(M)

MR. WALSH ON THE MEASUREMENT OF

EXCHANGE-VALUE 1

[THE ground on which I ventured to criticise Mr. Pierson's attack on index-numbers, namely, the not to be ignored connection of the subject with Probabilities, is also the main ground of my differences with Mr. Correa Walsh. They are expressed in the following paper, which appeared in the ECONOMIC JOURNAL, 1901, under the title, "Mr. Walsh on the Measurement of General Exchange-Value." Mr. Walsh does not accept my view, and has replied with vigour in a brochure entitled "The Problem of Estimation," of which an account is given in the Journal of the Statistical Society for 1921, in a review bearing the wellknown initials G. U. Y.

A rejoinder to Mr. Walsh's replies is published in two parts, one in the ECONOMIC JOURNAL, September 1923, the other in the Journal of the. Royal Statistical Society, July 1923. Cp. above, p. 198.]

The capacity of taking boundless trouble, which is a characteristic of solid talent, distinguishes the work of Mr. Walsh. Whether he searches the writings of others or elaborates his original ideas, the thorough student and close thinker is manifest on every page.

The literature of the subject has never been examined so fully. Every devious path in the field where index-numbers flourish has been traversed in order to form an unrivalled collection of methods for measuring changes in the value of money. Many of the specimens here exhibited are probably new even to specialists. Or if the form was known, its origin and evolution were unknown. Who ever heard, for instance, of Carli and of Dutot as authorities on the subject? The bibliography would alone be sufficient to impart a lasting value to this work.

But Mr. Walsh is much more than a collector of specimens, 1 The Measurement of General Exchange-Value, by Correa Moylan Walsh. New York: Macmillan & Co., 1901.

VOL. I.

369

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The powers of a systematic botanist are also his. He classifies the material which he has collected. For example, it is doubtless a great improvement in logical arrangement to distinguish indexnumbers in which, as usual, a single system of weights is used for the relative prices, from those typified by Lehr's and Drobisch's methods in which "double weighting" is practised. Again, among methods of weighting each article according to the expenditure thereon, there is a distinction between those which in effect compare the money value of the same set of articles at different times and those typified by Mr. Palgrave's method. I give the essence, as I conceive it, rather than the wording of some passages in the author's learned and logical Appendix C.

Mr. Walsh has not contented himself with classifying the specimens which he has collected. He has also attempted to penetrate to the structure and function of an index-number by a new microscopical analysis. Having observed the properties of the different kinds, by skilfully crossing the "arithmetic with the "geometric " type he has produced a new variety which may claim to excel in certain respects the existing species.

Limits of space prevent me from tracing these general characteristics through the contents of Mr. Walsh's volume. In truth, it might be feared that my reader's patience would give out if I attempted to reproduce in anything like their original, almost Kantian, elaborateness discussions to which the term "exhaustive," with all its suggestions, is particularly applicable. I will, therefore, select a few points which seem to be of special and permanent interest. Some solid and salient stepping stones may thus be afforded for traversing the flood of dialectic.

Mr. Walsh begins by defining different senses of value. He is specially happy in distinguishing cost value from other species. He complains not without justice, although great names fall under his condemnation, of those who have confounded the different quæsita. He well remarks that, if a measure pertaining to cost value is to be constructed, we should not confine our calculations to the consideration of wages, but include profits.1 His own investigation is confined to "general exchange value," which seems to have a certain parallelism with " final utility," as appears from its relation to Lehr's method :

"In this method [Lehr's] its author has made an effort to do what appears to be accomplished in the method here presented. He has tried to measure the variation in the average price of

1 Cp. Section on the "Labour Standard in the Memorandum attached to the third Report of the British Association Committee (above, p. 293).

mass-units, in all the classes, that have the same exchange-value over both the periods together-to which equivalent mass-units he has given the not inappropriate name of pleasure-units (p. 386).

