Herfindahl–Hirschman Index

For instance, we consider two cases in which the six largest firms produce 90% of the goods in a market. In either case, we will assume that the remaining 10% of output is divided among 10 equally sized producers.

• Case 1: All six of the largest firms produce 15% each.
• Case 2: The largest firm produces 80% and the next five largest firms produce 2% each.

The six-firm concentration ratio would equal 90% for both case 1 and case 2. But the first case would promote significant competition, where the second case approaches monopoly. The Herfindahl index for these two situations makes the lack of competition in the second case strikingly clear:

• Case 1: Herfindahl index = 6 * 0.15² + 10 * 0.01² = 0.136 (13.6%)
• Case 2: Herfindahl index = 0.80² + 5 * 0.02² + 10 * 0.01² = 0.643 (64.3%)

This behavior rests in the fact that the market shares are squared prior to being summed, giving additional weight to firms with larger size.

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). If the resulting figure is above a certain threshold then economists 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.,[1] 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.[4] 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 0.25).

${\displaystyle H=\sum _{i=1}^{N}s_{i}^{2}}$

where s_i is the market share of firm i in the market, and N is the number of firms. Thus, in a market with two firms that each have 50 percent market share, the Herfindahl index equals 0.50²+0.50² = 1/2.

The Herfindahl Index (H) ranges from 1/N to one, 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 100², or 10,000.

An H below 0.01 (or 100) indicates a highly competitive industry.
An H below 0.15 (or 1,500) indicates an unconcentrated industry.
An H between 0.15 to 0.25 (or 1,500 to 2,500) indicates moderate concentration.
An H above 0.25 (above 2,500) indicates high concentration.[5]

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:

${\displaystyle H^{*}={\left(H-1/N\right) \over 1-1/N}}$ for N > 1 and

${\displaystyle H^{*}=1}$ for N = 1

where again, N is the number of firms in the market, and H is the usual Herfindahl Index, as above. Using the normed 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; H = 1/N = 1/2 = 0.5 and H* = 0. Now compare that to a situation with three players and again an equally distributed market share; H = 1/N = 1/3 = 0.333..., note that H* = 0 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.

The usefulness of this statistic to detect monopoly formation, however, is directly dependent on a proper definition of a particular market (which hinges primarily on the notion of substitutability).

• 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, people would be suffering due to a market 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 Department of Justice and the Federal Trade Commission use the Herfindahl index as a screening tool to determine whether a proposed merger is likely to raise antitrust concerns. Increases of over 0.01 generally provoke scrutiny, although this varies from case to case. The Antitrust Division of the Department of Justice considers Herfindahl indices between 0.15 and 0.25 to be "moderately concentrated" and indices above 0.25 to be "highly concentrated".[6]

When all the firms in an industry have equal market shares, H = N(1/N)² = 1/N. 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_eqi or N_eff.[7] 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 H=0.658). A higher Herfindahl signifies a less competitive 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.[8] Suppose that we have a Cournot model for competition between ${\displaystyle n}$ firms with different linear marginal costs and a homogeneous product. Then the profit of the ${\displaystyle i}$th firm ${\displaystyle \pi _{i}}$ is:

$${\displaystyle \pi _{i}=P(Q)q_{i}-c_{i}q_{i},\quad Q=\sum _{i=1}^{n}q_{i}}$$

where ${\displaystyle q_{i}}$ is the quantity produced by each firm, ${\displaystyle c_{i}}$ is the marginal cost of production for each firm, and ${\displaystyle P(Q)}$ is the price of the product. Taking the derivative of the firm's profit function with respect to its output in order to maximize its profit gives us:

$${\displaystyle {\partial \pi _{i} \over {\partial q_{i}}}=P'(Q)q_{i}+P(Q)-c_{i}\implies -{dP \over {dQ}}q_{i}=P-c_{i}}$$

