Editing Herfindahl–Hirschman index (section)

==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]].