# The Purchasing Power of Money/chapter 10

CHAPTER X
THE BEST INDEX NUMBERS OF PURCHASING POWER
§ 1 X.1 IN the previous chapter the necessity for an index number (P) was shown and a particular form of index number was suggested. This form of index number had been shown in Chapter II and its appendix to meet certain conditions (of proportionality of price level to quantity of money, etc.) required by the equation of exchange, MV+M'V'=PT. In the present chapter, this index number will be compared with others and the general purposes of index numbers discussed, including purposes having little direct concern with the equation of exchange.
X.2 Index numbers may be compared in respect to (1) form, under which term are included methods of weighting and of determining the "base" prices; (2) the selection of elements to be included. In this section we shall consider the question of form.
X.3 The number of possible forms of index numbers is infinite. They differ enormously in complexity, in ease of calculation, and in conformity to various other tests. A few of the simplest may here be mentioned. Their discussion will be brief and will in many cases be dogmatic. Full proofs and discussions are contained in the mathematical appendix.*18
X.4 If in 1900 the average price per pound of sugar was 6 cents, and in 1910 it was 8 cents, the ratio of the price in 1910 to that in 1900 must have been 8/6 or 133 1/3 per cent. If, in the same period, the average price of coal per ton had changed from $4 to $6, the corresponding ratio for coal must have been 6/4 or 150 per cent. If the price of a given grade of cloth, on the other hand, fell from 10 cents to 8 cents a yard, the ratio for cloth must have been 8/10 or 80 per cent. P is an average of all these three price ratios and other price ratios, that is, an average of 133 1/3 per cent, 150 per cent, 80 per cent, etc. The simple arithmetical average of these three ratios specified would be
or 121 per cent. The simple geometric average would be
, or 117 per cent.
X.5 These are examples of simple or unweighted averages. Since, however, weighted averages have many advantages in theory and some advantages in practice, we shall proceed to consider them.
X.6 There are innumerable methods of weighting*19 and of averaging. None of them is perfectly satisfactory from a theoretical standpoint. We must choose what seems to be best from a practical standpoint. The effect of changed volume of currency or changed velocity of circulation on the whole series of prices is complex, and cannot, even in theory, be compressed into one figure representing all price changes, any more than a lens can be constructed which will focus in one point all the rays of light reaching it from a given point. But, although in the science of optics we learn that a perfect lens is theoretically impossible, nevertheless, for all practical purposes lenses may be constructed so nearly perfect that it is well worth while to study and construct them. So, also, while it seems theoretically impossible to devise an index number, P, which shall satisfy all of the tests we should like to impose,*20 it is, nevertheless, possible to construct index numbers which satisfy these tests so well for practical purposes that we may profitably devote serious attention to the study and construction of index numbers.
X.7 The index number mentioned in Chapter IX may be constructed by the following process: Suppose that the year 1910 is the period to be considered in our equation of exchange MV + M'V' = SpQ = PT. We select another year (say 1900) and call it the "base" year. This means that the prices of 1910 are to be expressed as a percentage of the prices of the equation of exchange for 1900.
X.8 Next we obtain an expression for trade (or T). As shown in the appendix to this chapter, every form of index number, P, for prices implies a correlative form of index for trade, T, and vice versa. It is convenient to select T first. We observe that trade (or T) is not the value of transactions measured at the actual prices of the year 1910, for this value is PT or SpQ, that is, the entire right side of the equation. Trade (T) by itself must be divorced from the price level (P); it may be conceived as the value which the total transactions would have had if the actual quantities sold had been sold at the base prices. It is thus the sum of a number of terms, each term being the product of the quantity, or Q, pertaining to 1910 and the price, or p, pertaining to the base year 1900. Algebraically it is p0Q + p'0Q' + p0Q + etc., or, more briefly, Sp0Q, where the prices of 1910 are expressed simply as p, p', p, etc., and those of the base year, 1900, are expressed as p0, p'0, p0, etc.
X.9 Having defined this ideal value (T), we now define P as the ratio of the real value of transactions in 1910 (SpQ) to that ideal (Sp0Q). More fully expressed, P is the ratio of a real value (the value of the trade of 1910 at the prices of 1910) to an ideal value (the value of the trade of 1910 at the prices of 1900). This ratio is really a weighted arithmetical average of price ratios.*21 The foregoing method is simple both in conception and in mathematical expression,*22 and appears to furnish, theoretically at least, the best form of P, or index number of prices. The particular form of P (viz. SpQ ÷ Sp0Q) which we have just described is, then, associated with and dependent on a particular form of T (viz. Sp0Q). T may be called an index number of trade, and we may say that the particular form of T (viz. Sp0Q) is the best form of index or barometer of trade.
X.10 Another method of conceiving the same form of index number of prices is that mentioned at the close of the preceding chapter, as follows: Conceive each kind of goods to be measured in a new physical unit—viz. the amount which was worth one dollar in the base year (1900)—and let us use this unit for each other year (as 1910). Thus instead of a pound as the unit for sugar we take as the unit whatever amount of sugar was a dollar's worth in 1900. Hence the price of sugar in the base year, 1900, was $1, as of course was the price of everything else. If, now, the price of sugar in any other year (as 1910) is $1.25 in terms of the new unit (viz. the amount which was a dollar's worth in 1900), we know that the price has risen 25 per cent. In this way P may be defined simply as an average price instead of as an average price ratio and T as the total number of the new units of goods sold of all kinds. The right side of the equation is now simply the product of the total number of units sold, multiplied by their average price.
X.11 The two definitions of P which have been given (viz. the ratio of real to ideal values, and the average price in 1910 of all goods when measured in dollar's worth of 1900) are interchangeable; and both definitions of T (ideal values of transactions in 1910 at prices of 1900, and total number of units sold in 1910, the units being each a dollar's worth in 1900) are interchangeable. There are other ways of defining P and T without changing their meanings. Thus "P is the weighted arithmetical average of the ratios of prices of goods in 1910 to those of 1900, when these ratios are weighted according to the values of the goods exchanged in 1910 reckoned at the prices of 1900." Whichever of these definitions we prefer, the system of index numbers is the same and has advantages over most other systems. Above all, it enables us to say without qualification that if the quantities sold remain unchanged, so that T will remain unchanged, P will vary directly as the left side of the equation of exchange.*23
X.12 We choose, then, as one of the best index numbers of prices, the average price of the goods sold, those goods being measured in units worth a dollar in the base year; in other words, the ratio of the value of sales at actual prices to the value of the same sales if made at base prices; in still other words, the weighted arithmetical average of all price ratios, each ratio being weighted according to the values sold, reckoned at base prices.