Dividing by ${\displaystyle P}$ gives us each firm's profit margin:

$${\displaystyle {P-c_{i} \over {P}}=-{dP \over {dQ}}{q_{i} \over {P}}=-{dP/P \over {dQ/Q}}{q_{i} \over {Q}}={s_{i} \over {\eta }}}$$

where ${\displaystyle s_{i}=q_{i}/Q}$ is the market share and ${\displaystyle \eta =-d\log Q/d\log P}$ is the elasticity of demand. Multiplying each firm's profit margin by its market share gives us:

$${\displaystyle s_{1}\left({P-c_{1} \over {P}}\right)+\cdots +s_{n}\left({P-c_{n} \over {P}}\right)={H \over {\eta }}}$$

where ${\displaystyle H}$ 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 concentration.[9] In portfolio theory, the Herfindahl index is related to the effective number of positions ${\displaystyle N_{\text{eff}}=1/H}$[10] held in a portfolio, where ${\displaystyle H=\|w\|^{2}}$ 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 ${\displaystyle H=0.02}$ is equivalent to a portfolio with ${\displaystyle N_{\text{eff}}=50}$ equally weighted positions. The H-index has been shown to be one of the most efficient measures of portfolio diversification.[11]

It may also be used as a constraint to force a portfolio to hold a minimum number of effective assets:

$${\displaystyle \|w\|^{2}\leq N_{\text{eff}}^{-1}}$$

For commonly used portfolio optimization techniques, such as mean-variance and CVaR, the optimal solution may be found using second-order cone programming.

Supposing that ${\displaystyle N}$ firms share all the market, each one with a participation of ${\displaystyle x_{i}}$ and market share ${\displaystyle s_{i}=x_{i}/\sum _{j=1}^{N}x_{j}}$, then the index can be expressed as ${\displaystyle H={\frac {1}{N}}+N\sigma ^{2}}$, where ${\displaystyle \sigma ^{2}}$ is the statistical variance of the firm shares, defined as ${\displaystyle \sigma ^{2}={\frac {1}{N}}\sum _{i=1}^{N}\left(s_{i}-\mu \right)^{2}}$ where ${\displaystyle \mu ={\frac {1}{N}}}$ is the mean of participations. If all firms have equal (identical) shares (that is, if the market structure is completely symmetric, in which case ${\displaystyle s_{i}=1/N}$) then ${\displaystyle \sigma ^{2}}$ is zero and ${\displaystyle H}$ equals ${\displaystyle 1/N}$. 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.

• Marketing strategy
• Microeconomics
• N50 statistic - a measure of concentration used in genomics
• Small but significant and non-transitory increase in price - a test to determine the relevant market

• Brown, Donald M.; Warren-Boulton, Frederick R. (May 11, 1988). "Testing the Structure-Competition Relationship on Cross-Sectional Firm Data". Discussion paper 88-6. Economic Analysis Group, U.S. Department of Justice. Cite journal requires |journal= (help)
• Capozza, Dennis R.; Lee, Sohan (1996). "Portfolio Characteristics and Net Asset Values in REITs". The Canadian Journal of Economics. 29 (Special Issue: Part 2): S520–S526. doi:10.2307/136100. JSTOR 136100.
• Hirschman, Albert O. (1964). "The Paternity of an Index". The American Economic Review. American Economic Association. 54 (5): 761–762. JSTOR 1818582.
• Kwoka, John E., Jr. (1977). "Large Firm Dominance and Price-Cost Margins in Manufacturing Industries". Southern Economic Journal. 44 (1): 183–189. doi:10.2307/1057315. JSTOR 1057315.
• Warren-Boulton, Frederick R. (1990). "Implications of U.S. Experience with Horizontal Mergers and Takeovers for Canadian Competition Policy". In Mathewson, G. Franklin; et al. (eds.). The Law and Economics of Competition Policy. Vancouver, B.C.: The Fraser Institute. ISBN 978-0-88975-121-7.

• World Integrated Trade Solution, Calculate Herfindahl-Hirschman Index using UNSD COMTRADE data
• US Department of Justice market concentration cutoffs.
• Herfindahl-Hirschman Index Calculator. Web tool for calculating pre- and post-merger Herfindahl index.
• Department of Justice and Federal Trade Commission 2010 Horizontal Merger Guidelines. More detailed information about mergers, market concentration, and competition (from the Department of Justice).