Moreover, since upon Ricardian principles the value in exchange of commodities is proportioned to the "comparative quantity of labour expended on each," there may be expected some correspondence between the two expressions, not only as to their general form, but also as to the constants which they involve, the weights with which the variations of wages and prices are respectively to be affected. But the idea of such a correspondence is marred by the fact that the denominations of finished products do not coincide with the classification of wages. Also the suggested analogy is vitiated by a circumstance which is of great theoretical importance that values in exchange-and accordingly the proportions which form the weights of the Consumption Standard -depend not only on quantity of labour, but also on interest, according to the different degrees of durability of the capital employed in producing them. This circumstance, as it creates a difficulty with regard to Ricardo's first principles, so it suggests a scruple about the method which is here connected with those principles. When we "hypothetically argue and speak" of an invariable commodity "which at all times requires the same sacrifice of toil and labour to produce it," 2 should we include in the idea of "sacrifice" not only bodily and mental labour, but also abstinence?* Shall we introduce into our Index-number the variation in the rate of Interest, weighted by the total amount paid in the way of Interest? Or shall we follow the example of the great theorist himself, and omit the consideration of Interest as often as convenience and rotundity of statement and the purpose of a rough approximation may require? The management of these and other difficulties connected with the Labour Standard must be resigned to the abler hand which has already touched this part of the subject.

1

1 Cp. Sidgwick, Political Economy, Book I. ch. ii. "It is rather a perplexing question how Ricardo and M'Culloch could deliberately adhere to the statements above quoted [that labour is the measure of the real value of things, etc.], while they at the same time drew attention to the differences in the value of different products, due to the different degrees of durability of the capital employed in producing them.”

2 Ricardo, loc. cit.

* I should now answer to this question "yes;" having regard to Marshall's above cited definition of "appreciation," with which may be compared the conception introduced in his later work with reference to international trade, of representative "bales," "each of which represents uniform aggregate investments of her [country's] labour (of various qualities) and of her capital." I have altered some passages in the context, so as to exclude the term "wages" which, designating a share in the total product, is not germane to the RicardoMarshall conception.

CONCLUSION.

In conclusion it may be useful to enumerate and summarily characterise the principal definitions of the problem, or "Standards," 1 which have been discussed in this and the preceding Memorandum. An alphabetical order will be adopted, the order of merit being not only invidious, but also impossible in so far as different methods are the best for different purposes.

1. The Capital Standard takes for the measure of appreciation or depreciation the change in the monetary value of a certain set of articles. This set of articles consists of all purchasable things in existence in the community, either at the earlier epoch or at the later epoch, or some mean between those sets. This standard is due to Professor Nicholson. It is stated by him (in terms a little less general than those here adopted) in his book on Money. It is discussed in the sixth and the tenth sections of the former Memorandum.

2. The Consumption Standard takes for the measure of appreciation or depreciation the change in the monetary value on a certain set of articles. This set of articles consists of all the commodities consumed yearly by the community either at the earlier or the later epoch, or some mean between those two sets. This standard has been recommended by many eminent writers, in particular by Professor Marshall in the Contemporary Review of 1887. It is proposed by the Committee as the principal standard. It is discussed in the second section of the former Memorandum.

3. The Currency Standard takes as the measure of appreciation or depreciation the change in the monetary value which changes hands in a certain set of sales. These sales comprise all the commodities bought and sold yearly at the earlier epoch or at the later epoch, or some mean between those quantities. This standard appears to be implicit in much that has been written on the subject, but to have been most clearly stated by Professor Foxwell. It is discussed in the second section of this Memorandum.

4. The Income Standard takes as the measure of the appreciation or depreciation the change in the monetary value of the average consumption, or in the income per head, of the community. This standard is proposed in the fourth section of the former Memorandum.

1 The methods discussed in connection with the names of Mr. Giffen, Mr. Bourne, and Sir Rawson Rawson are rather solutions than statements of the problem.

5. The Indefinite Standard takes as the measure of appreciation or depreciation a simple unweighted average of the ratios formed by dividing the price of each commodity at the later period by the price of the same commodity at the earlier period. The average employed may be the Arithmetic Mean used by Soetbeer and many others, or the Geometric Mean used by Jevons, or the Median recommended by the present writer. This standard is recommended by the practice of Jevons 1 and the theory of Cournot. It is discussed in the eighth and ninth sections of the former Memorandum, and the fifth section of the present one. 6. The Production Standard takes as the measure of appreciation or depreciation the change in the pecuniary remuneration of a certain set of services, namely, all (or the principal) which are rendered in the course of production, throughout the community, during a year, either at the initial or the final epoch; or some expression intermediate between the two

1 Most of Jevons' celebrated calculations (Currency and Finance, II., III., and IV.), and in particular his calculation of the Probable Error incident to his result (ibid., p. 157), involve this conception.

