Measurement of market (industry) concentration based on value validity

Abstract

As measures of concentration, especially for market (industry) concentration based on market shares, a variety of different measures or indices have been proposed. However, the various indices, including the two most widely used ones, the concentration ratio and the Herfindahl-Hirschman index (HHI), lack an important property: the value-validity property. An alternative index with this and other desirable properties is introduced. The new index makes it permissible to properly assess the extent of the concentration and make order and difference comparisons between index values as being true representations of the real concentration characteristic (attribute). Computer simulation data and real market-share data are used in the analysis. It is shown that the new index has a close functional relationship with the HHI index and has a firm theoretical relationship with market power as measured by the price-cost margin. Corresponding modifications to existing merger guidelines are presented.

1. Introduction

When dealing with nominal data, as with all types of data, it is often of interest to use some summary measures to represent certain characteristics or attributes associated with the data. One such characteristic is the concentration reflected by the probabilities or proportions p₁,…,p_n of n mutually exclusive and exhaustive categories or events. Concentration is then considered high when one or a few of the p_i‘s are relatively large and decreases as the p_i‘s become increasingly equal. The converse characteristic, qualitative variation (evenness), is measured by indices whose values increase as the p_i‘s become increasingly even, equal, or uniform (e.g., [1, 2]).

One area in which the measurement of concentration has a long history is that of measuring market concentration or industrial concentration where the p_i (i = 1,…,n) are the market shares of the n firms within a market or industry (e.g., [3, Ch. 4], [4, 5, pp. 221–238], [6, Ch. 8], [7–9]). Increasing market concentration tends to decrease competition and efficiency and increase market power. Any such trends are of concern to the business community and are being monitored by the U.S. Department of Justice (DOJ) and the Federal Trade Commission (FTC) in the case of antitrust. While such potentially negative consequences from increasing market concentration is the generally accepted proposition, it does not necessarily apply universally. In fact, a number of empirically-based exceptions have been documented and reported [7, 10–12]. However, no one disputes the fact that concentration is an important market indicator.

A number of different indices or measures of market (industry) concentration have been proposed over the years as outlined in this paper. However, two such indices have become by far the most popular ones: the m-firm concentration ratio and the Herfindahl-Hirschman index HHI (e.g., [13, pp. 116–118], [14, pp. 97–101], [6, Ch. 8], [15]). The m-firm concentration ratio (CR_m) is simply defined as the sum of the market shares of the m largest firms in a market, with the 4-firm CR₄ being the most commonly used member of the CR_m -family [16, p. 255]. The HHI, after Herfindahl [17] and Hirschman [18], includes all market shares and is defined as the sum of their squares. While CR₄ was used in the earliest 1968 Merger Guidelines by the DOJ and FTC [19], the later guidelines, with the most recent being the 2010 Horizontal Merger Guidelines [19], have been using the HHI as a screening tool for potential antitrust concerns from mergers. The preference for HHI over CR₄, both for practical screening and public policy as well as for research and analysis, is based on the much more comprehensive form of HHI (e.g., [7]).

This paper points out a serious limitation of the various concentration indices, including CR_m and HHI, that is specifically related to the numerical values taken on by the indices for different market-share distributions. While those indices have some highly desirable mathematical properties, no concern seems to have been raised so far about an important question concerning the actual values of an index: does a concentration index C take on numerical values throughout its range that can be rigorously explained or justified as providing realistic, true, or valid representations of the concentration characteristic or attribute? That is, does C have value validity?

A reason for this concern is the fact that different indices with similar mathematical properties can produce wildly different values for the same data sets. Comparisons between index values can therefore result in potentially misleading and unreliable interpretations, findings, and conclusions. What is needed of a concentration index C is an additional property requirement and its basic condition to provide C with value validity. This is the focus of this paper.