66

But Mr. Walsh's exchange-value is more objective (9). The properties of general exchange-value are set forth in a series of propositions, which may deserve the epithet "expletive," in so far as they are mostly self-evident yet render our instructive knowledge fuller and clearer. Among original points may be noticed the distinction between the exchange-value of a thing (e.g., money) in relation to all other things, and in relation to all things including itself (13). When first the reader learns that exchange-value is considered as objective, he may be disposed to expect that it is an affair only of ratios abstracted from the quantities produced and consumed. Insensibly, however, as we ascend the gentle steps formed by the series of more or less expletive" propositions, there is borne in on us the need of weighting. We dimly descry a unit, sometimes called an “economic individual" (102, 301), an "exchange-value quantum" (302); we are directed to contemplate" mass-units ideally constructed " (285), "considered as equal, not as weights or capacities, but as exchange-values" (284), in relation to which it is sought to determine the value of money at different times (and places). The data for this determination are prices and quantities of commodity; the problem is properly to combine these data. Two main questions arise :-What importance or weight" is to be assigned to each of the given prices which enters into the combination? and What should be the method of combination? These questions are first considered separately as far as possible, and then in their necessary connection. I will not follow the preliminary separate inquiries through the windings of Mr. Walsh's exhaustive discussion. Suffice it to notice that materials are not to be included in our index-number along with finished goods (78, 96), apparently for a reason usually given, that the factors of production are counted in the products. Nor is it the quantity of each exchangeable thing that is actually exchanged for money (85), but rather, as I understand, the quantity that is used, which concerns us. As to the method of combining the data we are practically restricted to the three classic Means, the Arithmetic, Harmonic, and Geometric. The author compares the properties of these means, showing certain grounds for the preference of the Geometric :

"If the exchange-value of money in [B] rises by more than 100

per cent. the compensatory fall of the exchange-value of money in [A] should be to below zero according to the arithmetic method of averaging, which therefore is inapplicable in this case [where [A] and [B] are two equally important classes of things.] And if the exchange-value of money in [A] falls to less than half, the exchange-value of money in [B] should rise from below zero, according to the harmonic method of averaging, which therefore is inapplicable here. But in the use of the geometric compensation there are no such impossible cases (249).

This passage illustrates certain properties of the compared means, to which the author attaches importance. In the simple case of two extremes, between which a Mean is taken, the distance of the Arithmetic Mean from one extreme, per cent. of the Arithmetic Mean, is equal to the distance of the Arithmetic Mean from the other extreme, per cent. of the Arithmetic Mean. The distance of one extreme from the Harmonic Mean, per cent. of that one extreme, is equal to the distance of the other extreme from the Harmonic Mean, per cent. of that extreme. The distance of one extreme from the Geometric Mean, per cent. of that extreme, is equal to the distance of the Geometric Mean from the other extreme, per cent. of the Geometric Mean. This last proposition cannot be extended from the case of two to that of many variables, from the geometric mean, in Mr. Walsh's very peculiar phraseology, to the geometric average. To the same class of properties, true of the "mean," but not the average, belongs the following, which Mr. Walsh considers important :

Confining myself to the general and concrete case of plural data, I hasten on to the latter stages in which the question of weights and means, at first separated, are considered in their real connection. We have now to consider penultimately the two simplified cases in which either (1) the sums of money expended on each commodity remain constant at the two periods (or places) compared, or (2) the quantities of each commodity are thus constant; and finally (3) the general concrete case in which both expenditure and quantities vary. In the first case I think most people would be disposed to answer off-hand that the sums supposed constant form the proper weights for an arithmetic combination. The author, however, seems to rightly judge that the ideal of comparing the money-values of the same number of exchange units or "economic individuals" would not be

realised by this procedure; for a reason which he thus assigns with respect to the proposal of taking the arithmetic mean of the sums when supposed different :—

"If it happens that the exchange-value of money has fallen or prices in general have risen, greater influence upon the result would be given to the weighting of the second period. . . . Or in a comparison between two countries greater influence would be given to the weighting of the country with the higher level of prices. But it is plain that the one period or the one country is as important in our comparison between them as the other, and the weighting in the averaging of their weights should really be even (105).

To avoid the difficulty thus indicated, the following formula is proposed in the case of constant sums being expended on each commodity. Let a1a2; B1,B2; be the prices at the first and second epoch respectively, and x1‚¤; Y1,2; ... the corresponding quantities of commodity; the required index-number is

; or, as by hypothesis xa1 = X2а2,

this may be written, 212 + B2 142 +

(310). The

transition is easy from this formula, "Scrope's emended method,” as Mr. Walsh calls it, to Scrope's method pure and simple, which is proper to the second abstract case, in which the quantities of each commodity are constant, say x, y... We have only to substitute in the last written formula, x for x12, and so on (360). These prolusions lead up to the general concrete case in which neither the sums nor the quantities remain constant. Guarding against the difficulties encountered in the simpler cases, the author proposes this "universal formula

This form is shown to have a certain theoretical advantage over other species of index-number, in particular those which, as affected with "double weighting," most challenge comparison with it, namely Drobisch's and Lehr's methods. The universal formula satisfies some of the criteria which Mr. Walsh has laid down. It does not, however, in general, satisfy what he has called Professor Westergaard's test that (e.g.) " prices measured from 1860 to 1870 and from 1870 to 1880 ought to show the same variation from 1860 to 1880 as would be shown by comparing