By David J. Smyth
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That is, the differences in the estimates in earlier studies may be explained purely in terms of differences in the measures of firm size used. SIZE AND GROWTH OF FIRMS 37 These differences are no help to us, however, in explaining our own results for while the estimated values of A vary from measure to measure they are all significantly less than unity. Two possible explanations come to mind. First, our firms are large firms, the largest 500 industrial firms. Both Hart and Samuels make use of a sample of firms including firms much smaller than these, and the rates of growth of these firms may be slower than of the firms in the top 500.
1. 1) above, with the value of a being negative. Thus we know that the four concentration indices are not proportional to each other, for proportionality requires a = 0; hence the conditions for interchangeability are not met and conclusions of empirical studies will not be independent of the measure of concentration used. 3) will be. As a is negative the elasticity of any index CRn with another index CRn+k, where k is positive, is greater than one. 3), the larger k is in the CRn+k measure used as C 2, the larger, in absolute terms, is the negative coefficient a and hence the larger will be the elasticity of Y with respect to CRn+k relative to the elasticity of Y with respect to CRn.
6 CONCLUSIONS For empirical results to be independent of the choice of measure of concentration a quite stringent condition must hold. It is not sufficient that the measures are highly correlated. The necessary and sufficient condition for measures to be interchangeable is that they MEASURES OF CONCENTRATION: THEORY AND EVIDENCE 47 be proportional. It follows simply from the definitions of the measures that measures cannot be proportional to each other and so this condition cannot be fulfilled.