# Market concentration

In economics, market concentration is a function of the number of firms and their respective shares of the total production (alternatively, total capacity or total reserves) in a market. Alternative terms are Industry concentration and Seller concentration.[1]

Market concentration is related to industrial concentration, which concerns the distribution of production within an industry, as opposed to a market. In industrial organization, market concentration may be used as a measure of competition, theorized to be positively related to the rate of profit in the industry, for example in the work of Joe S. Bain.

## Metrics

Commonly used market concentration measures are the Herfindahl index (HHI or simply H) and the concentration ratio (CR)[citation needed]. The Hannah-Kay (1971) index has the general form

$\text{HK}_\alpha(x) = \begin{cases} & \left(\sum s_i^\alpha\right)^\frac {1}{\alpha-1} \text{ if }\alpha > 0, \alpha \ne 1 \\ & \prod s_i^{s_i} \text{ if }\alpha = 1\\ \end{cases}$.

Note, $\prod s_i^{s_i} = \exp\left(\sum s_i \log(s_i)\right)$, which is the exponential index.

## Uses

When antitrust agencies are evaluating a potential violation of competition laws, they will typically make a determination of the relevant market and attempt to measure market concentration within the relevant market.

## Motivation

As an economic tool market concentration is useful because it reflects the degree of competition in the market. Tirole (1988, p. 247) notes that:

Bain's (1956) original concern with market concentration was based on an intuitive relationship between high concentration and collusion.

There are game theoretic models of market interaction (e.g. among oligopolists) that predict that an increase in market concentration will result in higher prices and lower consumer welfare even when collusion in the sense of cartelization (i.e. explicit collusion) is absent. Examples are Cournot oligopoly, and Bertrand oligopoly for differentiated products.

## Empirical tests

Empirical studies that are designed to test the relationship between market concentration and prices are collectively known as price-concentration studies; see Weiss (1989).

Typically, any study that claims to test the relationship between price and the level of market concentration is also (jointly, that is, simultaneously) testing whether the market definition (according to which market concentration is being calculated) is relevant; that is, whether the boundaries of each market is not being determined either too narrowly or too broadly so as to make the defined "market" meaningless from the point of the competitive interactions of the firms that it includes (or is made of).

## Alternative definition

In economics, market concentration is a criterion that can be used to rank order various distributions of firms' shares of the total production (alternatively, total capacity or total reserves) in a market.

## Further Examples

Section 1 of the Department of Justice and the Federal Trade Commission's Horizontal Merger Guidelines is entitled "Market Definition, Measurement and Concentration." Herfindahl index is the measure of concentration that these Guidelines state that will be used.

A simple measure of market concentration is 1/N where N is the number of firms in the market. This measure of concentration ignores the dispersion among the firms' shares. It is decreasing in the number of firms and nonincreasing in the degree of symmetry between them. This measure is practically useful only if a sample of firms' market shares is believed to be random, rather than determined by the firms' inherent characteristics.

Any criterion that can be used to compare or rank distributions (e.g. probability distribution, frequency distribution or size distribution) can be used as a market concentration criterion. Examples are stochastic dominance and Gini coefficient.

Curry and George (1981) enlist the following "alternative" measures of concentration:

(a) The mean of the first moment distribution (Niehans, 1958); Hannah and Kay (1977) call this an "absolute concentration" index:

$\bar{X}_1 = \sum x_i^2/x_i = \sum x_i H$

(b) The Rosenbluth (1961) index (also Hall and Tideman, 1967):

$R = \frac{1}{2\sum is_i - 1}$ where symbol i indicates the firm's rank position.

(c) Comprehensive concentration index (Horvath 1970):

$CCI = s_1 + \sum_{i=2}^N s_i^2(2 - s_i)$ where s1 is the share of the largest firm. The index is similar to $2\text{H} - \sum s_i^3$ except that greater weight is assigned to the share of the largest firm.

(d) The Pareto slope (Ijiri and Simon, 1971). If the Pareto distribution is plotted on double logarithmic scales, [then] the distribution function is linear, and its slope can be calculated if it is fitted to an observed size-distribution.

(e) The Linda index (1976)

$L=\frac 1 {N(N-1)}\sum_{i=1}^{N-1}Q_i$ where Qi is the ratio between the average share of the first $i$ firms and the average share of the remaining $N - i$ firms. This index is designed to measure the degree of inequality between values of the size variable accounted for by various sub-samples of firms. It is also intended to define the boundary between the oligopolists within an industry and other firms. It has been used by the European Union.

(f) The U Index (Davies, 1980):

$U = I^{*a}N^{-1}$ where $I^{*}$ is an accepted measure of inequality (in practice the coefficient of variation is suggested), $a$ is a constant or a parameter (to be estimated empirically) and N is the number of firms. Davies (1979) suggests that a concentration index should in general depend on both N and the inequality of firms' shares.

The "number of effective competitors" is the inverse of the Herfindahl index.

Terrence Kavyu Muthoka defines distribution just as functionals in the Swartz space which is the space of functions with compact support and with all derivatives existing.The Media:Dirac Distribution or the Dirac function is a good example .