Herfindahl–Hirschman Index


The Herfindahl index is a measure of the size of firms in relation to the industry and an indicator of the amount of competition among them. Named after economists Orris C. Herfindahl and Albert O. Hirschman, it is an economic concept widely applied in competition law, antitrust and also technology management. It is defined as the sum of the squares of the market shares of the firms within the industry, where the market shares are expressed as fractions. 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 monopolistic producer. Increases in the Herfindahl index generally indicate a decrease in competition and an increase of market power, whereas decreases indicate the opposite. Alternatively, if whole percentages are used, the index ranges from 0 to 10,000 "points". For example, an index of.25 is the same as 2,500 points.
The major benefit of the Herfindahl index in relationship to such measures as the concentration ratio is that it gives more weight to larger firms.
The measure is essentially equivalent to the Simpson diversity index, which is a diversity index used in ecology; the inverse participation ratio in physics; and the effective number of parties index in politics.

Example

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.
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:
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. If the resulting figure is above a certain threshold then economists consider the market to have a high concentration. This threshold is considered to be 0.25 in the U.S., 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. So to take the example, if in market X company B suddenly bought out the shares of company C 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. or in the EU.

Formula

where si 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 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 indicates a highly competitive industry.
An H below 0.15 indicates an unconcentrated industry.
An H between 0.15 to 0.25 indicates moderate concentration.
An H above 0.25 indicates high concentration.
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:
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 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.

Problems

The usefulness of this statistic to detect monopoly formation, however, is directly dependent on a proper definition of a particular market.
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".

Intuition

When all the firms in an industry have equal market shares, H = N2 = 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 number of firms in the industry", Neqi or Neff. 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. 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. Suppose that we have a Cournot model for competition between firms with different linear marginal costs and a homogeneous product. Then the profit of the th firm is:where is the quantity produced by each firm, is the marginal cost of production for each firm, and 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:Dividing by gives us each firm's profit margin:where is the market share and is the elasticity of demand. Multiplying each firm's profit margin by its market share gives us:where 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. In portfolio theory, the Herfindahl index is related to the effective number of positions held in a portfolio, where 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 is equivalent to a portfolio with equally weighted positions. The H-index has been shown to be one of the most efficient measures of portfolio diversification.
It may also be used as a constraint to force a portfolio to hold a minimum number of effective assets:For commonly used portfolio optimization techniques, such as mean-variance and CVaR, the optimal solution may be found using second-order cone programming.

Decomposition

Supposing that firms share all the market, each one with a participation of and market share, then the index can be expressed as, where is the statistical variance of the firm shares, defined as where is the mean of participations. If all firms have equal shares then is zero and equals. 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 will result in a higher index value. See the Brown and Warren-Boulton and Warren-Boulton texts cited below.