X.13 We have still to consider the selection of the base. It makes a difference to the above index numbers, not only absolutely, but also relatively, whether the base year is, for instance, 1900 or 1860.
X.14 Excepting Jevons's index numbers, which were geometric averages, there are few index numbers which are not vitiated in some degree by having a base remote from the years for which comparisons are most needed. As Professor Marshall has maintained and as Professor Flux has emphasized, the best base for any year seems to be the previous year.
X.15 Instead, then, of employing a fixed base year for which all prices are called 100 per cent and in terms of which all other prices are expressed in percentages, each year may be taken as the base for the succeeding year. Thus we obtain a chain of index numbers, each number being connected with the preceding year instead of with a common base year.
X.16 The great advantage of this chain system is that it yields its best comparison for the cases in which comparison is most used and needed. Each year we are interested in Sauerbeck's index number in order to compare it with the number of the preceding year, and only to a less extent with other years. The number, however, as actually constructed, affords something quite different. It gives us, as its best or most accurate comparison, the ratio between the current year and the years 1867-1877. This comparison is of little or no interest to any one. What all users of these statistics actually do is to compare two comparisons. The index numbers for 1909 and 1910 (each calculated in terms of 1867-1877) are compared with each other. But direct comparison between 1909 and 1910 would give a different and more valuable result. To use a common base is like comparing the relative heights of two men by measuring the height of each above the floor, instead of putting them back to back and directly measuring the difference of level between the tops of their heads. The direct comparison is more accurate, although in the case of the men's heights both methods would theoretically agree. In the case of price levels, unfortunately, few index numbers will even theoretically give consistent results when the base is shifted;*24 and those few will fail to meet other equally important tests.
X.17 It may be said that the cardinal virtue of the successive base or chain system is the facility it affords for the introduction of new commodities, the dropping out of obsolete commodities, and the continual readjustment of the system of weighting to new conditions. A fixed base system soon gets behind the times in every sense of the word.
§ 2 X.18 Our next question is: What prices should be selected in constructing an index number? The answer to this question largely depends on the purpose of the index number. Hitherto we have considered only one purpose of an index number, viz. to best meet the requirements of the equation of exchange. But index numbers may be used for many other purposes, of which the two chief are to measure capital and to measure income. Each of the three purposes mentioned (viz. exchange, capital, and income) may be subclassified according as the comparison desired is between places or times. Thus index numbers may be used for comparisons between places with respect to their exchange of goods, their capital, or their income. When, for instance, the British Board of Trade*25 tries to compare the cost of living in various towns of England, Germany, and the United States the comparison is with reference to the prices of living (or income) of the working classes.
X.19 We thus have at least six large classes of purposes for which index numbers may be used, viz. to compare the prices, in different places, of the goods exchanged; of the capital goods; and of the income goods; and of the same three groups of goods at different times.
X.20 In each of the six cases, prices of goods and quantities of goods will be associated with each other, and an index number (P) for one will imply an index number (T) for the other (we here use T in the general sense of an index of quantities of goods whether they are exchanged goods, as hitherto, or capital goods, or income goods).
X.21 Evidently there will be a great difference in the selection of the prices to be compared according to which of the six comparisions we wish to make. Suppose, for instance, that we wish to measure changes in the general level of prices of capital goods,*26—railways, ships, real estate, etc.,—and also to measure the relative changes in the amounts of those goods. The prices of some kinds of capital may have increased, and of others decreased, and some may have increased at different rates from others. How shall we measure the general change in prices of capital goods? Again, the quantity of some kinds of capital, as railroads, may have increased faster than that of other kinds, as sailing ships. Still other kinds of capital may have decreased. How shall it be determined whether capital in general has increased, and how much? These two problems (of prices of capital and quantities of capital) may be said to consist in measuring the average change in the price of the same quantity of capital, and the average change of quantity of capital taken at the same prices.
X.22 For either index (since only capital is under consideration, and not income or other designated goods, whether stocks or flows), the index numbers should relate, not to general prices and quantities, but only to prices and quantities of capital goods. Thus, the prices and quantities of all labor services should be omitted. The use of capital and the rents paid for that use, such as the rent paid for house shelter, should be omitted. Only capital instruments, and not the services yielded by these, should be included. We may obtain the price index first and then obtain a quantity index by dividing the value of capital in any year by the price index, or we may proceed in the reverse direction.*27 In making index numbers of prices of capital and quantities of capital, we naturally select for our list articles which are important as capital, and weight them accordingly.
X.23 To determine this general change in prices of capital, we should weight each ratio by the value of the particular capital to which that ratio relates. In this case each ratio should be weighted, not according to the annual sales, but according to the existing capital. Obviously, the difference between these two modes of weighting may be great. Thus, real estate forms a large part of all existing capital, but sales of real estate are a relatively unimportant part of all sales. Food products, on the other hand, contribute little to capital and much to exchange. Consequently, prices and quantities of food products would not figure in capital index numbers, but would figure largely in the index numbers relating to the equation of exchange.
X.24 Again, suppose the purpose of the index numbers of prices is to measure the quantities and prices of income, not of elements of capital. In this case the list of articles and their weights will be quite different from those in a capital index.
X.25 If the income of workingmen is under consideration, we have to deal with index numbers for prices of those goods entering into workingmen's budgets, and with index numbers of the quantities of such goods. The first will show the cost of living of the workingman, or the purchasing power of a workingman's dollar; the second will show what is called his "real wages" or "consumption." In this case the aim is to compare, not stocks existing at two points of time, but flows through two periods of time. One way to obtain an index of real wages is to correct the nominal or money wages by using the index number of prices of goods for which wages are spent. Thus, if money wages for 1908 were twice those for 1900, but money prices of the necessaries and comforts of life had also doubled, real wages would be unchanged.
