The Linda Index

In my opinion, it would be more correct to name the article The Linda Indices, because the use of the econometric methodology developed in the late 1960s and 1970s by Remo Linda implies the calculation of a number of indices, but the term Linda index in the singular is more common. The Linda indices calculated by market share (for example, by sales volume) are more often found, but in the original 1976 paper¹, Linda analyzed the structure of markets by calculating an index system for leading companies based on seven parameters

• sales volume
• employment
• wages and salaries
• net profit
• gross income
• gross investment
• equity

You can read about the reasoning for creation of the index, the formulas for calculating it and the interpretation of the results in the article below the calculator.

The Linda Index

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The Linda Indices

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The Linda Indices

The Linda indices are a methodology for analyzing the concentration of industries and markets developed by Remo Linda, Head of the Market Structure Division of the Commission of the European Communities, and used by the Commission of the European Communities since 1972 to study changes in the concentration of industries and markets in the European Union.

Background

Since 1970, the European Commission, on the initiative of the Directorate-General for Competition, has been launching a research program to study changes in the concentration of production in different countries of the European Union. This was due to the growing importance of these processes in the emergence of common markets, and the lack of consistency in the data of statistical agencies of different countries, hindering the study of concentration and competition.

The program was to develop uniform systematic requirements to the collected statistical information, which could be used in the research of market structures and regulation of competition in different countries.

Statistical data were studied for all member countries and a large number of industries since 1962. At the same time, the studies did not include obviously concentrated or atomized industries, as well as industries for which it was difficult to obtain data for a number of reasons (for example, due to the high cost of data collection).

The second phase of the research involved the calculation of econometric indicators to obtain data on the evolution of concentration of the industries and markets in question, and numerical correlations between concentration, its change and the productivity of companies.

Basic Metrics

The following basic industry metrics were considered:

• number of enterprises n
• average size of enterprises by the number of employees in the industry as the total number of employees divided by the number of enterprises
• concentration ratios (CR) CR4, CR8 or CR10 - as a share of the first 4, 8 or 10 largest enterprises in the industry, expressed as a percentage of the total (sales or number of employees).

These metrics have a number of disadvantages, the main one being that they say nothing about the degree of unevenness of the distribution, although it is one of the most important indicators of concentration. For example, using them, it is difficult to compare the state of the industry in different time periods.

Concentration indices.

The following, more complex indicators, which can be described as "concentration indices" in theory should eliminate the disadvantages described above. These include:

• coefficient of variation
• Gini index
• Herfindahl-Hirschman Index
• entropy index

Despite their usefulness, these indices also have a significant disadvantage: they assume that the data for all industry is known, that is, for example, sales volume for all industry enterprises is known and total sales can be calculated.

This is quite an essential requirement, especially when it comes to analyzing the industry by more than one indicator. Attempting to collect such data on small and medium-sized businesses is almost impossible. As an example, Linda cites 1971 data for the Italian food industry: the total number of businesses is about 40,000, but only about 2,000 of them have more than 20 employees and collectively generate 55% of sales. The data for the remaining enterprises suffer from inaccuracy. Thus, the results of the calculation of the concentration indices above will vary greatly depending on how many enterprises are included in the analysis.

The Linda Index

To overcome these disadvantages, the indices should be defined in such a way that the inclusion or non-inclusion of additional small companies in the calculations does not have a decisive effect on the indices characterizing concentration in the industry. This approach involves analyzing only some subset of large companies, focusing on oligopolistic interdependence, which is what Linda proposed the new index system for.

This system focuses on the oligopolistic aspect of concentration, involves working with a system (rather than one) of indices, and applies to a sample of the n largest companies. Generally, it is recommended to include in the sample all large companies occupying at least ⅔ of the market by sales or employment, and exclude companies occupying less than 1% of the total value of the sample, as they can hardly be oligopolists. In most cases, the minimum number of companies in the sample is 6-8, and the maximum is 60-70.

Before making the calculations, the companies in question must be sorted in descending order of the analyzed indicator.

The Linda index for n companies in the sample:
,
where
,
A_i - - total share of the first i companies from the total of the indicator for n companies
A_n - 100% or 1
in other words, A_i in the last formula is expressed as fractions of one from the value of A_n

EO_i - oligopolistic equilibrium indicator, the ratio of the average value of the analyzed indicator for the first i companies to the average value of the indicator for the remaining n-i companies.

L_n - the arithmetic mean of n-1 values of the oligopolistic equilibrium divided by n

The Linda indices are calculated sequentially starting from L₂, and both the individual index values and the shape of the resulting curve are of interest here. The special point here is the first minimum, that is, the first index value that is smaller than the two neighboring ones, the previous and the next, denoted by the index n_m.

Linda also identifies the L_s index, the arithmetic average of the indices from the second to n_m inclusive, as the degree of equilibrium and concentration of the first n_m companies in the industry.

Interpretation

n_m also expresses the quantitative boundary between the big companies, which Linda characterized by the term "oligopolistic arena" and everyone else, i.e. n_m is the number of oligopolists in an industry. The minimum is formed when the value of the share of the next company decreases significantly compared to the previous ones.

Also a special point is the first maximum. In case it is the L₂ index, and the value of this index is greater than 1, the first company has a significant lead, because it shows that its share is at least twice as large as the share of the next-largest company.

According to the data obtained in the above-mentioned study, the maximum is most often observed for an index value of 2. Less often for 3 (5%), and even more rarely for 4 (less than 1%).

In general, if the share of the first-leading company exceeds the share of the next company by m times, the value of the L₂ index will be m/2.

In the case of a mono-duopolistic market structure, the maximum will be on the third index, after which the curve will decrease without forming a minimum.

As a result of the research it was empirically established that L_s value of about 0.2 corresponds to a fairly spacious and balanced "oligopolistic arena" where competition works satisfactorily, values over 0.5 indicate an increased density of oligopolistic companies, and values above 1.0 indicate the existence of dominance in the market.

In the case of a perfect market equilibrium, when all companies in the market have an equal share, the index value L_n will be exactly 1/n, and the curve will contain no inflection points.

The ideal equilibrium curve provides an important reference point for interpreting the empirical index curve.

In conclusion, I would like to note that, apparently, the methodology described above is rarely used outside the European Union - at least in the English-language Internet there are few scientific papers that use the Linda indices, and those that exist are mainly from the authors from Eastern Europe and the former Soviet Union. Perhaps one reason for this is the relative complexity of the calculations and interpretation of the results.