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I was recently reading about the Herfindahl Index and from what I've learned so far, is that HHI is preferred over the Concentration Ratio. However, I didn't quite understand the reasoning that lead to this conclusion. What impact does squaring have on the final result? Any hints or suggestions for further reading will be appreciated.

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From the Industrial Organization by Belleflame and Petiz (Page 34/35, Chapter 2):

While the m-firm concentration ratio adds market shares of a small number of firms in the market, the so-called Herfindahl index (also known as Herfindahl–Hirschman index) considers the full distribution of market shares.

We can conclude that the mathematical approach in HHI is more adequate for a full market setup, rather than observing a particular concentration ratio in one firm. Also from the same book

the Herfindahl index provides a better measure of concentration as it captures both the number of firms and the dispersion of the market shares.

Hence the squared market shares.

For your information, there is another concentration measure called the Lerner Index, although the Lerner index is a snapshot of the intensity of competition. You can calculate L by finding the difference between price and marginal costs as a percentage of the price. More formally: $L=\frac{p-C'}{p}$

But it is easily observed that L ignores some dynamics of the markets. Lower prices do not necessarily mean high levels of competition.

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