Editing Herfindahl–Hirschman index

{{Short description|Measure of the relative size of firms}}
{{About|the economic measure|the author-level metric|H-index}}
{{Competition law}}

The '''Herfindahl index''' (also known as '''Herfindahl–Hirschman Index''', '''HHI''', or sometimes '''HHI-score''') is a measure of the size of [[corporation|firm]]s in relation to the [[Industry (economics)|industry]] they are in and is an indicator of the amount of competition among them.  Named after economists [[Orris C. Herfindahl]] and [[Albert O. Hirschman]], it is an [[economics|economic]] concept widely applied in [[competition law]], [[antitrust]] regulation,<ref>{{harvnb|United States Department of Justice|2010|loc=§ 5.3}}</ref> and technology management.<ref>{{cite journal|title=Inventive concentration in the production of green technology: A comparative analysis of fuel cell patents|last1=Liston-Heyes|first1=Catherine|author-link1=Catherine Liston-Heyes|last2=Pilkington|first2=Alan|journal=[[Science and Public Policy]]|volume=34|issue=1|pages=15–25|date=February 2004|publisher=Beech Tree Publishing|doi=10.3152/147154304781780190}}</ref> HHI has continued to be used by antitrust authorities, primarily to evaluate and understand how mergers will affect their associated markets.<ref name=":0">{{Cite journal|last=Roberts|first=Toby|date=April 2014|title=When Bigger Is Better: A Critique of the Herfindahl-Hirschman Index's Use to Evaluate Mergers in Network Industries|url=https://digitalcommons.pace.edu/cgi/viewcontent.cgi?article=1863&context=plr|format=PDF|journal=Pace Law Review|volume=34|issue=2|pages=894–946|issn=0272-2410}}</ref>

HHI is calculated by squaring the market share of each competing firm in the industry and then summing the resulting numbers<ref name="dojhhi">{{Cite web|date=July 31, 2018|title=Herfindahl-Hirschman Index|url=https://www.justice.gov/atr/herfindahl-hirschman-index|access-date=January 19, 2023|website=justice.gov|publisher=[[United States Department of Justice]]}}</ref> (sometimes limited to the 50 largest firms<ref>{{Cite book|chapter-url=https://wps.pearsoned.com/wps/media/objects/3844/3936850/protected/ch09/micsg09.pdf|chapter=Chapter 9: Organizing Production|title=Economics|last=Parkin|first=Michael|publisher=[[Addison-Wesley]]/[[Pearson Education]]|year=2002|edition=6th|pages=155–166|location=[[Boston|Boston, MA]]|isbn=9780321112057|access-date=January 19, 2023}}</ref><ref>{{cite book|title=Economics: Canada in the Global Environment – Solutions to Problems|edition=6th|chapter=Chapter 9: Organizing Production – Solutions to Problems|chapter-url=https://people.stfx.ca/gholmes/Micro%20101/Answers%20to%20Text%20Questions/chpt%209.pdf|last1=Parkin|first1=Michael|last2=Bade|first2=Robin|year=2006|pages=30–32|publisher=[[Pearson Education|Pearson Education Canada]]|via=[[St. Francis Xavier University]]|isbn=978-0321312686}}</ref>). The result is proportional to the average market share, weighted by market share. As such, it can range from 0 to 1.0, moving from a huge number of very small firms to a single [[monopoly|monopolistic]] producer. Increases in the HHI generally indicate a decrease in competition and an increase of [[market power]], whereas decreases indicate the opposite. Alternatively, the index can be expressed per 10,000 "[[Basis point|points]]".  For example, an index of .25 is the same as 2,500 points.

The major benefit of the Herfindahl index in relation to measures such as the [[concentration ratio]] is that the HHI gives more weight to larger firms. Other advantages of the HHI include its simple calculation method and the small amount of often easily obtainable data required for the calculation.<ref>{{Cite web|last=Maverick|first=J. B.|title=What Are the Benefits and Shortfalls of the Herfindahl-Hirschman Index?|url=https://www.investopedia.com/ask/answers/051415/what-are-benefits-and-shortfalls-herfindahlhirschman-index.asp|date=September 21, 2021|access-date=January 19, 2023|website=[[Investopedia]]}}</ref>

The HHI has the same formula as the [[Simpson diversity index]], which is a [[diversity index]] used in ecology; the [[Purity (quantum mechanics)#Inverse Participation Ratio (IPR)|inverse participation ratio (IPR)]] in physics; and the inverse of the [[effective number of parties]] index in political science.

