If we assume for a moment that no changes in value occurred during the period studied, we could use weights of the first year alone. Weighting by total sales value, we should then obtain:17

This is the formula usually called an index of the "physical volume" of production. It is not a physical index in any thoroughgoing sense, because it uses value weights. But, although taking values into account, it gives no effect to the comparative values of goods in the second year. If these changes in value are significant as in our stretch of seventeen years they surely aresome method must be devised for incorporating them.

We may construct an exactly similar formula to use weights of the second or terminal year of our comparison. But in doing so it will be more appropriate to use the terminal rather than the beginning year as the base year of the index.18 The similar formula giving effect to 1924 values in place of those of 1907 is:

In order both to give effect to the change of values and to the change in physical output, an average of these two obviously suggests itself. This is not possible so long as one is expressed as a

ferent possible shapes of demand curves, especially those to the left of the point of equilibrium. The result is that the total area under the curve which might otherwise be an indication of want-satisfaction is not necessarily proportional to the rectangular area representing total sales value. The indeterminate remainder is the roughly triangular area which Marshall took to indicate consumer's surplus. The individual errors arising from the shifts from one shape of schedule to another, however, probably go far to offset each other.

17

'q=physical quantity produced of a single good; v=aggregate sale value produced of the same good; subscript o indicates the first year in the comparison; subscript I indicates the second year in the comparison.

18 This is a special form of the general rule that magnitudes must be expressed in the same type of units before they can be averaged. If London production is being expressed in terms of that of Paris (Paris base), the weights of Paris and not those of London must be used in order to be consistent. Cf. Fisher, op. cit., pp. 270-76.

afterward may be seen. Although there had been slight increases in the number of paupers during the latter part of July, it is clear that, beginning with the week ending August 8, there was a definite and considerable increase in the number of both indoor and outdoor paupers, which was continued until the week of August 29. In the quarterly statement issued by the Local Government Board showing the number of persons in receipt of poor relief during this period, attention is called to the fact that there is a customary seasonal increase in the number of persons relieved during the third quarter of every year. The Local Government Board statement, however, called attention to the fact that the seasonal increase

which ordinarily occurs at some point in the third quarter of the year began rather earlier than usual, viz., on the second Saturday in July, and the normal course of variation in the volume of pauperism has during the last two months of the quarter been considerably modified by the war. After the outbreak of war, considerable increases forthwith occurred during the greater part of August in the number of persons in receipt of relief. The number rose from 618,685 on Saturday, 1st August, to 650,737 on the 22nd of August, an increase of 32,052. . . After the 22nd of August the total number of persons in receipt of relief declined week by week, and on the 26th September it amounted to 641,028 (or 17.2 per 1,000 of the population) as compared with 611,448 (or 16.6 per 1,000 of the population) on the corresponding day in the previous year.

....

The increase between 27th June and 26th September amounted to 22,713, or 3.7 per cent, compared with a decrease of 2,032, or 0.3 per cent, in the corresponding period of the previous year.

The increases or decreases as compared with the corresponding weeks of 1913 should be studied, since the weekly variations may be regular seasonal changes. Table IV, therefore, has been prepared showing the total number of persons relieved in corresponding weeks of 1914 and 1913. This table shows that, during the month before the declaration of hostilities, there had been an increase in the total number of paupers as compared with corresponding weeks of 1913, and that the increases not only continued but rose sharply during the first three weeks of August and were maintained well through the months of September and October, although the high-water mark, an increase of 41,261 over the

hence has incidentally the formal advantages he has shown for that form of index.22 The form developed is more convenient than Fisher's for use with census data, however, for it is not necessary to cross-multiply the price of one year by the quantity of the other. A single calculation of quantity ratios and their reciprocals has merely to be weighted in turn by the aggregate values for each of the two years.

III. GENERAL RESULTS FOR BRITAIN COMPARED WITH

PREVIOUS INDICES

We are now ready to answer the query as to how large was British production in 1924 compared with that in 1907. Our definition of production is expressed in the formula which follows for reasons we have just examined:

By applying this formula to the data of the 1907 and 1924 censuses,23 we secure an index for 1924 of 123. This is a minimum estimate, since we used methods which would tend to understate Simplifying:

This is the form of index for quantities of production exactly analogous to Fisher's "ideal index" for prices. Cf. Fisher, op. cit., pp. 3, 233-34.