X.26 Evidently the index numbers used in the case of quantities and prices of workmen's living are not the same as those used in the case of capital. Goods should have each an importance in the index number, dependent upon its importance in workingmen's budgets. The goods in this case are flows, while in the case of capital the goods considered were stocks. In comparing capital, the index numbers must relate to capital; and in comparing income, the index numbers must relate to income.
§ 3 X.27 Perhaps the most important purpose of index numbers is to serve as a basis of loan contracts.*28 It is desirable to determine the particular form and weighting best suited to this purpose as well as the best selection of prices to be included.
X.28 An index number which serves the purpose of measuring the appreciation or depreciation of loan contracts—or what is called "deferred payments"—evidently belongs to the time rather than the place group of comparisons. But to which if any of the three sub-groups (exchange, capital, or income) it most properly belongs is not at first clear. But, before considering this question and as a preliminary to finding the best index number for contracts between borrower and lender, we must arrive at some opinion as to what is the ideal basis for loan contracts.
X.29 In the first place, it should be pointed out that though there is a gain and loss there is not necessarily any "injustice" wrought because of a change in the level of prices. Thus, if a man borrows $1000, contracting to pay it back with $40 additional as interest at the end of five years, and meanwhile prices unexpectedly double, he is a decided gainer. Though he has to pay, to be sure, the same number of dollars, he needs to sell only about half as much of his stock of goods as he expected. He pays back, in the principal, only half of the real purchasing power borrowed. The lender, on the other hand, is a loser by the change.
X.30 Yet the contract was perfectly fair. Each party knew or should have known that the price level might change, and took the risk. There was no fraud any more than when wheat has been ordered for future delivery at a certain price and the market unexpectedly turns; or when an insurance company loses a "risk" prematurely.
X.31 Indeed, for a government to attempt by legislation to deprive the gainer of his profit would itself be in general unfair.*29 To protect themselves from losses the risk of which they took upon themselves, the losing parties cannot justly betake themselves to legislation after the making of contracts.
X.32 The unfairness of so doing becomes the more manifest, when it is considered that, if the change in price level is at all expected, there is apt to be some compensation by means of an adjustment in the rate of interest.*30 If the price level is rising, the nominal rate of interest will probably be a little higher, compensating the lender somewhat for the loss of part value in his principal; while, if the price level is falling, the borrower is likely to be partly compensated for his loss by a lower nominal rate of interest. It is not right that either side should use its influence with government to impair the obligations of contracts already made.*31 It is, however, sound public policy to lessen in advance the risk element, as rapidly as may be, so that future contracts may be made by all parties on the most certain basis possible. In the problem of time contracts between borrowers and lenders, the ideal is that neither debtor nor creditor should be worse off from having been deceived by unforeseen changes. Experience shows that the rate of interest will seldom adjust itself perfectly to changes in price level, because these changes are only in part foreseen. The aim should be to make the currency as certain or dependable as possible. Practically speaking, this means that it shall be as nearly constant as possible.
X.33 In an ideal standard of value, the index number of prices would continually register 100 per cent.*32 But as long as an absolutely stable currency does not exist, and cannot be had, the index number is itself a possible standard for long-time contracts. It is called the "tabular standard," as it depends on a table of prices. Thus, if a man borrows $1000 when the index number is 100, he might agree to pay back, not the same dollars, but the same general purchasing power, with interest. If, at the time of repayment, the index number had gone to 150, the principal of the debt would be understood to be $1500, since this represents the same purchasing power that was borrowed. If, on the other hand, the level of prices had fallen to 80, the principal would automatically become $800. Thus, both parties would be protected against fluctuations in the value of money. The same correction would apply to the interest payments, each of which would be adjusted according to the index number relating to the time of payment.
X.34 We are now ready to consider the question of what are the goods the prices of which should be included in an index number to serve the purposes of measuring changes in loan contracts.
X.35 If all goods kept the same ratios of prices among themselves, it would make no difference whether a loan contract were made in terms of one index number or another or even whether it were made in terms of wheat, tons of coal, or pounds of sugar. But because prices do not vary in the same, but in different proportions, an index number measuring the general level of prices is necessary. If repayment is made in equivalent purchasing power (plus interest), over one kind of goods, this may be either more or less than equivalent purchasing power over other kinds. Hence, one party or the other is a loser according to the kind of goods he handles as a producer or prefers to use as a consumer. Even if each contracting party could arrange to receive or pay back purchasing power over an amount of goods of the kinds which most concerned him, equivalent to what he lent or borrowed, with interest, the speculative element resulting in gain or loss to one or the other, though decreased, would not be entirely removed.*33
X.36 Suppose, for example, that a lender receives back purchasing power over an amount of goods of the kinds he wished to use, equivalent to what he lent plus interest.*34 Suppose also that, during the period of the loan, these goods appreciate relatively to others. Then the lender really gains, since he can now get more of other goods in exchange for those it was his original purpose to use,—a course which he may now be tempted to take while otherwise he would not. To the borrower, however, the appreciation of the goods on the basis of which repayment is made, relatively to the goods he is engaged in producing, might be regarded as causing him loss. The same purchasing power over the goods, on the basis of which he is to make payment, means in such a case, a greater purchasing power over the goods he is engaged in producing.
X.37 It is clear that no one kind of goods is a fair standard. An index number intended to serve as a standard for deferred payments must have a broad basis.
X.38 Were all borrowers and all lenders interested merely as consumers—lenders denying themselves in immediate consumption in order to lend, with the idea of consuming more on repayment, and borrowers planning to consume more immediately with the intention of later consuming less—an exactly satisfactory index number for each individual would seem impossible. The goods which interested a lender in any given case might not be those of most importance to the borrower. Only a rough average could be struck and an index number found to be used by all parties in their contracts. Such an average would doubtless be one in which each price ratio would be weighted according to the total consumption of the goods to which it related,—the total consumption of all borrowers and all lenders in the country considered.