==Example==
Consider an example of 3 firms before and after a merger, with the top 2 firms producing 40% of goods each, and the other firm producing 20%. 

'''Prior to Merger:''' <math>0.4^2+0.4^2+0.2^2=0.36=36\%</math><ref name=":1">{{Cite journal|last=Rhoades|first=Stephen A.|date=March 1993|title=The Herfindahl-Hirschman Index|url=https://fraser.stlouisfed.org/files/docs/publications/FRB/pages/1990-1994/33101_1990-1994.pdf|journal=[[Federal Reserve Bulletin]]|volume=79|issue=March|pages=188–189|publisher=[[Federal Reserve Bank of St. Louis]]}}</ref>

Now let's consider the top 2 firms merging.

'''Post Merger:''' <math>(0.4 + 0.4)^2+0.2^2=0.68=68\%</math><ref name=":1" />

As can be seen prior to the merger, the HHI, while not low, is in a range that allows for strong competition. However, post merger the HHI reaches 68%, approaching a HHI consistent with monopolies. This high HHI would lead to weak competition.<ref name=":1" /> 

This demonstrates how the HHI enables antitrust authorities to understand the impact that mergers have on the market.<ref name=":0" />

The index involves taking the market share of the respective market competitors, squaring it, and adding them together (e.g. in the market for X, company A has 30%, B, C, D, E and F have 10% each and G through to Z have 1% each). When calculating HHI the post merger level of the HHI score and the total increase of the HHI score are considered when reviewing the outcome. If the resulting figure is above a certain threshold then economists will consider the market to have a high concentration (e.g. market X's concentration is 0.142 or 14.2%). This threshold is considered to be 0.25 in the U.S.,<ref>{{harvnb|United States Department of Justice|2010|loc=§ 5.3}}</ref> while the EU prefers to focus on the level of change, for instance that concern is raised if there is a 0.025 change when the index already shows a concentration of 0.1.<ref name="dojguidelines"/> So to take the example, if in market X company B (with 10% market share) suddenly bought out the shares of company C (with 10% also) then this new [[market concentration]] would make the index jump to 0.162. Here it can be seen that it would not be relevant for merger law in the U.S. (being under 0.18) or in the EU (because there is not a change over 0.025).

==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.

==Problems==
The usefulness of this statistic to detect monopoly formation is directly dependent on a proper definition of a particular market (which hinges primarily on the notion of substitutability). The index fails to take into consideration the complex nature of the market being tested.<ref>{{Cite web|last=Bromberg|first=Michael|title=Herfindahl-Hirschman Index (HHI) Definition, Formula, and Example|url=https://www.investopedia.com/terms/h/hhi.asp|date=November 21, 2022|access-date=January 19, 2023|website=[[Investopedia]]}}</ref>

* For example, if the statistic were to look at a hypothetical financial services industry as a whole, and found that it contained 6 main firms with 15% market share apiece, then the industry would look non-monopolistic. However, suppose one of those firms handles 90% of the checking and savings accounts and physical branches (and overcharges for them because of its monopoly), and the others primarily do commercial banking and investments. In this scenario, the index hints at [[Dominance (economics)|dominance]] by one firm. The market is not properly defined because checking accounts are not substitutable with commercial and investment banking. The problems of defining a market work the other way as well. To take another example, one cinema may have 90% of the movie market, but if movie theaters compete against video stores, pubs and nightclubs then people are less likely to be suffering due to market dominance.
* Another typical problem in defining the market is choosing a geographic scope. For example, firms may have 20% market share each, but may occupy five areas of the country in which they are monopoly providers and thus do not compete against each other. A service provider or manufacturer in one city is not necessarily substitutable with a service provider or manufacturer in another city, depending on the importance of being local for the business—for example, telemarketing services are rather global in scope, while shoe repair services are local.