22 In essentials, what Fisher demonstrated was that this formula satisfied more closely than any other these two conditions: (1) It is consistent with the truism for individual items that p.q=v. In this formula the p's and q's are interchangeable to give either a price or a quantity index. Notice that this means that the result would have been algebraically the same if we had used the first method suggested for our purpose (p. 5, supra), dividing aggregate sale value Σ by Fisher's index of price level (cf. note 35, infra). (2) It is a consistent formula from whichever point of time it is figured, the one ratio being the exact reciprocal of the other. For the data Fisher used, he estimated that the "instrumental error" in fulfilling these conditions was less than 0.8 per cent.

23 The detailed methods and supplementary estimates necessary to do this are treated here in sections IV and V.

production in the latter year whenever an arbitrary choice has to be made. We may say roughly, then, that total production was one-fourth greater in 1924 than in 1907. Let us compare this estimate with the indices hitherto accepted.

The most systematic attempt2 to estimate British production has been made by Mr. J. F. W. Rowe of the London School of Economics.25 His published index for 1924 is 97.5. This index, however, includes agriculture, which does not concern us, and is calculated on a composite base of years 1907-13 instead of 1907 alone. His figures may be recalculated, however, and his result for industrial production in 1924 based on 1907 then becomes 98.1. Surely it makes a good deal of difference whether production declined 2 per cent, as most people have thus been led to believe, or has increased by about one-quarter, as our index indi

cates!

Our index is higher largely because we now have data for more commodities. The only complete production data available for Mr. Rowe previous to the 1924 census were for coal, pig iron and steel, and ship-building tonnage launched. All these are products of especially depressed industries. The lack was partly made up by some data secured from private sources, but also from export and import returns. The latter also underestimate production because of the particularly depressed condition of Britain's foreign trade, and because an increased product is now made from a given quantity of basic material imported, as compared

*There have been many impressionistic estimates, of course. For example, the Westminister Bank Review in 1923 estimated 1922 production at 22 per cent less than 1913, based on export and import returns. See criticism in the Economist, XCVII, 78. For an able criticism of several of these indirect methods of measuring production, see A. W. Flux, "An Index of Production," Journal of the Royal Statistical Society, XC, Part II, 226-38.

*He has published an annual index for the years 1907 to 1913 and from 1920 to date, and in addition a quarterly index since 1920. The quarterly index, which is independent of the annual and is based on less adequate data, will not concern us here. See London and Cambridge Economic Service, Monthly Bulletin Nos. 1, 4, 6, and 10 each year, 1923 to date. Also Special Memorandum No. 8, where Mr. Rowe summarizes his methods and results up to 1923. For general discussion, see J.F. W. Rowe, "An Index of the Physical Volume of Production," Economic Journal, XXXVII, 173-87.

to pre-war.26 Our figures, on the other hand, are drawn from the Census of Production itself and hence are far more complete.

So great, indeed, was the disparity between Mr. Rowe's results and other indications of production, such as coal consumed, railway goods carried, and income tax returns that the editor of the Economist was moved to protest even before the 1924 census was available. Mr. Layton estimated that 1925 production was about equal to the "pre-war level of 1913," as against Mr. Rowe's figures, which show it to be 5.5 per cent below 1907 and 14.4 per cent below 1913.27

Perhaps it was Mr. Layton's criticism that led Mr. Rowe to say, in commenting on his tables, that his index might "register as much as five per cent too low" in post-war years.28 It is not clear whether this correction would be sufficient to satisfy Mr. Layton or not.29 Nevertheless, it is interesting to notice that the maximum index for 1924 which Mr. Rowe would have conceded is 103.30 This substitutes a 3 per cent increase in production for the 2 per cent decrease which his table itself indicates. But even this is hardly more than one-eighth the increase which is shown by our index of 123.

If the statement of Mr. Layton quoted in the foregoing be given an interpretation most favorable to a high 1924 index, then his guess comes much nearer to our result. It is possible in this

"W. T. Layton, discussion in the Journal of the Royal Statistical Society, XC, Part II, 259-61.

28

23 L. & C. Econ. Service, Monthly Bulletin No. 6 (1927), p. 203.

"If Mr. Layton meant that the 1925 production of industry and agriculture equaled that of 1907, Rowe's correction suffices, since 105 per cent of this result for 1925 (97.5) equals 101.6. But at the other extreme, if Mr. Layton meant that 1925 industrial production (excluding agriculture) equaled 1913 industrial production, then even this maximum estimate by Mr. Rowe leaves him 10 per cent behind Layton's estimate. Thus, using Rowe's data, we get: