Herfindahl index

The Herfindahl index , also Hirschman Index , or Herfindahl-Hirschman Index called, is a frequently used measure for concentration measurement , for example in economics for measuring a concentration of companies . It is named after Orris Clemens Herfindahl (1918–1972) and Albert O. Hirschman (1915–2012) and is often abbreviated as HHI or . ${\ displaystyle H}$

When calculating the Herfindahl index, it is assumed that objects are distributed over several groups: For example, the entire sales of a product in a certain market is divided between a certain number of producers who manufacture the product. However, sales are usually not evenly distributed among all producers. The Herfindahl index now provides information on the extent to which sales are concentrated on one or a few providers . ${\ displaystyle N}$ ${\ displaystyle H}$

calculation

HHI

The HHI is calculated as follows:

{\ displaystyle H: = \ sum _ {i = 1} ^ {N} {a_ {i} ^ {2}}}

With

{\ displaystyle a_ {i}: = {\ frac {x_ {i}} {\ sum _ {j = 1} ^ {N} {x_ {j}}}}}

The -th denotes the (positive) characteristic value. ${\ displaystyle x_ {i}}$ ${\ displaystyle i}$

In words, the HHI is the sum of all the squared market shares of the competitors in a market. Mathematically, this value lies between , where 1 corresponds to a monopoly. In economics, it is common to lose the percentage sign (or to multiply the percentage market shares by 100 beforehand), the HHI obtained in this way can then assume values between . The values can be converted into each other by multiplication or division by 10,000. ${\ displaystyle {\ tfrac {1} {N}} \ leq H \ leq 1}$ ${\ displaystyle {\ tfrac {10000} {N}} \ leq H \ leq 10 \, 000}$

The European Commission gives the following simplified example for a better understanding: The HHI measures the market concentration by adding the squared market shares of all providers in an economic sector. For example, if five companies hold a 20% stake in a market, the HHI is 400 + 400 + 400 + 400 + 400 = 2000. The higher the HHI for a given market, the greater the proportion of production that applies to a small number of companies. In general, if the HHI is below 1000, the market concentration can be described as low, from 1000 to 1800 as moderate and above 1800 as high.

Normalized HHI

The normalized HHI is calculated as follows:

{\ displaystyle H: = {\ frac {\ sum _ {i = 1} ^ {N} {a_ {i} ^ {2}}} {(\ sum _ {i = 1} ^ {N} {a_ { i}}) ^ {2}}} = \ sum _ {i = 1} ^ {N} {\ left ({\ frac {a_ {i}} {N \ cdot {\ overline {a}}}} \ right) ^ {2}}}

or

{\ displaystyle H: = {\ frac {\ sum _ {i = 1} ^ {N} {a_ {i} ^ {2}}} {(\ sum _ {i = 1} ^ {N} {a_ { i}}) ^ {2}}} = {\ frac {\ sum {x_ {i} ^ {2}}} {(\ sum {x_ {i}}) ^ {2}}} = \ sum \ left ({\ frac {x_ {i}} {N {\ overline {x}}}} \ right) ^ {2}}

Here means the sales attributable to producers . ${\ displaystyle a_ {i} (i \ in [1; N])}$ ${\ displaystyle i}$

The Herfindahl index is therefore the normalized sum of the squared proportional values and can assume values from to , whereby the minimum value with even distribution of sales across all providers (= minimum concentration), the maximum value, however, with maximum concentration (i.e. when the entire sales are a single provider is not required). ${\ displaystyle {\ tfrac {1} {N}}}$ ${\ displaystyle 1}$

The Herfindahl index is not invariant to the number of providers. If invariance with respect to the number of providers is required, other unequal distribution measures are an alternative . ${\ displaystyle N}$

application areas

The Herfindahl index is mainly used in the following areas:

In economics as well as in the area of antitrust law to prove the dominant position of one or more providers: High values of suggest that one or a few manufacturers control large parts of the market while standing for a monopoly . ${\ displaystyle H}$ ${\ displaystyle H = 1}$

In market research as a measure of the brand loyalty of a customer or a group of customers: the higher the value, the more the customer focuses on one or a few brands when making purchases . Always buying the same brand is the same . ${\ displaystyle H}$ ${\ displaystyle H}$ ${\ displaystyle 1}$

In operating system theory when it comes to the fair distribution of resources .

In political science , when it comes to party fragmentation, with a lower Herfindahl index representing a more fragmented party system.

In transplant medicine to assess the success of therapy depending on the frequency of operations in different hospitals

In religious studies , to measure the degree of religious plurality in a society.

literature

Josef Bleymüller, Günther Gehlert, Herbert Gülicher: Statistics for economists. WiSt study course. Vahlen, Munich ¹⁰ 1996. ISBN 3-8006-2081-2 . Chapter 26.
Albert Otto Hirschman: The Paternity of an Index. In: The American Economic Review. Vol. 54, No. 5 (Sept. 1964), pp. 761 f
Ronald Rousseau: The repeat rate: from Hirschman to Stirling. In: Scientometrics (April 2018): 1-9. doi : 10.1007 / s11192-018-2724-8

Individual evidence

^ Pearson: CFA Program Curriculum. Vol. 2 - Economics.

^ Fahrmeir, Künstler, Pigeot, Tutz: Statistics , 3rd edition, Springer-Verlag (2001), ISBN 3-540-67826-3 , Chapter 2.3.2: Alternative concentration measures

↑ Quotation from: EK, Glossary of EU Competition Policy (2004), 25.

^ Wagschal, Uwe 1999. Statistics for political scientists. Munich: Oldenbourg.

↑ Vidano P. Nguyen et al .: "Effect of regional competition on heart transplant waiting list outcomes", in: "The Journal of Heart and Lung Transplantation", August 2016, volume 35, issue 8, pages 986-994.

↑ Jörg Stolz: How does plurality affect individual religiosity? A confrontation between the sociology of knowledge and rational choice . In: Martin Baumann; Samuel M. Behloul, (Ed.): Religious Pluralism. Empirical studies and analytical perspectives. transcript, Bielefeld 2005, p. 211 .

[1] Pearson: CFA Program Curriculum. Vol. 2 - Economics.