The United States federal anti-trust authorities such as the [[United States Department of Justice|Department of Justice]] and the [[Federal Trade Commission]] use the Herfindahl index as a screening tool to determine whether a proposed merger or acquisition is likely to raise antitrust concerns. Increases of over 0.01 (100) generally provoke scrutiny, although this varies from case to case. The [[United States Department of Justice Antitrust Division|Antitrust Division]] of the Department of Justice considers Herfindahl indices between 0.15 (1,500) and 0.25 (2,500) to be "moderately concentrated" and indices above 0.25 to be "highly concentrated".<ref name="dojhhi"/> However, these indices scores are not rigid guidelines that must be followed, while high levels of concentration is concerning, they indices scores provide ways to identify which mergers and acquisitions are potentially noncompetitive. There are other factors that need to be considered that will either help reinforce or counter the harmful effects of higher market concentration. The Herfindahl-Hirschman index is used as a starting point to gauge initial market power and then determine if additional information is needed to conduct further analysis on any potential anti-competitive concerns.<ref name="dojguidelines"/>

==Intuition==
When all the firms in an industry have equal market shares, <math display="inline">H = N\left( \dfrac{1}{N} \right)^2 = \dfrac{1}{N}</math>.  The Herfindahl is correlated with the number of firms in an industry because its lower bound when there are ''N'' firms is 1/''N''.  In the more general case of unequal market share, 1/''H'' is called "equivalent (or effective) number of firms in the industry", ''N''<sub>eqi</sub> or ''N''<sub>eff</sub>.<ref>{{cite journal|title=A Generalized Interpretation of the Herfindahl Index|last=Kelly, Jr.|first=William A.|journal=Southern Economic Journal|date=July 1981|volume=48|number=1|pages=50–57|publisher=[[Southern Economic Association]]|doi=10.2307/1058595 |jstor=1058595}}</ref><ref>{{cite journal|title=Comment on the "H" Concentration Measure as a Numbers-Equivalent|last=Adelman|first=M. A.|date=February 1969|volume=51|number=1|journal=[[The Review of Economics and Statistics]]|publisher=[[MIT Press|The MIT Press]]|pages=99–101|doi=10.2307/1926955 |jstor=1926955}}</ref><ref>{{cite journal|title=Elasticities, Cross-Elasticities, and Market Relationships|last=Bishop|first=Robert L.|journal=[[The American Economic Review]]|date=December 1952|volume=42|number=5|pages=780–803|publisher=[[American Economic Association]]|jstor=1812527}}</ref> An industry with 3 firms cannot have a lower Herfindahl than an industry with 20 firms when firms have equal market shares.  But as market shares of the 20-firm industry diverge from equality the Herfindahl can exceed that of the equal-market-share 3-firm industry (e.g., if one firm has 81% of the market and the remaining 19 have 1% each, then <math>H=0.658</math>).  A higher Herfindahl signifies a less competitive (i.e., more concentrated) industry.

