There must, however, be a way of measuring the value of money and of ascertaining it at any given time. Since it is not possible to measure or ascertain the exchange value of money for all commodities at any given time, economists have agreed upon lists of commodities whose prices are to be taken as measuring the value of money. Thus, for instance, if at any given moment we say that a bushel of wheat is worth $1, a ton of pig iron $20, a yard of cloth $3.50, and so on up to a total of $100, we may say that the value of the precious metal contained in $100 is represented by the sum total of the units of different goods thus selected for comparison. The purpose of thus reversing the idea of value is, however, primarily that of making possible comparisons of value at different times.

The working of the index-number plan for the measurement of prices is comparatively simple. Its use is necessitated simply by the fact that it is possible to combine or add dissimilar units only after reducing them to a mathematical base or equivalent. For instance, if it be desired to average the price of ships and clothing and vegetables, no success can be had without first getting a mathematical expression for each price. The index-number plan involves further the choice of a definite base period from which to reckon. For instance, assume that it is desired to compare the level of prices in the year 1910 with that of the year 1900. The first problem is to select the commodities which are to be taken as representative of all others. Suppose that ten such commodities are chosen and that they are, say, pig iron, copper, wool, woolen cloth, raw cotton, cotton cloth, wheat, corn, anthracite coal, and a standard grade of lumber. Evidently these prices are entirely incomparable. They will run from, say, $25 a ton for pig iron to $1 a bushel for wheat or $15 a ton for coal. The first step, therefore, is to regard these various prices as merely the base from which to figure, assigning to each one of them an abstract value of 100. Evidently, then, the combined price of these commodities in the year 1900 will be 1,000 - that is to say, ten times 100, the base or equivalent price of each article. Now, if in the year 1910 pig iron instead of being $25 per ton is $30, the index number for that article will be 5-25 above what it was in the year 1900 - that is to say, it will be 120. Again, if the price of wheat was $1 per bushel in 1900 and in the year 1910 turns out to be, say, 90 cents per bushel, there has been a decline of 10 per cent in value, and the index number for that article will be 90. So also if anthracite coal has risen from $15 to $20 the index number for the latter year is 133 1/3. It is possible that in adding the ten index numbers for the year 1910 the declines in some articles will offset the advances in others, leaving the index number for 1910 1,000 just as it was in 1900. This, however, is very unlikely. The figures will probably show an increase or a decline. Suppose it is 1,100 instead of 1,000. Evidently the standard increase in the general level of prices during the ten years in question is 10 per cent, which we may express by saying that with the year 1900 as a base, there has been a 10-percent increase in prices during the decade, leaving the index number in contrast with 1910 at 110. It will be evident to the careful reader that there are many sources of error in this way of measuring prices. For instance, the commodities chosen as representative may have been badly selected, or consumption habits may have changed during the ten years, so that, although they were well chosen at the outset, they are not very representative in the later year. There may be difficulties in the actual getting of the prices on an accurate basis. Quotations may have been readily available in the earlier year, but difficult to obtain ten years later. Some commodities may have gone out of current use. This is largely true of cotton and woolen textiles, in which methods of weaving have greatly altered during recent years. It is also true that a simple arithmetic average of the kind used in our illustration may be open to very serious question. For instance, pig iron does not figure largely in the ordinary consumption of the individual, while anthracite coal may be an important element in family expenses, but is only a minor item of cost to the in-habitants of a hotel. From all this it follows that the index-number mode of measurement is always very uncertain in its working, and at the best is never more than approximate in its adaptation to actual conditions. It is, however, the only way of measuring general changes in prices. It is always possible to trace the movement of one particular price, but to say that prices as a whole have risen or fallen is feasible only through the use, as already explained, of some mathematical means of reducing them to a common base. That is the service which the index number performs, and that is the reason why there has been an increase in the use of these numbers - they render possible comparisons which otherwise would be out of the question. In some sets of index numbers there are employed elaborate mathematical methods of computation designed to allow for or to eliminate error. These may have a more or less sound base and hence may be more or less desirable. The general opinion of writers on the subject undoubtedly is that elaborate mathematical modifications in methods of computing index numbers have not, on the whole, justified themselves and that the simple arithmetic plan of computation is on many occasions the best plan, while in others the introduction of mathematical refinements must be sought only after the most careful analysis of the various factors entering into the problem.