# Herfindahl index

The Herfindahl index, also known as Herfindahl-Hirschman Index or HHI, 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 but widely applied in competition law and antitrust. It is defined as the sum of the squares of the market shares of the 50 largest firms (or summed over all the firms if there are fewer than 50)[1] within the industry, where the market shares are expressed as percentages. The result is proportional to the average market share, weighted by market share. As such, it can range from 0 to 10,000, 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.

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.

## Example

For instance, two cases in which the six largest firms produce 90 % of the output:

• Case 1: All six firms produce 15% each, and
• Case 2: One firm produces 80 % while the five others produce 2 % each.

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 in the first case competition would be fierce 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 * 152 + 10 * 12 = 1360
• Case 2: Herfindahl index = 802 + 5 * 22 + 10 * 12 = 6430

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 "1420" if you multiply percentages in whole figures). This threshold is considered to be "0.18" in the US,[2] while the EU prefers to focus on the level of change, for instance that concern is raised if there's a "0.025" change when the index already shows a concentration of "0.1".[3] 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 "0.172". Here it can be seen that it would not be relevant for merger law in the U.S. (being under 0.18) but would in the EU (because there's a change of over 0.025). Put simply, now two firms control half the market, so serious competition questions are raised.

## Formula

$H =\sum_{i=1}^N s_i^2$

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 (H) ranges from 1 / N to one, where N is the number of firms in the market. Equivalently, the index can range up to 10,000, if percents are used as whole numbers, as in 75 instead of 0.75. The maximum in this case is 1002 = 10,000.

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 normalised 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:

$H* = {\left ( H - 1/N \right ) \over 1-1/N }$

where again, N is the number of firms in the market, and H is the usual Herfindahl Index, as above.

A H* index below 0.1 (or 1,000) indicates an unconcentrated index.
A H* index between 0.1 to 0.18 (or 1,000 to 1,800) indicates moderate concentration.
A H* index above 0.18 (above 1,800) indicates high concentration[1].

## Problems

The usefulness of this statistic to detect and stop harmful monopolies 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, 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 theatres 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 uses the Herfindahl index to determine whether mergers are equitable to society; increases of over 0.0100 points generally provoke scrutiny, although this varies from case to case. The Antitrust Division of the Department of Justice considers Herfindahl indices between 0.1000 and 0.1800 to be moderately concentrated and indices above 0.1800 to be concentrated. As the market concentration increases, competition and efficiency decrease and the chances of collusion and monopoly increase.

## Intuition

When all the firms in an industry have equal market shares, H = 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. An industry with 3 firms cannot have a lower Herfindahl than an industry with 10 firms when firms have equal market shares. But as market shares of the 10-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=0.83). A higher Herfindahl signifies a less competitive industry.

## Decomposition

The index can be expressed as $H =\frac1n+n V$ where n is the number of firms and V is the statistical variance of the firm shares, defined as $V=\frac{\sum_{i=1}^n\left(s_i-1/n\right)^2}n$. If all firms have equal (identical) shares (that is, if the market structure is completely symmetric, in which case si = 1/n for all i) then V is zero and H equals 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 Brown and Warren-Boulton (1988), also see Warren-Boulton (1990).

## References

1. ^ Chapter 9 Organizing Production
2. ^ Merger Guidelines § 1.51
3. ^ However, it gets far more complicated than that. See para. 16-21 Guidelines on horizontal mergers