=== Appearance in market structure ===
It can be shown that the Herfindahl index arises as a natural consequence of assuming that a given market's structure is described by [[Cournot competition]].<ref>{{Cite book|url=https://mitpress.mit.edu/9780262038065/economics-of-regulation-and-antitrust/|url-access=subscription|title=Economics of Regulation and Antitrust|last1=Viscusi|first1=W. Kip|author-link1=W. Kip Viscusi|last2=Harrington, Jr.|first2=Joseph Emmett|last3=Sappington|first3=David Edward Michael|publisher=[[MIT Press|The MIT Press]]|year=2018|isbn=9780262038065|edition=Fifth|location=[[Cambridge, Massachusetts]]|pages=177–178|lccn=2017056198}}</ref> Suppose that we have a Cournot model for competition between <math>n</math> firms with different linear marginal costs and a homogeneous product. Then the profit of the <math>i</math>-th firm <math>\pi_{i}</math> is:
<math display="block">\pi_{i} = P(Q)q_{i} - c_{i}q_{i}, \quad Q = \sum_{i=1}^{n}q_{i} </math>
where <math>q_{i}</math> is the quantity produced by each firm, <math>c_{i}</math> is the [[marginal cost]] of production for each firm, and <math>P(Q)</math> is the price of the product. Taking the derivative of the firm's profit function with respect to its output to maximize its profit gives us:
<math display="block">\frac{\partial\pi_i}{\partial q_i} = 0 \implies P'(Q)q_{i} + P(Q) - c_{i} = 0 \implies - \frac{dP}{dQ} q_{i} = P-c_{i} </math>
Dividing by <math>P</math> gives us each firm's [[profit margin]]:
<math display="block">{P-c_{i}\over{P}} = -{dP\over{dQ}}{q_{i}\over{P}} = -{dP/P\over{dQ/Q}} {q_{i}\over{Q}} = {s_{i}\over{\eta}} </math>
where <math>s_{i} = q_{i}/Q</math> is the market share and <math>\eta = -d\log Q/d\log P</math> is the [[price elasticity of demand]]. Multiplying each firm's profit margin by its market share gives us:
<math display="block">s_{1}\left( {P-c_{1}\over{P}} \right) + \cdots + s_{n}\left( {P-c_{n}\over{P}} \right) = {H\over{\eta}}</math>
where <math>H</math> is the Herfindahl index. Therefore, the Herfindahl index is directly related to the weighted average of the profit margins of firms under Cournot competition with linear marginal costs.

=== Effective assets in a portfolio ===
The Herfindahl index is also a widely used metric for [[Portfolio (finance)|portfolio]] concentration.<ref>{{cite book|last=Lovett|first=William Anthony|title=Banking and Financial Institutions Law in a Nutshell|year=1988|publisher=West Publishing Company|edition=2nd|isbn=9780314414434}}</ref> In portfolio theory, the Herfindahl index is related to the effective number of positions <math>N_{\text{eff}} = 1/H</math><ref>{{cite arXiv|last1=Bouchaud|first1=Jean-Philippe|last2=Potters|first2=Marc|last3=Aguilar|first3=Jean-Pierre|title=Missing Information and Asset Allocation|date=July 1997|eprint=cond-mat/9707042}}{{bibcode|1997cond.mat..7042B}}</ref> held in a portfolio, where <math display="inline">H = \sum \|w\|^{2}</math> is computed as the sum of the squares of the proportion of market value invested in each security. A low H-index implies a very diversified portfolio: as an example, a portfolio with <math>H = 0.02</math> is equivalent to a portfolio with <math>N_{\text{eff}} = 50</math> equally weighted positions. The H-index has been shown to be one of the most efficient measures of portfolio diversification.<ref>{{cite journal|last1=Woerheide|first1=Walt J.|last2=Persson|first2=Don|year=1993|title=An Index of Portfolio Diversification|url=https://pdfs.semanticscholar.org/0a5e/ec924dae3ea30b6cae8e66f7070344d47631.pdf|archive-url=https://web.archive.org/web/20180323220210/https://pdfs.semanticscholar.org/0a5e/ec924dae3ea30b6cae8e66f7070344d47631.pdf|url-access=subscription|url-status=dead|archive-date=2018-03-23| journal=Financial Services Review|volume=2|issue=2|pages=73–85|doi=10.1016/1057-0810(92)90003-U|s2cid=18548005|issn=1057-0810}}</ref>

It may also be used as a [[Constraint (mathematics)|constraint]] to force a portfolio to hold a minimum number of effective assets:
<math display="block">\|w\|^{2} \leq N_{\text{eff}}^{-1}</math>
For commonly used [[portfolio optimization]] techniques, such as [[Modern portfolio theory|mean-variance]] and [[Expected shortfall|CVaR]], the optimal solution may be found using [[second-order cone programming]].