X.39 The case is even more complicated, however; for many borrowers and lenders are interested less in consumption than in investment.*35 The choice is as much between lending and other investing as between lending and consuming. Similarly, the borrower may borrow to invest as well as to consume and may raise the money for repayment by curtailing investment rather than by curtailing consumption. Borrowers and lenders, in other words, may be more interested in purchasing factories, railroads, land, durable houses, etc., which yield services during a long future, than in purchasing more or better food, shelter and entertainments, which yield immediate satisfactions. To base our index number for, time contracts solely on services and immediately consumable goods would therefore be illogical. Though the practical differences may amount to little, yet, in theory at least, they are important.
X.40 Let us suppose each price ratio to be weighted by the value (at standard prices) of the services of quickly consumable goods enjoyed during a given period, purchases of durable capital being omitted. Suppose also that before the time of repayment arrives the rate of interest has risen. With higher interest, the value of land, railroads, and other durable capital will be lower because the value depends on future earnings or future services, and these are now discounted at a higher rate.*36 The borrower, in paying back an equal purchasing power over consumable goods and services, is paying back a much higher purchasing power over such things as land, houses, and factories—a much higher purchasing power over future income—than he borrowed. The lender is receiving back, therefore, a larger purchasing power over these durable items of capital than he loaned, though not a larger purchasing power (except for the interest) over immediately consumable goods and services. He gets back no more control over present income, but he gets a purchasing power over a greater amount of deferred income. Had he invested in land at the start, instead of lending, the rise of interest would have left him with the same amount of land, but a less value. As it is, he gets back a purchasing power over a greater amount and the same value of land. An accident has made the lender better off than he expected, and better off than he would have been had he invested instead of loaning.
X.41 If, on the other hand, the rate of interest should fall, the borrower will be benefited, and the lender injured. The value of land and of any other property, the income of which extends far into the future, would rise in comparison with the value of food, shelter, and so on. The value of a house is the discounted value of its future rent or service in affording shelter. The rate of interest having fallen, the value of the house will be higher, in comparison with the yearly rental value, than before. To repay the same amount of purchasing power over shelter as was borrowed is to repay less than the same amount of purchasing power over houses. The borrower is benefited to the extent that he has to curtail investments to repay, since he repays a less investing power although as great a spending power. He need not, therefore, curtail his investments in land and machinery quite so much as he otherwise would have to do. The lender, on the other hand, is in the same degree injured. If he wishes to invest in durable capital, such as an office building, a mine, or shares in a railroad, he cannot purchase as much of these with the returned principal as he could have purchased with the same principal at the time of the loan. Had he foreseen the fall in interest, he might have refused to make the loan and invested instead. He would then have had, in place of interest on a loan, a return on his investment and a larger amount of capital on which to realize future income. The effect of the fall in interest would then have been, not to decrease the returns on his investment, but to increase the capitalized value of the investment.
X.42 It appears, then, that while an index number based on services and the less durable commodities may be adapted to time contracts between a borrower intending to indulge in immediate consumption, and a lender intending to postpone consumption until the repayment of the loan, such an index number is not entirely suited to contracts one or both parties to which are interested in more permanent investment.
X.43 Instead, therefore, of basing our index number on consumable goods and services enjoyed during a period, we ought rather to base it partly on these and partly on the amount of durable capital. Each borrower and each lender may wish to make a different distribution in time of his income stream.*37 One man, that he may have a large income in the future, wants to invest; another, that he may enjoy a large income soon, does not want to invest. One lender is, therefore, interested in getting back as much durable capital as he lent; another lender is interested in receiving as great purchasing power over services and consumable goods as he lent.
X.44 Now different persons, with different intentions as to the spending of their money, nevertheless make loan contracts with each other. Even if a separate index number, specially weighted, could be used for each couple, such a standard would not be equally fitted to both parties. Yet the same debt cannot be paid in two different standards. Therefore, absolute equalization is out of the question. We can mitigate the evils of a fluctuating money standard, but we cannot entirely remove the element of speculation from time contracts.
X.45 Although different persons and different classes might establish different standards for special contracts, yet for the great mass of business contracts involving postponed payments, a single series of index numbers including articles used and purchased by all classes, and including also services, would probably be found advisable. This index number would be best suited to contracts between different classes, between individuals of differing habits of consumption, and to fix the money payments on bonds which are securities sold to the public in general.
X.46 Without attempting to construct index numbers which particular persons and classes might sometimes wish to take as standard, we shall merely inquire regarding the formation of such a general index number. It must, as has been pointed out, include all goods and services. But in what proportion shall these be weighted? How shall we decide how much weight should be given, in forming the index, to the stock of durable capital and how much weight to the flow of goods and services through a period of time,—the flow to individuals, which mirrors consumption? The two things are incommensurable. Shall we count the railways of the country as equally important with a month's consumption of sugar, or with a year's?
§ 4 X.47 To cut these Gordian knots, perhaps the best and most practical scheme is that which has been used in the explanation of the P in our equation of exchange, an index number in which every article and service is weighted according to the value of it exchanged at base prices in the year whose level of prices it is desired to find.*38 By this means, goods bought for immediate consumption are included in the weighting, as are also all durable capital goods exchanged during the period covered by the index number. What is repaid in contracts so measured is the same general purchasing power. This includes purchasing power over everything purchased and purchasable, including real estate, securities, labor, other services, such as the services rendered by corporations, and commodities.
X.48 There has been much discussion as to the propriety of the inclusion of services of human beings, or so-called "labor." In one way the question solves itself, since the inclusion or exclusion on the basis of piece work will make little or no difference to the results.
X.49 It is well known that we may measure wages either "by the piece" or by "time." In either case they enter into and affect the general index number expressing the price level, but the influence is different in the two cases. If we take hours of labor as the basis and measure the wages paid by the hour or by the day, then we are likely to find that, during a period of improvement in the arts, money wages are rising while the prices of goods are falling, or that money wages are rising faster than the prices of goods, or are falling more slowly. But if we measure wages by the piece, we shall find less inconsistency of results. If goods increase faster than currency, so that prices tend to fall, piece wages will tend to fall, on the average, in very much the same proportion. As improvements in machinery make the output per hour of labor, i.e. the piece work, increase, the price per piece may decrease.
X.50 The two methods of measurement giving these different results for price indexes make opposite differences in the volume of trade. The volume of piece work increases with progress in invention faster than the volume of time work.
