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of corn, and of pig iron in 1913 as 100 and find that in a certain later year their prices have changed so that the index of wheat is then 130, of corn 140, and of pig iron 190, the arithmetic average in the later year is the sum of these indexes, 460, divided by their number, 3, which gives 153. This index can be weighted by merely adding in more than once the commodity which deserves higher importance. Of this method it may be said that it does not give results accurate enough for precise and close measurements. Indeed, Fisher advocates “the total abandonment of the simple arithmetic type of index number.” 4 This extreme position, however, is unwarranted since for many purposes, only fair approximation to perfect accuracy is needed. Moreover, the ease in the use of the arithmetic average, and the fact that everybody is familiar with it make it a valuable instrument of measurement. It is valuable, that is, provided the user understands that the results allow for a margin of error of possibly 5 and sometimes as much as 10 per cent.

Instead of adding together the price relatives of wheat, corn, and pig iron, we may multiply them together and take the cube root. The result would be the geometric average. If we have ten commodities, we multiply them all together and take the tenth root. Thus, to compute the geometric average, we multiply together all the individual relative prices for a given date and extract the nth root, n standing for the number of commodities included. The geometric average yields more accurate results than the arithmetic, and where weights are not available, the geometric average offers fairly satisfactory results. For general usage, it has the slight disadvantage that it is not as easy to calculate as the arithmetic average.

Instead of adding or multiplying the separate price relatives, we may simply select the middlemost number of the list. Thus the middlemost number of the following list of numbers, 2, 5, 6, 10, 30, is 6, and 6 is the median. This formula of index number is the simple median. It is more accurate than the arithmetic average and just as good as the geometric. It has the advantage of being comparatively easy to calculate. Where data are unavailable for weighting, the median may be used, and may be expected to give sufficiently accurate results to serve most practical purposes.

The foregoing types of formulæ require first calculating the percentage of price change individually for each commodity, and then taking an average or median of these various individual percentages. These individual percentages are price relatives. If instead of converting each commodity singly to a price relative we simply add together the actual prices of commodities, we arrive at a sum total of original price quotations. Comparison of such a sum total in a given year with the sum total of a base year gives a price index based upon the aggregate prices of the two years. This aggregative index gives good results when properly weighted, but when weights are not known, it is unreliable. There are two additional types of index number, the mode

4 The Making of Index Numbers, p. 30.

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and the harmonic, but they are so slightly used that they need not be described here.

Where data on weights are not available, preference on grounds of accuracy is given to the simple median and simple geometric. The use of a large number of commodities will offset the disadvantage of a lack of data on "weighting.” Some statisticians reject the arithmetic altogether, whereas others admit its use with cautions against possible margins of error. Where data on weights are available, it is possible to adopt a formula which reduces mathematical error to a negligible point. It would be out of place in a treatise of this character to describe the mathematical complications of “crossing weights” or “crossing formula," but it may be pointed out that an index formula is available which reduces the mathematical error in price averages to one-tenth, or even one-hundredth, of one per cent."

Fisher analyzes 134 different formulæ for index numbers, and concludes that fully thirty of these are within less than one-half of one per cent of the ideal and that any of the thirty, so far as accuracy is concerned, is good enough to serve for all practical purposes. Fisher further shows that "none outside of this list need ever be used for any purpose where great accuracy is demanded, although about as many other formulæ are accurate enough for most purposes. As an instrument of measurement, the index number is just as accurate as any other devices of physical measurement, such as the pound or the foot. For all practical purposes the index can be considered as absolutely accurate, when properly computed.

Moreover, even some of the simplest formulæ are more accurate than were formerly supposed, and indeed accurate enough to show the primary facts of price movements. The simple median and the simple geometric, for instance, are usually correct within 6 per cent. Errors and discrepancies in these index numbers are less than those which result from most attempts to measure economic quantities. As Snyder states, “It is clear that even what is called a simple or unweighted index number may be, if the quotations are properly chosen and of sufficiently large number, about as well weighted and the index about as accurate as if to them a formidable array of laboriously ascertained weights has been assigned."'? And to quote Mitchell, “About the major facts of price history, the testimony of the leading American index numbers is unanimous. If prices are accurate, the particular method used in computing the index is of secondary importance.

Spoor. Spolo 5 The so-called "ideal” formula, formula number 353, is Pa =

Spoqo Epogo where indicates “the sum of such terms as" Pn the price of any commodity in a given year or period զո the quantity of that commodity in the given year Po the price of that commodity in the base year

the quantity of that commodity in the base year. 6 The Making of Index Numbers, pp. 244-248, 265. 7 The New Republic, June 18, 1924, p. 106. 8 Bulletin 284, United States Bureau of Labor Statistics, pp. 96, 104, 110.

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(7) The Purpose of the Index Number. As has been pointed out in the foregoing treatment of various phases of the index number, the purpose for which the index number is intended makes all the difference in the world in the method of its construction. The base period, the assortment of commodities, the number of commodities, the weighting, the collection of prices, and the selection of the formula,' depend upon the purpose in hand. The original purpose of index numbers was to measure the purchasing power of money and this promises to continue cne of the leading purposes. But more recently the uses of index numbers have expanded greatly. We have indexes of the cost of living, of workingmen's budgets, of retail prices, of stock and bond prices, of foreign exchange rates, of quantity of production, of national income, of interest, of rents, of wages, of volume of employment, of freight traffic, of the business cycle, and of many other business factors. We have indexes for business at large, for separate trades and industries, for individual plants in a given trade, for separate departments of the individual plant. The index number has become an indispensable tool for economic thinking. Wherever we attempt to verify our theory or test out our hypothesis we are dependent upon the index number as a measuring device. By adapting the method of its construction to the particular purpose in hand, we are able to secure interpretations which are accurate and valid. The purpose is everything. In the present chapter we are chiefly interested in index numbers of prices, and particularly in index numbers of the general price level as a measurement of the general purchasing power, or value, of money.

The Secular Trend of Prices.—The secular trend of the price level is the long-time trend prevailing over a period of years. The history of price levels shows clearly that over a period of a decade, or a quarter of a century, or more, the purchasing power of money may be gradually declining, whereas at another time and for a period of a decade, or a quarter of a century, or more, the purchasing power of money may be gradually increasing. The chart of wholesale price movements in England and the United States (see page 532) shows these long swings of the purchasing power of money during more than a century and a third.

The striking similarity of price movements in both England and the United States is simply a manifestation of the fact that the price levels of all countries using the same money standard tend to fluctuate in unison. The forces fixing the secular trend of the price level are worldwide. All countries using gold money are bound together by common forces which govern the purchasing power of their money. What sways the price level of one gold standard country sways the price level of all. If a country temporarily abandons the gold standard, its price level runs an erratic course governed by its internal supply and demand of paper money. To understand the forces involved in long-time and world-wide price movements, a brief survey of the history of price trends is indispensable.

9 Statisticians are not agreed as to whether in the selection of the formula the index number that is best for one purpose is equally good for all purposes. Fisher claims that there is one best index number, and that the formula as a formula is good or bad regardless of the purpose for which the index is to be used. Mitchell and others, on the other hand, insist that even in the selection of the formula, the purpose is all important. See The Making of Index Numbers, pp. 229-234, and Bulletin 284, United States Bureau of Labor Statistics, p. 23.

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* From index figures computed by the Federal Reserve Bank of New York, Indexes for the United States from 1860 to 1923 are derived from the United States Bureau of Labor Statistics. Indexes for England are based upon the Sauerbeck-Statist computations.

During the sixteenth and seventeenth centuries prices rose in Western Europe from 200 to 300 per cent. This rise was attributable in large part to the discovery of gold and silver in the New World and the consequent flow of these precious metals into the monetary circulation of the European countries. These centuries of inflation owed their depreciation of money primarily to this new source of supply of specie.

Looking at price history over a period of several centuries, we find that although prices have sometimes fallen they have more often risen. In general there has gone on a persistent and progressive depreciation of money. In 1924, for instance, prices in France were four to five times as high as they were in 1914. In 1914, prices were four to six times as high as they were five hundred years ago and upwards of ten times as high as they were one thousand years ago.

The 134 years of price movements recorded in the above price chart may be studied from the standpoint of three chief kinds of periods : periods of wars, periods of slowly falling prices, periods of slowly rising prices.

The periods of wars were times of extreme high points in price movements. Extremes of inflation and depreciation were products of

war financing. The first high point of inflation coincides with the Napoleonic Wars in Europe and the War of 1812 in the United States. Excessive issues of inconvertible paper money were characteristic of these wars. The second high point of inflation coincides with the Civil War in the United States. Since England was not a participant in that struggle, the English price level does not show as violent an incline as that of the United States. Excessive issue of inconvertible paper money in the form of the Greenback was the distinguishing characteristic of this war period. The third high point of inflation coincides with the World War. Excessive issue of inconvertible paper money was the characteristic of the period, with one notable exception. That exception was the United States. In the United States, paper money was at all times convertible into gold, but inflation took place none the less. The gold of other countries was dumped into the United States, and on this gold as a reserve backing, note issue and bank credit expanded excessively and brought about inflationary results just as truly as though the gold standard had been abandoned outright. Gold inflation in the United States and paper inflation in the rest of the world characterized the price movements of the World War period. The cardinal lesson of a century and a third of recorded prices is that war produces inflation. Where war is, there inflation, violent and extreme, ensues.

The periods of falling prices are found in three major groups of years. The first of these was from the War of 1812 to 1850. Prices fell more than one-half and the value of money doubled. Owing to the exhaustion of gold and silver mines, very little of the precious metals was mined. At the same time that there was a falling off in production of new specie, there was a steady growth in demand for new money due to the normal growth and expansion of the trade and population of the country. The scarcity of gold and silver supply, coupled with the increase of demand for a medium of exchange, increased the value of the dollar. The second period of falling prices dated roughly from the end of the Civil War down to 1896. Again the cause was a slump in the rate of production of specie coupled with an increase in demand. This time gold was the precious metal of primary importance because by the end of the period the gold standard had been almost universally adopted. Gold supply suffered from the exhaustion of old mines. Gold demand steadily grew, not only because the widespread adoption of the gold standard called for the metal but also because the steady growth of trade and of population called for more money to do business with. The third period of falling prices is the period since 1920. The early years of this period have shown a price decline accompanied by a contraction of war time note issue and bank credit. Whether the next decade or next quarter of a century is to be a continued period of falling prices depends upon a number of factors which at present are unknown. The future rate of gold production and the future demand for gold are the chief unknown factors which are of concern in this connection.

The periods of rising prices may also be considered in three main

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