This paper is organized as follows. First, the set of properties generally required of a concentration index C are defined and discussed, followed by a definition and discussion of the value-validity property of C. Second, a concise historical account is given of the various indices that have been proposed over the years and their lacking properties are identified, with particular emphasis on the two indices CR₄ and HHI. Third, since existing indices lack the value-validity property, a new index C_K with this and other necessary properties is introduced and discussed as an extension of the author’s earlier and preliminary effort to measure the homogeneity of categorical data [20]. Fourth, C_K is compared with other indices, notably CR₄ and more so HHI, using both computer simulation data and real market-share data. Fifth, an approximate functional relationship between C_K and HHI is established with impressive accuracy and serving various pruposes: (i) the behavior and changes in HHI can be compared with those of C_K, (ii) reported values of HHI can be converted to those of C_K, (iii) the theoretical relationship between industry profitability and HHI can be extended to that of C_K, and (iv) the DOJ and FTC merger guidelines based on HHI can be equivalently expressed in terms of C_K.

Since the new index C_K has the value-validity property as proved in this paper, C_K is equally sensitive to changes in a market-share distribution throughout its potential range of values, permitting order and difference comparisons between index values. By contrast, other indices such as the most widely used HHI lack this property so that different types of comparisons between index values reflect those of an index itself rather than of the concentration attribute being measured. In the case of HHI, the effect of lacking the value-validity property implies that HHI lacks adequate sensitivity to changes in the market-share distribution for smaller values of HHI whereas it has excessive sensitivity for larger values of HHI. This important difference between C_K other indices, notably HHI, is indeed relevant when considering the anticompetitive effects of mergers and the ongoing debate about whether or not and to what extent concentration and market power have increased over the recent decades (e.g., [21–24]).

2. Required properties

2.1 General requirements

In order for C to qualify as a concentration index, it has to have a number of specific properties as discussed extensively in the published literature (e.g., [3, pp.47-58] [4, 25, 26]). First, however, a comment on notation: while the strictly mathematically correct notation would be to refer to C(P_n) as being the value of the index or function C for the distribution P_n = (p₁,…,p_n), C and C(P_n) will both be used throughout this paper to denote both an index (function) and its value in order to simplify the notation when there is no chance of ambiguity. Required properties of C may then be outlined as follows:

• (P1) Continuity: C is a continuous function of all the p_i (i = 1,…,n).

• (P2) Symmetry: C is (permutation) symmetric in p₁,…,p_n.

• (P3) Zero-indifference: C is unaffected if one or more firms with zero market share enter the market, i.e., C(p₁,…,p_n, 0…,0) = C(p₁,…,p_n).

• (P4) Schur-convexity: C is strictly Schur-convex.

• (P5) Value validity: C has value validity.

The continuity Property (P1) ensures that small changes in some of the p_i (i = 1,…,n) result in only a small change in the value of C. Property (P2) simply states that C is invariant with respect to the order in which the original p_i (i = 1,…,n) are given (e.g., [27], [5, p. 222]). According to Property (P3), the addition or deletion of one or more p_i = 0 components to or from the distribution P_n = (p_i,…,p_n) has no effect on the value of C. Property (P3) has two important implications. First, with respect to market concentration, (P3) together with (P1) and (P4), has the effect of slightly decreasing the value of C if one or more additional small firms enter the market (or the reverse effect if the small firms leave the market) (e.g., [28, Ch. 8], [3, p. 53]). Second, Property (P3) implies that C cannot be an explicit function of n such as if C were to be normed to some specific range by adjusting (controlling) for n. Properties (P4)-(P5) are subsequently defined and discussed in detail.

Another property advocated by some is the so-called replication property. First suggested by Hall and Tideman [29], this property means that if each p_i is split into k equal parts p_i/k,…,p_i/k for i = 1,…,n, resulting in the kn component distribution denoted by P_n/k, then C should be reduced by a factor of 1/k, i.e., C(P_n/k) = C(P_n)/k. Replication, which is different from the homogenous property of a function, may be an interesting and novel mathematical concept, but is entirely unrealistic of any practical situation involving concentration. While favored by some (e.g., [29, 30]), others do not view replication as a necessary property (e.g., [3, pp.47-58]), some do not even mention replication among the properties of a concentration index (e.g., [5, pp.221-223], [27]), and some feel that it is not self-evident that this property is desirable [31, pp. 63–64]. The index C and its normalized form C*∈[0, 1] cannot both have the replication property, but no rigorous explanation has been offered as to which one should have this property and why. It is also worth noting that the (sample) standard deviation s_n (with devisor n) of p₁,…,p_n has the replication property, but s_n−1 (with devisor n-1) does not (nor does the variance $s_n^2$). However, nobody would argue that s_n is to be preferred over s_n−1 because of the replication principle even though s_n−1 has the statistical property of unbiasedness, nor would s_n be preferred over $s_n^2$ for this reason.