X.51 In considering the index number as a standard for deferred payments, the desirability of assuming piece wages to change like commodity prices is based largely on the difficulty and consequent impracticability of including wages on a time basis. On the piece-wage basis, changes in money prices of other goods furnish an approximate measure of changes in money prices of labor.
X.52 Those who make time contracts on the basis of such an index number know that they will pay back, or receive back, purchasing power over the same quantities of goods, the purchasing power over which they borrowed or lent. This form of index number is an objective standard of goods.
X.53 If an index number were to be constructed from time wages alone (not including goods at all), debtors would pay back and creditors receive an equivalent purchasing power over hours of labor. When the time wages and prices of goods are both included, the problem is how much weight to give to each. Kemmerer weights wages 3 per cent out of a total of 100 per cent. Its influence, therefore, would in any case not be felt greatly, while, if we take piece wages, it will not be felt at all; that is, it will not greatly matter whether wages are included or not. Since practically we have no statistics of relative piece wages, and few good statistics for time wages, we may, in general, as well omit wages altogether.
X.54 This procedure has another advantage. In an index number intended to serve as a basis for deferred payments for wage earners, it is clear that wages should be excluded. A wage earner does not judge his purchasing power on the basis of how much labor he can buy.*39
X.55 In this connection it may be well to call attention to another standard of purchasing power of money which has sometimes been suggested for adjusting contracts. This is the utility standard. According to this, each person would be expected to receive or pay back marginal utility equivalent to what he had lent or borrowed. But the marginal utility of the same goods is different for different persons and different for the same person at different periods of his life. Hence, no such standard could be practically applied.
X.56 A price is an objective datum, susceptible of measurement, and the same for all men. Marginal utilities, on the other hand, not only are impossible to measure, but are unequal and vary unequally among individuals. The purchasing power of money in the objective sense is, therefore, an ascertainable magnitude with a meaning common to all men. It is of course true that marginal utility of money is a fundamental magnitude and that it depends in part on the purchasing power of money. But it depends also on each man's income. The marginal utilities of money will vary directly with the purchasing power of money if all prices and all money incomes change in the same ratio, or (roughly at least) if incomes change in the ratio of the average price change. Ideally, this fixed ratio between marginal utilities and purchasing power should hold true when the quantity of money varies (assuming that deposits vary equally and that velocities of circulation and volume of trade remain unchanged) after transition periods are over. Practically, however, all these elements vary, and vary unequally. Money incomes sometimes increase faster, often more slowly, than prices. The result is that changes in the purchasing power of money do not correspond to changes in marginal utilities of money.
X.57 Society may become more prosperous, or it may become less so, during the time a contract has to run. This fact, it may be thought, should influence the relation of the amount repaid to the amount borrowed. It has been claimed that the benefits of progress should be equably distributed between borrowers and lenders.*40
X.58 But while loan contracts are made with reference to marginal utilities, it is here contended that corrections in a monetary standard not only cannot, but should not, include variations in the subjective value of money due to changes in incomes, but should be confined to variations in objective purchasing power. At any rate, to obtain a measure of objective purchasing power is a step which may properly be taken by itself before any step more ambitious is considered. The search for a standard of deferred payments which shall automatically provide for the just distribution of the "benefits of progress" seems as fatuous a quest as the search for the philosopher's stone. Since we cannot measure utility statistically, we cannot measure the corrections in utility required to redistribute the "benefits of progress." In the absence of statistical measurement, any practicable correction is out of the question. The "utility standard" is therefore impracticable, even if the theory of such a standard were tenable.
X.59 Somewhat similar theories of a perfect standard of deferred payments are based on the idea that a dollar should require always the same amount of labor to produce it. In one sense, since marginal utility and marginal effort are normally equal, the labor standard is identical with the marginal utility standard. But in whatever sense "labor" is defined, it is an elusive magnitude, quite impracticable as a measurable basis for statistics of purchasing power. Seemingly labor may be measured in terms of time and, on such a basis, "a day's labor" has been suggested as a proper unit for measuring deferred payments. But even "a day's labor" is not a sufficiently definite unit in which to measure with any considerable degree of accuracy the purchasing power of money. Days' labor differ in hours, in intensity, and disagreeability of effort as well as in the quality of labor performed—whether it be manual, mental, etc. A magnitude which offers so many theoretical difficulties in measurement can never serve as a practical standard of deferred payments.
X.60 We see then that the attempt to set up a utility or labor standard is too ambitious to be practicable. *41 We should content ourselves with securing the maximum attainable improvement in the standard of deferred payments, without attempting to secure an ideal distribution of "the benefits of progress."
X.61 It will also simplify our problem if we remember that our ideal is not primarily constancy of the dollar but rather dependability. Fluctuations which can be foreseen and allowed for are not evils. Each man may presumably be depended on to allow for changes in his own fortunes, utility, and labor, and perhaps even to a large extent on the general effects of invention and progress. At any rate he should not expect the monetary unit to insure him against every wind that blows.
X.62 The manner in which each person allows for such future changes as he can foresee is by adjusting the size of loans he makes or takes and the rate of interest thereon. If the average income is rising, the borrower can afford to repay more and the lender should receive more; while, if the average income is falling, the amount paid should be less. The fact is that such are the tendencies where the rise or fall of average income is foreseen. If the average income is rising, the lender will be less anxious to deplete his present income, which is relatively meager, in order to increase his future income, which he sees will probably be larger anyway. Thus, increasing prosperity (by which is meant, not great prosperity, but growing prosperity) tends to restrict the supply of loans. At the same time, it tends to increase the demand and so raise interest. Conversely, a decreasing average income will tend to lower interest.*42
X.63 All this follows only in case the rise or fall of incomes is foreseen. If not foreseen, it can exercise no influence of importance on the interest rate. To the extent that such changes come unexpectedly after loan contracts have been entered into which take no account of them,—to that extent loan contracts are speculative. If incomes fall, the lender has gained relatively to the borrower, because he has realized a higher interest than he could have realized had the change been foreseen. The chief burden of the change falls on the borrower. If incomes experience an unexpected rise, the relative positions are reversed; the whole gain goes to the borrower. The normal effect of continuous extension of income is to raise the rate of interest.
