The Statistical Conditions for a Change in Business Concentration

Abstract

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.