2 Cournot has considered our problem in each of the five volumes in which he has treated of, or touched on, Political Economy (Dictionary of Political Economy, Art. Cournot). It is sufficient here to refer to the first and the last of those works, the Recherches of 1838 and the Revue Sommaire of 1876-the Alpha and almost the Omega of economic wisdom. From these it is clear that variation in the absolute " or "intrinsic" value of money, in Cournot's view, corresponds to the "Indefinite Standard " as defined in Section viii. of the predecessor to this Memorandum. Cournot illustrates the variation due to a change on the part of money, by that change in the position of the earth with respect to the stars, which is due to the motion of the earth. In this analogy the stars are treated as "points" (Recherches, Art. 9). No account is taken of their mass. The context shows that Cournot contemplates a simple average of distances between the earth and each star; not a weighted average, or the distance between the earth and the centre of gravity of the stars. In his later works he expressly declares against, or at least thinks unbefitting highest place, the measure of what he calls the "power of money (Review Sommaire, Sect. 3), that is, in our terms, the Consumption Standard; the analogy of which is the distance of the earth from the centre of gravity of the stars, or rather of certain select stars-say those which are nearest to our human sphere. The Currency Standard, of which the analogy is the distance of the earth from the centre of gravity of all stars whatever, does not seem to have been entertained by Cournot.

Cournot, alluding to Jevons' treatment of the problem in Money, not unjustly takes him to task for not having distinguished assez nettement variations in the "intrinsic value of money" [of which the measure is our Indefinite Standard] from variations in the " power of money " [of which the measure is our Consumption Standard] (Revue Sommaire, p. 121). Referring to Jevons' proposal to construct a Tabular Standard of Value, Cournot expresses his approbation in words which may fittingly conclude the present study :-" Ce sont là des idées qu'il faut laisser mûrir. Quand le moment sera venu de construire effectivement l'étalon monétaire les géomètres pourront y trouver une application intéressante de leur Théorie des Moyennes, telles qu'ils l'ont déjà construite pour les besoins de l'astronomie et de la physique."

specified. The theoretical basis and practical construction of such a standard are indicated in Ricardo's Principles of Political Economy (ch. xx. and elsewhere), in Professor Marshall's evidence before the Gold and Silver Commission (Parl. Papers 1888, C. 5,512, Question 9,625), and in the papers contributed by Mr. Giffen to the second volume of the bulletin of the International Statistical Institute. The standard is discussed in the last section of this Memorandum. It is akin to the method suggested in the fifth section of the First Memorandum, and to the standard proposed by Professor Simon Newcomb which was discussed in the first section of this (third) Memorandum.

(I)

TESTS OF ACCURATE MEASUREMENT.

[THIS is the second of the Memoranda on Variations in the Value of Money, prepared for the British Association in the latter 'eighties of last century. A separate place is assigned to the second Memorandum, which purports to test the accuracy of the calculations prescribed in the first (and third). The verification is effected by the use of that leading principle of Probabilities known as the Law of Error. Caution is required in applying the test. The beginning of wisdom in this matter is to recognise the analogy between the grouping of heterogeneous price-variations and the dispersion of errors-of-observation.

But it is not safe to treat each relative price as an independent observation. In the case of prices there is no doubt a good deal of independence in the sense proper to Probabilities. But there is alsoas perhaps more than is commonly assumed in the case of physical observations-a good deal of interdependence or correlation among the factors which compose or cause the given observations. The subjoined computations are made in the supposition that each element of an index-number, each percentage representing a comparative price, is "subject to a presumably independent error (par. 5). So far as this hypothesis does not hold good in the concrete, the inverse square root of n, the number of the data, which continually figures in the measure of possible inaccuracy must be taken cum grano (cp. infra, p. 324).

Attention may be called to the advocacy of the Median (for the computation of certain index-numbers) on the score not only of its peculiar facility, but also (in certain cases) its comparative accuracy. The Weighted Median, Laplace's Method of Situation, is not so familiar an operation but that its exemplification may be useful.

The reader may be assisted in following the computations by having before him an example to which they are often referred, namely, the model index-number proposed by the British Association Committee. It is reprinted here together with the explanations attached to it by Giffen, who took a principal part