X.64 Our present problem, however, is not to safeguard the interests of debtors and creditors against all possible elements of change, but only against those elements which are purely monetary. Industrial changes are in a class by themselves, and contracting parties must be trusted to work out their own salvation. We are merely concerned in providing them with a stable or reliable monetary standard. A secure monetary standard cannot guarantee against earthquake nor insure the equable distribution of prosperity. It can, however, mitigate the losses now suffered from changes in the relation of money to other goods.
X.65 Statistics of nominal or money interest rates and virtual or commodity interest rates prove that the latter fluctuate much more than the former.*43 The effects of this lack of compensation are evil. In the first place, the situation interferes with the normal distribution of wealth and income. If the level of prices is rising, since nominal interest does not for a considerable time rise enough to compensate, the lender gets back a less amount of wealth or services than he might reasonably have expected. Creditors lose and debtors gain. It should be noted also that all persons with relatively fixed money salaries lose by this rise of prices. When the level of prices falls, on the other hand, creditors and persons with relatively fixed incomes gain at the expense of debtors. The distribution of wealth is changed in either case from purely monetary causes, and the change can be averted by making the standard of deferred payments more stable.
§ 5 X.66 We are brought back again, therefore, to the conclusion that on the whole the best index number for the purpose of a standard of deferred payments in business is the same index number which we found the best to indicate the changes in prices of all business done;—in other words, it is the P on the right side of the equation of exchange.*44
X.67 It is, of course, utterly impossible to secure data for all exchanges, nor would this be advisable. Only articles which are standardized, and only those the use of which remains through many years, are available and important enough to include. These specifications exclude real estate, and to some extent wages, retail prices, and securities, thus leaving practically nothing but wholesale prices of commodities to be included in the list of goods, the prices of which are to be compounded into an index number. These restrictions, however, are not as important as might be supposed. The total real estate transactions of New York City (Manhattan and the Bronx) in 1909 (an active year) measured by assessed valuations (probably 4/5 of the market valuation) amounted to only $620,000,000. This is utterly insignificant if compared only with the 104 billions of bank clearings in New York City. Yet real estate transactions probably constitute a higher percentage of total transactions in New York than in the United States.*45 In the United States we feel safe, therefore, in saying that they amount only to a fraction of 1 per cent of the total transactions. As to exchanges in securities, Kemmerer estimates, on the basis of the transactions of the New York Stock Exchange, that about 8 per cent of the total transactions of the country consist in the transfer of securities.*46 As already stated, he also estimates that wages amount to about 3 per cent.*47 As to the comparative importance of retail as compared with wholesale prices, we have some figures of Professor Kinley, of the Monetary Commission.*48 On this basis, and because wholesale and retail prices roughly correspond in their movements, *49 we may omit retail prices altogether. It is true that retail prices usually lag behind wholesale prices; but part of the lagging is more apparent than real. Expert testimony of those who have collected such statistics shows that when, as at present, prices are rising rapidly, retailers obviate the necessity of confronting their customers with too frequent and rapid increases in prices by quoting the same prices and substituting inferior grades or, in some instances, smaller loaves or packages.
X.68 It is true that wholesale transactions constitute a minority of all transactions, perhaps only a fifth.*50 Nevertheless, wholesale prices are more typical than any other.
X.69 They are to a large extent typical of producers' prices which precede them, and of retail prices which succeed them. They are typical of many large and often nondescript groups which go to make up the total transactions, such as are classed together in Kinley's Report to the Monetary Commission under the head "other deposits," including hotel charges, fees of professional men, etc., as well as wages. Among items of which wholesale prices may not be very typical are the transactions in securities (speculative and other), railway and other transportation charges, and insurance. Latterly, prices of stock securities have advanced faster than wholesale prices, while transportation and insurance charges have not advanced as fast. The attempts of Kemmerer and myself (Chapter XII) to combine in one average wholesale prices and prices of stocks and wages yield results differing only slightly from those based on wholesale prices. From a practical standpoint, wholesale prices of commodities are the only prices which are yet sufficiently standardized, and the use of the goods sufficiently stable through a long period of time, to make them serviceable for general use.
X.70 Not only may we consider that wholesale prices roughly represent all prices, but we may, with even more confidence, confine our statistics for wholesale prices to a relatively small number. Edgeworth and others have shown, both practically and theoretically, that a large number of articles is needless and may even be detrimental. The 22 commodities employed by "The Economist" afford an index number of considerable value; the 45 of Sauerbeck have given us a standard of great value; and the 200 and more commodities used in the Aldrich Report and the bulletins of the Bureau of Labor are certainly numerous enough, if not too numerous, to give a most accurate index number of prices.
X.71 The recommendations of the Committee of the British Association for the Advancement of Science were that the index number should include six groups, comprehending twenty-seven classes of articles, and that the prices should be weighted in round numbers representing approximately the relative expenditures of the community in these objects. The groups and classes with weighting were as follows:*51—
Breadstuffs (wheat 5, barley 5, oats 5, potatoes, rice, etc., 5) 20 Meat and dairy (meat 10, fish 2½, cheese, butter, milk, 7½) 20 Luxuries (sugar 2½, tea 2½, beer 9, spirits 2½, wine 1, tobacco 2½) 20 Clothing (cotton 2½, wool 2½, silk 2½, leather 2½) 10 Minerals (coal 10, iron 5, copper 2½, lead, zinc, tin, etc., 2½) 20 Miscellaneous (timber 3, petroleum 1, indigo 1, flax and linseed 3, palm oil 1, caoutchouc 1) 10
X.72 This report was made after very thorough consideration by a remarkably competent committee consisting of Mr. S. Bourne, Professor F. Y. Edgeworth (Sec.), Professor H. S. Foxwell, Mr. Robert Giffen, Professor Alfred Marshall, Mr. J. B. Martin, Professor J. S. Nicholson, Mr. R. H. Inglis Palgrave, and Professor H. Sedgwick. The report also gives the precise technical description of the articles the price quotations of which are to be used (the iron, for instance, being "Scotch pig iron"), and also the price list or other source for price quotations (the wheat, for instance, being the "Gazette Average").
