ONE of the ways of tracing changes in business concentration is to compare the rates of growth of firms of different sizes. If firms that are large at a certain date subsequently grow on the average more rapidly than small firms, business concentration will obviously have increased. It is not however always realized that the obverse proposition does not hold; that is, if the large firms have grown less rapidly than the small firms then -however paradoxical it may seem -concentration does not necessarily decrease. A related paradox -that associated with what is known as jobbing arises if we look at the average rate of growth achieved in the past by firms that are large today, and compare it with that for smaller firms. This differs from the previous example, since we are now looking backward in time instead of forward; as will be seen below, the two points of view are symmetrically related to one another. The matter is more complicated in its logic than appears at first sight, and it is perhaps not surprising that many careful empirical investigations are to be found in the literature which are vitiated by a logical fault in the inferences drawn from them. The errors are generally pointed out subsequently, only to be repeated by the next generation. The present note attempts to give a simplified but systematic exposition of the necessary conditions for changes in concentration; the basic algebra has already been set out from a different point of view in an earlier paper,' but the approach adopted here may be easier to follow. The statistician will recognize all that follows as being no more than the simple theory of in the sense of Galton (not in the modern sense, where regression is often taken as equivalent to the procedure of fitting a line by least-squares). Our understanding of the problem owes much to the review by Hotelling in the Journal of the American Statistical Association, xvIII (933), 463-65 of Secrist's The Triumph of Mediocrity in Business (Chicago, I933), and the subsequent discussion, ibid., XIX (I934), I96-2 00.2 Similar arguments are to be found in other contexts in the writings of Professor Milton Friedman.

On the basis of this information we may make the flat statement: If the retirement distribution can be assumed to resemble the curve shown in Chart I, and the trend of installations over a past period of double the average service life of the assets can be roughly represented by a constant rate of growth, then, whatever that growth rate and whatever the life average, the gross survivor value computed by the crude method does not deviate by more than 6 per cent from what would be obtained by the correct method.5 In the case of a structurally similar retirement distribution with less relative dispersion the divergence for any given r n, hence also the maximum divergence, would be smaller.6 As far as they go, these findings indicate that the simple cumulation method does yield an acceptably close approximation to the correct result. To secure a fully generalized answer, covering most cases likely to occur in practice, it would be necessary (and probably sufficient) to add a similar analysis for the other f (x) and g (t) types listed above. If survival rates based on a skew retirement distribution curve were applied to the g (t)function underlying the preceding analysis, the divergence between the two S would have different values and different maxima, depending on the degree of skewness and dispersion of f (x). But a maximum divergence for some specifiable rnproduct may again be expected to exist for any assumed f (x). If the assumed g (t)-pattern is anything other than the simple exponential growth function we have used, the percentage deviation between the two gross stock values must be expected to depend also on the specific contour of the installation flow during the past period indicated by the range s, hence on the length of that range and thus, in general, on the life average n as well. 'The margin between 6 per cent and the 5.26 per cent we have derived is certainly sufficient to allow for any possible difference of our result from what would be obtained if annual rather than continuous functions were used. 8In our analysis we have purposely experimented with a retirement distribution curve having a fairly high coefficient of variation, which of course tends to increase the relative disparity between Si and S2. If retirements are completely concentrated at the average service life, the gross survivor values obtained by the two methods are always equal. In this as in any similar analysis, minor erratic oscillations of actual installations around a generally realistic g (t)-trend, or of actual retirements around a generally realistic f (x) -curve, will hardly affect the reliability of the results, except perhaps in the case of very short average service lHves.