==Decomposition==
Supposing that <math>N</math>  firms share all the market, each one with a participation of <math>x_i</math> and market share <math display="inline">s_i = x_i/\sum_{j=1}^N x_j</math>, then the index can be expressed as <math display="inline">H = \frac 1 N + (N-1)\sigma^2</math>, where <math>\sigma^2</math> is the [[variance|statistical variance]] of the firm shares, defined as <math display="inline">\sigma^2 = \frac1{N-1} \sum_{i=1}^N \left(s_i-\mu\right)^2</math> where <math display="inline">\mu = \frac 1 N</math> is the mean of participations. If all firms have equal (identical) shares (that is, if the market structure is completely ''[[symmetric]]'', in which case <math>s_i=1/N</math>) then <math>\sigma^2</math> is zero and <math>H</math> equals <math>1/N</math>. If the number of firms in the market is held constant, then a higher variance due to a higher level of asymmetry between firms' shares (that is, a higher ''share dispersion'') will result in a higher index value. See the Brown and Warren-Boulton (1988) and Warren-Boulton (1990) texts cited below.

==See also==
*[[Lerner index]]
*[[Concentration ratio]]
*[[Marketing strategy]]
*[[Microeconomics]]
*[[N50 statistic]] &ndash; a measure of concentration used in genomics
*[[Small but significant and non-transitory increase in price]] &ndash; a test to determine the relevant market

==References==
{{reflist|30em}}

==Further reading==
*{{cite report|last=Brown|first=Donald M.|author2=Warren-Boulton, Frederick R.|title=Testing the Structure-Competition Relationship on Cross-Sectional Firm Data|version=Discussion paper 88-6|publisher=Economic Analysis Group, U.S. Department of Justice, Antitrust Division|location=[[Washington, D.C.]]|date=May 11, 1988|oclc=221796344}}
*{{cite journal|last=Capozza|first=Dennis R.|author2=Lee, Sohan |year=1996 |title=Portfolio Characteristics and Net Asset Values in REITs|journal=The Canadian Journal of Economics|volume=29|issue=Special Issue: Part 2 |pages=S520&ndash;S526 |doi=10.2307/136100|jstor=136100}}
*{{cite journal|last=Hirschman|first=Albert O.|author-link=Albert O. Hirschman|year=1964|title=The Paternity of an Index|journal=[[The American Economic Review]]|volume=54|issue=5|pages=761–762|jstor= 1818582|publisher=[[American Economic Association]]}}
*{{cite journal|last=Kwoka|first=John E. Jr.|year=1977|title=Large Firm Dominance and Price-Cost Margins in Manufacturing Industries|journal=[[Southern Economic Journal]]|volume=44|issue=1|pages=183&ndash;189 |doi=10.2307/1057315|jstor=1057315}}
*{{cite book|title=The Law and Economics of Competition Policy|chapter=Implications of U.S. Experience with Horizontal Mergers and Takeovers for Canadian Competition Policy|last=Warren-Boulton|first=Frederick R.|editor-last1=Mathewson|editor-first1=Frank|editor-last2=Trebilcock|editor-first2=Michael|editor-last3=Walker|editor-first3=Michael|year=1990 |publisher=[[The Fraser Institute]]|location=[[Vancouver|Vancouver, B.C.]]|pages=337–368|isbn=978-0-88975-121-7|url-access=registration|url=https://archive.org/details/laweconomicsofco0000unse}}

==External links==
* Orris Herfindahl, "Concentration in the steel industry", 1950, published on Archive.org with consent of his heirs in June 2021, [https://archive.org/details/herfindahl-concentration-in-the-steel-industry-1950-publish/ Dissertation on Archive.org]
* [http://wits.worldbank.org World Integrated Trade Solution], ''Calculate Herfindahl-Hirschman Index using UNSD COMTRADE data''
* [https://www.justice.gov/atr/herfindahl-hirschman-index US Department of Justice market concentration cutoffs].
* [http://www.unclaw.com/chin/teaching/antitrust/herfindahl.htm Herfindahl-Hirschman Index Calculator]. Web tool for calculating pre- and post-merger Herfindahl index.
* [https://www.justice.gov/atr/public/guidelines/hmg-2010.html Department of Justice and Federal Trade Commission 2010 Horizontal Merger Guidelines]. More detailed information about mergers, market concentration, and competition (from the [[United States Department of Justice|Department of Justice]]).

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