X.73 With slight modifications this recommendation of the British Committee could be made to apply to American figures. In America we have had a number of index numbers of wholesale prices, the most important being (1) those of Roland P. Falkner in the Aldrich Senate Report, covering a period from 1840 to 1891, in which, beginning with 1860, there were 223 commodities included, the results being given in two ways, viz., weighted, the weighting being arranged according to relative expenditures on these articles or their congeners used by workmen, and also unweighted; (2) those of the United States Labor Bureau for 251 to 261 commodities beginning with 1890, and now, it is understood, to be published every year; (3) Dun's index numbers from 1860 to 1906 continued recently for Gibson by Dr. J. P. Norton; and (4) Bradstreet's index numbers since 1895 for 96 commodities.
X.74 We need not go into detailed criticism of these index numbers. On the whole they seem to include too many commodities, while they all employ the objectionable fixed-base system. It would be a great advantage if we could fix upon a system in America which would be not only authoritative, but would give out its results at least yearly and promptly.
X.75 For practical purposes the median is one of the best index numbers. It may be computed in a small fraction of the time required for computing the more theoretically accurate index numbers, and it meets many of the tests of a good index number remarkably well. It also has the advantage of easily exhibiting (by means of the "quartiles") the tendency to dispersion of prices (from each year as a base to the next) on either side of the median. The median should be weighted in round numbers analogously to the weighting already discussed for the more theoretically perfect index numbers.*52 The median of a series of numbers is a number such that there are as many numbers above as below it in the series. If the number of terms in the series is odd, the median is the middle term of the series of numbers arranged in the order of magnitude. If the number of terms is even, the median falls between two terms. If these are equal, the median is identical with them both; if they are unequal, the median lies between them and may then be taken as their simple arithmetical, geometric, or any other average. Practically the two middle terms are almost inevitably so close together that it would make no appreciable difference whatmethod of averaging the two middle terms is adopted. The method of weighting the terms from which a median is computed consists in counting each term the number of times indicated by its weight. To illustrate these statements, it is evident that the median of the numbers 3, 4, 4, 5, 6, 6, 7, arranged in order of magnitude is 5; and the median of 3, 4, 4, 5, 6, 6 is 4 ½.
X.76 If the weights to be attached to these latter numbers are
for number 3 weight 1 for number 4 weight 2 for number 4 weight 3 for number 5 weight 4 for number 6 weight 2 for number 6 weight 1 the median is then found from the following:—
of which the median is 5. The arithmetical averages corresponding to the three medians mentioned (5, 4½ and 5) are 5., 4.67 and 4.69 respectively.
X.77 Practically it is not necessary to arrange the terms in exact order of size. Terms easily recognized as low can readily be paired off against those easily recognized as high and only the remaining few central terms need be arranged in exact order. The terms near the middle being usually almost or quite equal, make the selection of the median extremely easy.
X.78 In order to use the median for an index number of prices, we first arrange our price ratios and then select the median ratio.
§ 6 X.79 In this chapter we have aimed to show that an excellent form of index number of prices is the ratio of real values to ideal values at base prices; and that the elements entering into the construction of index numbers differ according to the different purposes for which they are desired. If the purpose is to measure capital, the prices of services should not be included, but only the prices of the different articles of wealth existing at any point of time. If the purpose is to obtain means to measure real wages, only those things should be included which workingmen buy; and they should be included according to the values bought during a given period, these values being measured at standard prices.
X.80 The question of justice between borrower and lender, where the purpose is to fix on the best index number as a standard for deferred payments, was also considered. It was seen to be not an infringement of justice that one man should gain from another on account of fluctuations in the money standard; for the contract is a free one in which normally each should assume whatever risk there may be of loss for the sake of whatever chance there may be of gain. It was maintained, also, that it would be wrong for the government deliberately to take his gain away from a person who had assumed a risk of loss in the first place. Nevertheless, it was urged that a means by which contracts made in the future could be made less speculative, is desirable.
X.81 It was urged that it is no part of the function of an index number of general prices to guard against rising and falling real income. The function of such an index number is to measure the change in the level of prices, in order that, in contracts involving deferred payments, there shall be no element of risk so far as money is concerned. Without the index number as a standard, such contracts are quite highly speculative. The adjustment of the rate of interest compensates to some extent, but not nearly enough, for the fluctuations in the value of money. These fluctuations influence the distribution of wealth among persons and classes, and bring about crises and business depressions. It is desirable that some basis for time contracts should be fixed upon, which will remedy these evils. It is believed that an index number expressing the price level entering into the equation of exchange might be adopted as such a basis. The ideal set forth is that neither debtor nor creditor should be the worse off by being deceived through changes in the level of prices of goods bought and sold. Some system is to be sought, therefore, by which the actual results of the contract should closely approximate the expected results in nearly all cases.
X.82 It was shown that different persons and different classes might be interested in having for their time contracts index numbers somewhat differently constructed, because different persons are interested in consuming different kinds of commodities and because they desire to invest larger or smaller proportions of their earnings. But for general purposes, as the best compromise to fit the needs of different classes, what was suggested was an index number based on the prices of all goods exchanged during a given period. It was pointed out, however, that the different forms of index numbers which had gained reputation lead to practically the same results.
X.83 Finally, it was shown that for rough and ready computations the median has advantages over all other forms of index numbers.

## Endnotes

18. See Appendix to (this) Chapter X, §§ 1-8, where 44 types of index numbers are compared.
19. For discussions of different ways that have been proposed, see Walsh, The Measurement of General Exchange Value, New York and London (Macmillan), 1901; Edgeworth, "Report on Best Methods of Ascertaining and Measuring Variations in the Value of the Monetary Standard"; Report of the British Association for the Advancement of Science for 1887, pp. 247-301; ditto for 1888, pp. 181-209; ditto for 1889, pp. 133-164. Nitti, La misura delle variazioni di valore della moneta, Turin, 624 pp.; also the Appendix to (this) Chapter X.
20. Cf. Mill, Political Economy, Book III, Chapter XV; Sidgwick, Principles of Political Economy, Book I, Chapter II; "Report of Committee on Value of Monetary Standard," Report of the British Association for the Advancement of Science, 1887; Wesley C. Mitchell, Gold, Prices, and Wages under the Greenback Standard, Berkeley, 1908 (University of California Press), p. 19; and Appendix to (this) Chapter X.
