator and the first term in the denominator of the right side would vary. Consequently the right fraction would be affected, except when the ratio of the first term, happened to be equal to, in which case by a well-known principle of proportion (the principle of "composition and division") the size of the terms p2Q2 and p0Q2 would be immaterial.

The first test, therefore, is fulfilled for the particular case where one of the two years compared is the base year, but is not fulfilled in general. Therefore, following our convention, we assign to formula 11 the number ½ as representing its degree of conformity to test 2.

TEST 2. Proportionality as to trade. This test, stated for the general case, is: If the quantities of all goods sold in the year 1 is k times the quantities of the corresponding goods sold in the year 2, the index number of trade for the year 1 (in terms of the year 0) should also be k times the index number for the year 2 (n terms of the year 0).

We shall find that this test is fulfilled in the general case, and therefore also in the particular case.

which was to have been proved. Therefore test 2 is completely fulfilled, and the formula is therefore assigned the full credit, "1," in the table.

TEST 3. Determinateness as to prices. This test is also completely fulfilled.

If, in the formula for P1, namely, some but not all of the prices, as p1, or one of the quantities, as Q1, should become zero, it is clear that the above expression would still be determinate, and lie between zero and infinity. It will merely happen that some of the numerous terms in the numerator will vanish, but all the other terms will remain.

Since the same reasoning applies to P2, it follows that P2/P1 must also be determinate, being the quotient of two finite, non-zero, and determinate numbers. Thus test 3 is completely fulfilled.

TEST 4. Determinateness as to trade. The fourth test is analogous to the third, and states that the trade index number must not be rendered indeterminate, zero, or infinity simply because some price or prices should become zero. The formula for T1 is always.

Since, as neither the numerator nor the denominator of this fraction becomes zero, infinity, or indeterminate by the vanishing of some but not all of the p's or Q's, the quotient must likewise be nonzero, finite, and determinate. Thus, test 4 is completely fulfilled.

TEST 5. Withdrawal or entry as to prices. Suppose that there are 100 specified commodities. If the general price level of one year s k times that of another, and if any good, the price of which in the one year is k times that in the other, be withdrawn from the 100 commodities, leaving 99, then the ratio of the price levels of the two years should remain unchanged.

This is a difficult test to meet, and the formula under discussion meets it only partially, that is, when one of the two years compared is the base year.

And, since, it follows by the principle of proportion ("composition and division") that that is, which was to have been proved.

If the missing commodity is reëntered, the ratio will evidently be undisturbed, so the rule works both ways, i.e. for entry as well as for withdrawal. Therefore, test five is fulfilled for the particular case.

When, however, we consider the general case for price ratios of two years neither of which is the base year, the test will not be fulfilled.

For, if this expression should happen to be equal to k in any particular instance, a slight change in any base year price, such as p'0, would disturb the equality, unless the variation in p'0 should affect the denominators of both numerator and denominator of the last expression in the same proportion. This would mean that the ratio

would be unaffected by a change in p'0, which in turn would assume (by the principle of "composition and division") that

This is not necessarily true, as it is evidently easy to assume values for (say) Q'1 which would render it untrue. Thus, a doubling of Q'1 would double the left side but not the right. Therefore test five is only partially met by our formula. This is therefore to be credited only with ½ as its degree of conformity to test 5.

TEST 6. Withdrawal or entry as to trade. If the index numbers for trade are in a given ratio, the inclusion or exclusion of a given good, the quantities of which are in the same ratio, ought not to disturb that ratio. This test is completely fulfilled by our formula.

TEST 7. Changing the base. This test 7 is not fulfilled by our formula even in the particular case. The particular case means here the case of reversing the base, as between (say) year 1 and year 0.

In order not to alter and thereby confuse the notation we shall have to use the subscript 0, which indicated the original base year, to indicate that same year, even when, for the moment, it is not considered as the base year; and likewise we shall use the subscript 1 to indicate the year 1, even when it is, for the moment, taken as the base year.

By the formula which we are testing, the price index ratio for year 1 compared to year 0 as the base is.

By analogy it is clear that the price index ratio for year 0, compared with year 1 considered as the base, is.

If these two expressions are reciprocals of each other, then

should be equal to.

That this is not necessarily true is evident, since there is no necessary relation between the Q0's and the Q1's. If the equation should accidentally hold true for a particular set of Q0's and Q1's, the change, even in the smallest degree, of a single letter, say Q0 or Q1, would evidently disturb the relation. Therefore the test is not fulfilled even for the particular case of reversing the base as between two years, and the formula must therefore be assigned a "0" or complete failure to conform to test 7.

TEST 8. Changing units of measurement. If the unit for indicating the price, say of coal, should be changed from a ton to a pound, the index number ought not to be affected thereby. We shall find that this test is met by formula (11).

Evidently a change of unit, say from a ton to a pound, applied to any particular goods (the prices of which are p1, p'1, p''1, the corresponding quantities being Q1, Q'1, Q''1) will magnify all the Q's 2000 times, but, on the other hand, will reduce all the p's in the reciprocal ratio (1/2000). Consequently, the products p1Q1, p'1, Q'1, p''1Q''1, will be unaffected. Hence the sum of such products constituting the numerator and denominator of the right side of the equation

will likewise be unaffected. Therefore the ratio of the index numbers P1/P2 will be unaffected. Hence the test is completely fulfilled.