The value of money is measured, like the value of everything else, by the quantity of other commodities for which it can be exchanged. But because the values of commodities, as well as of money, are constantly changing it is not easy to measure with precision the variations in the purchasing power of money. In our discussion of the relation of money to prices we have considered only the general level of prices and have assumed that all prices move up and down together. But this is seldom the case. Some goods are rising in price while others are falling. Changes in the general price level are slow and gradual but changes in the prices of individual goods, due to causes affecting demand and supply, may be sharp and sudden. Even when the fact of a rise or a fall in prices is evident, the extent of the change is difficult to measure.

1lbid., p. 282.

2 Report of the Director of the Mini (1913), p. 66.

3 Fisher: Purchasing Power of Money, p. 248 see also, American Economic Review, Sept., 1912, pp. 531-558.

To observe and register changes in the general trend of prices "index numbers" are used. An index number represents the price of a group of commodities, or the average price during a given period, which is used as a basis or standard with which to compare the price of these commodities at other dates. Suppose, for example, that the average price per bushel of barley for the period 1890-1899 was 48 cents, while the average for 1910 was 60 cents; then if the average price for the earlier period be represented by 100, called the "base," it will be seen that the relative price for 1910 is 125, that is, the index number shows a rise in price of 25 per cent. By grouping and comparing the prices of a large number of representative commodities, so that the influences affecting the value of different groups will counterbalance each other, a means is obtained of indicating the changes in the purchasing power of money from period to period.

The following simple example illustrates one method of constructing price tables and index numbers:

 1900 1914 Price Base Price Percentage to Base Steel rails, per ton...... \$20.00 100 \$28.00 140 Wheat, per bu .......................... .80 100 .96 120 Coal, per ton........... 4.00 100 3.60 90 Cotton, per lb.......... .12 100 .12 100 Sugar, per lb........... .05 100 .05 1/2 110 Total.................... 5)500 5)560 100 112

In this table we have two sets of hypothetical prices, one for 1900, the other for 1914. The prices for 1900 have been taken as the basis at 100 per cent and the changes in prices in 1914 calculated with reference to this base. Reduced to the simple arithmetical mean, the index number for 1900 is 100; that for 1914 is 112. It appears that while some prices have advanced and others have fallen, the general level of the commodities considered has risen 12 per cent. This rise of prices of 12 per cent indicates a decline in the value of money with respect to the commodities included in the table. It means that the purchasing power of the dollar in 1914 is 100/112, or 89 per cent of its purchasing power in 1900. In other words, a rise of 12 per cent in the general price level is equivalent to a fall of 11 per cent in the value of money. The general law may be expressed thus: Changes in the index number show direct variations in the general price level; changes in its reciprocal show variations in the value of money. If, instead of the five commodities used in the illustration, a table could be constructed including the prices of all commodities we should be able to derive index numbers which would register changes in the purchasing power of money. Most systems of price tables include a sufficiently large number of representative articles to show that though the prices of some articles may have declined while others have advanced, yet the general movement of prices has been in the direction indicated by the change in the index number.

The method of obtaining index numbers by the simple arithmetical average is open to the objection that it tends to exaggerate the influence of rising prices. Suppose, for example, that within a given period the price of a particular article has doubled while the price of another article has fallen by one-half. The index number after the change would be 125, indicating a decline of 80 per cent in the value of money, whereas it would appear that the value of money had not changed, since it had gained as much in the one case as it had lost in the other. To overcome this defect various methods have been suggested in computing index numbers, as, for example, the geometric mean, which is the square root of the product of two prices, the cube root of the product of three commodity prices, and so on for any number of articles. Another method uses the median, in which price quotations for a given period are arranged in numerical order and the figure which has an equal number of quotations above and below it is taken as the mean. Still another method is based on the harmonic mean, which is computed from the reciprocals of a series of index numbers. These methods are intended to offset the effect on the index number of a very high or low price of a single article or a small number of articles. In general, however, these various methods yield substantially the same results.

The method of the arithmetical mean is open to the objection, also, that it gives equal importance to all articles included in the price tables, whereas we know that our family budgets are much more seriously affected by an increase of fifty per cent in the price of wheat or coal than by a similar increase in the price of cutlery or silks. To correct this defect a system has been devised of "weighting" the articles according to their relative importance as determined by total consumption or production. The results obtained, however, by the weighting of price tables, especially where these embrace a large number and a wide range of commodities, are not materially different from those obtained by the simple or unweighted method. At best, price tables can only be approximate, indicating the general trend of prices.