Editing Herfindahl–Hirschman index (section)

=== 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.