Editing Herfindahl–Hirschman index (section)

==Formula==

<math display="block">HHI =\sum_{i=1}^N (MS_i)^2</math>
where <math display="inline">MS_i</math> is the market share of firm <math>i</math> in the market, and <math>N</math> is the number of firms.<ref name=":1" /> Therefore, in a market with 5 firms each producing 20%, the HHI would be <math>0.2^2 + 0.2^2 +  0.2^2 + 0.2^2 + 0.2^2= 0.20</math>.

The Herfindahl Index (''HHI'') ranges from 1/''N'' (in case of [[perfect competition]]) to 1 (in case of [[monopoly]]), where ''N'' is the number of firms in the market. Equivalently, if percents are used as whole numbers, as in 75 instead of 0.75, the index can range up to 100<sup>2</sup>, or 10,000.
[[File:Herfindahls index.jpg|thumb|Herfindahl-Hirschman Index]]
'''An ''HHI'' below 0.01 (or 100)''' indicates a highly competitive industry, Mergers and acquisitions with an increase of 100 points or less will usually not have any anti competitive effects and will require no further analysis.<ref name="dojguidelines">{{Cite web|date=August 19, 2010|title=Horizontal Merger Guidelines (08/19/2010)|url=https://www.justice.gov/atr/horizontal-merger-guidelines-08192010|access-date=January 19, 2023|website=justice.gov|publisher=[[United States Department of Justice]]|ref={{harvid|United States Department of Justice|2010}}}}</ref> <br>'''An ''HHI'' below 0.15 (or 1,500)''' indicates an unconcentrated industry. Mergers and acquisitions between 100 and 1500 points are unlikely to have anti-competitive effects and will most likely not need further analysis.<ref name="dojguidelines"/><br>'''An ''HHI'' between 0.15 and 0.25 (or 1,500 to 2,500)''' indicates moderate concentration. Mergers and acquisitions that result in moderate market concentration from HHI increases will raise anti-competitive concerns and will require further analysis.<ref name="dojguidelines"/><br>'''An ''HHI'' above 0.25 (above 2,500)''' indicates high concentration.<ref name="dojguidelines"/> Mergers and acquisitions with HHI scores of 2,500 or above will be considered anti competitive and an in-depth analysis produced, if the scores are well above 2,500 they are considered to enhance market power they may only be allowed to progress when significant evidence is shown that the merger or acquisition will not increase market power.<ref name="dojguidelines"/>

A small index indicates a competitive industry with no dominant players. If all firms have an equal share the reciprocal of the index shows the number of firms in the industry. When firms have unequal shares, the reciprocal of the index indicates the "equivalent" number of firms in the industry. Using case 2, we find that the [[market structure]] is equivalent to having 1.55521 firms of the same size.

There is also a normalized Herfindahl index. Whereas the Herfindahl index ranges from 1/''N'' to one, the normalized Herfindahl index ranges from 0 to 1. It is computed as:
*<math display="inline">HHI^* = {\cfrac{\left ( HHI - \dfrac{1}{N} \right )} {1-\dfrac{1}{N}} }</math> for ''N'' > 1 and 
*<math>HHI^* = 1</math> for ''N'' = 1
where again, ''N'' is the number of firms in the market, and ''HHI'' is the usual Herfindahl Index, as above.

Using the normalized Herfindahl index, information about the total number of players (''N'') is lost, as shown in the following example:

Assume a market with two players and equally distributed market share; <math display="inline">H=\dfrac{1}{N}=\dfrac{1}{2}=0.5</math> and <math>H^*=0</math>. Now compare that to a situation with three players and again an equally distributed market share; <math>H=\dfrac{1}{N}=\frac{1}{3}\approx 0.\overline{333}</math>, note that <math>H^*=0</math> like the situation with two players. The market with three players is less concentrated, but this is not obvious looking at just ''H*''. Thus, the normalized Herfindahl index can serve as a measure for the equality of distributions, but is less suitable for concentration.