21. See Appendix to (this) Chapter X for a table and discussion of forty-four sample types of index numbers.
22. It is formula 11 of the large table in the Appendix to (this) Chapter X.
23. See Appendix to Chapter II and the Appendix to (this) Chapter X, §§ 5, 6, 7.
24. See Appendix to (this) Chapter X, § 5, test 7.
25. See Report (to Parliament) of an Enquiry by the Board of Trade into Working Class Rents, Housing and Retail Prices, London (Darling), 1908, 1909.
26. This has been suggested by Nicholson, Journal of the Royal Statistical Society, March, 1887.
27. Giffen, in his Growth of Capital, London (Bell & Sons), 1899, pp. 50-54, makes correction for price changes although without attempting to construct a special index number for capital.
28. An early attempt to construct a series of index numbers expressing the general change in prices was made by Sir George Shuckburgh Evelyn, Bart., F.R.S. and A.S. in 1798 in an article entitled "An Account of Some Endeavors to Ascertain a Standard of Weight and Measure," in the Philosophical Transactions of the Royal Society of London, Vol. LXXXVIII, pp. 133-182, inclusive. Bishop William Fleetwood in 1707 in Chronicon Preciosum, an Account of English Money, the Price of Corn and Other Commodities for the Last Six Hundred Years, raises and discusses the question whether the holder of a fellowship founded between 1440 and 1460 and open only to persons having an estate of less value than £5 a year may rightly swear that he has less than that, if he has £6, the value of money, however, having meanwhile greatly depreciated.
The idea of using an index number or tabular standard of money value was later put forth by Joseph Lowe, The Present State of England in regard to Agriculture and Finance, London, 1822 (see pp. 261-291, Appendix, pp. 89-101), and afterwards by G. Poulett Scrope, Principles of Political Economy...applied to the Present State of Britain, London, 1833, pp. 405-408, although, as we have seen, the idea of an index number itself antedated them. See Correa Moylan Walsh, The Measurement of General Exchange Value, New York and London (Macmillan), 1901; Bibliography, p. 555.
29. Cf. Irving Fisher, "Appreciation and Interest," Part 3, Publications of the American Economic Association, 1896.
30. Rate of Interest, Chapter XIV.
31. Appreciation and Interest, Part 3, § 4.
32. It has been argued that an ideal standard ought to be such as to keep constant not objective, but subjective, prices, so that a debt would be repaid in a given amount of "labor" or "utility." But, aside from the practical difficulties of measuring such subjective magnitudes—which are insuperable and therefore render their discussion purely academic—there are even more serious theoretical objections because of the fact that the standard would increase with some persons and decrease with others as they grow poorer or richer and that these changes are anticipated in making loan contracts—in fact, are the instigating causes of such contracts. See § 4 infra and Irving Fisher, "Appreciation and Interest," Chapter XII, § 2, Publications of the American Economic Association, 1896.
33. Cf. Kinley, Money, New York (Macmillan), 1904, p. 267.
34. The argument of the remainder of this section is substantially the same as that in a paper by Harry G. Brown, in the Quarterly Journal of Economics, August, 1909, entitled, "A Problem in Deferred Payments and the Tabular Standard."
35. Cf. Kemmerer, Quarterly Journal of Economics, August, 1909.
36. For a discussion of the effect of a change in the rate of interest on prices, see Irving Fisher, Nature of Capital and Income, New York (Macmillan), 1906, p. 227, and Rate of Interest New York (Macmillan), 1907, pp. 226 and 227.
37. Rate of Interest, pp. 121-125.
38. The same conclusion as to the best standard for deferred payments is reached by Professor H. S. Foxwell by a somewhat different line of reasoning. See remarks of Professor F. Y. Edgeworth (as Secretary of the Committee on Variations of the Monetary Standard). Report of the British Association for the Advancement of Science, for 1889, pp. 134-139.
39. Cf. Edgeworth, in Palgrave's Dictionary of Political Economy, "Index Numbers."
40. See, for example, paper by Professor J. B. Clark, "The Gold Standard in Recent Theory," Political Science Quarterly, September, 1895. Compare with this "The Standard of Deferred Payments," by Professor Edward A. Ross, Annals of the American Academy of Political and Social Science, November, 1892; Lucius S. Merriam, "The Theory of Final Utility in its Relation to Money and the Standard of Deferred Payments," ibid., January, 1893; Professor Frank Fetter, "The Exploitation of Theories of Value in the Discussion of the Standard of Deferred Payments," ibid., May, 1895.
41. Cf. Charles A. Conant, The Principles of Money and Banking, Vol. II, Chapter VII.
42. Rate of Interest, pp. 95-98 and 304-306. This position is taken by Correa Moylan Walsh, The Fundamental Problem in Monetary Science, New York (Macmillan), 1903, p. 345, footnote.
43. Rate of Interest, Chapter XIV.
44. This is really the same conclusion as that reached by Walsh, The Fundamental Problem in Monetary Science, in which, after a thorough and critical review of the literature of the subject, the author concludes that the kind of stability desirable in the standard of deferred payment is "stability of exchange value."
45. At any rate the impression is strong that real estate is more "active" in New York than in most other cities, because of the rapid change in the character of sites on account of the narrowness of Manhattan Island and the consequent acceleration of growth in one direction (northward) and that, in general, cities have more trade in real estate than the country, not only absolutely, but relalively to other trade.
46. Money and Prices, 2d ed., New York (Holt), 1909, p. 138.
47. Ibid., p. 138.
48. Credit Instruments, 1910, 61st Congress, 2d Session, Senate Document 399, pp. 69, 73, 134, 136, would indicate that wholesale trade requires something like twice as much exchange work as retail trade.
49. The studies of the Bureau of Labor in retail prices seem to show a general sympathy in movement between retail or wholesale prices, as indeed might be expected.
50. See, in Report of National Monetary Commission on Credit Instruments, the figures of aggregate sums deposited in banks by wholesale merchants and others. While these do not afford an exact comparison, they aid in making a rough guess.
51. See Report of the Committee in Report of the British Association for the Advancement of Science, for 1888, p. 186.
52. See Appendix to (this) Chapter X, § 8.