Relationship between concentration ratio and Herfindahl-Hirschman index: A re-examination based on majorization theory

While the two most widely used measures of market (industrial) concentration, the m-firm concentration ratio [Formula: see text] and the Herfindahl-Hirschman index H, have no precise functional relationship, they can be related by means of boundary formulations. Such bounds and potential relationships, which have been considered in some earlier reported studies, are being re-examined, corrected, and reformulated in this paper. The underlying analysis uses a different approach based on majorization theory and the results are supported by computer simulation. Such boundary relationships make it possible to determine approximate values of H from those of [Formula: see text] and vice versa for any given set of market shares. Much more accurate predictions of H-values can be obtained with knowledge of the individual market shares of the m largest firms within a market (industry), with or without knowledge of the total number of firms.