Most importantly, however, is the fact that no generally accepted and rigorous basis appears to have been provided for the replication property as being of any importance to a concentration index. Also, from the value-validity property (P5) discussed below, it becomes apparent that replication is inconsistent with this property. It appears that the replication property lacks any real or solid justification.

2.2 Schur-convexity

The strict Schur-convexity Property (P4), which was also discussed by Hannah and Kay [3, Ch. 4], means that if the components of P_n = (p₁,…,p_n) are “less unevenly distributed” or “less spread out” than are those of another distribution Q_n = (q₁,…,q_n), then C(P_n)<C(Q_n). Such rather vague statements can be expressed more precisely in terms of majorization theory (e.g., Marshall et al. [32]). Thus, by definition, if the p_i‘s are ordered such that

$$p_1\ge p_2\ge \cdots \ge p_n$$ (1)

then P_n is majorized by Q_n (denoted by P_n≺Q_n) under the following condition:

$$P_n\prec Q_n\mathrm{i}\mathrm{f}{\displaystyle \sum _{i=1}^j{p}_i}\le {\displaystyle \sum _{i=1}^j{q}_i},j=1,\dots ,n-1$$ (2)

with ${\displaystyle \sum _{i=1}^n{p}_i}={\displaystyle \sum _{i=1}^n{q}_i}=1$. C is then strictly Schur-convex if

$$P_n\prec Q_n\mathrm{implies}C(P_n)<C(Q_n)$$ (3)

assuming P_n is not a permutation of Q_n [32, pp. 8, 80]. If the inequality in (3) is not strict, then C is Schur-convex.

The condition in (3), which is referred to by economists as the Dalton condition or the Pigou-Dalton condition, also implies that (a) C has the transfer property and (b) C preserves the Lorenz order [32, pp. 5–8, 560, 712–723]. The principle of “upward” transfers means that if p_i>p_j and an amount δ<p_i−p_j is transferred from p_j to p_i, the value of C increases. The fact that the strictly Schur-convex C preserves the Lorenz order means that if the distribution P_n is majorized by Q_n as defined in (2), then (a) the Lorenz curve (after [33]) for P_n falls nowhere below the Lorenz curve for Q_n and (b) C(P_n)<C(Q_n). By definition, if the p_i‘s are ordered such that p₍₁₎≤p₍₂₎≤…≤p_(n), with $F_{(i)}={\displaystyle \sum _{j=1}^i{p}_{(j)}}$ and F₍₀₎ = 0, then the Lorenz curve for P_n is obtained from the line segments between the consecutive points (i/n, F_(i) for i = 0,…,n.

2.3 Value validity

2.3.1 The importance of the value-validity property

In order to provide a simple confirmation of the need to impose a restriction on the values of a concentration index, consider first the complete lack of uniformity among the values taken on by different concentration indices for the same data sets. For some of the indices C_i (defined and discussed in Section 3 below) and for the distribution $P_{10}^{0.5}=(0.55,0.05,\dots ,0.05)$, the respective values of the indices C₁(CR₄), C₃(HHI), C₄, C₆, C₈, C₉, C₁₃, C₁₄, and C₁₇ become 0.70, 0.33, 0.18, 0.59, 0.48, 0.41, 0.19, 0.14, and 0.45. That is as much as 400% variation in index values for the same data set in spite of the fact that many of these indices have the same properties. Even the values of C₃(HHI) and C₁₃ differ by a factor of nearly 2 although both indices are members of the parameterized family C₁₈ (with α = 1 and α→0).

In fact, one could define a family of indices as the power function (HHI)^α with arbitrary parameter α>0 and with all members having Properties (P1)-(P4). Of course, the values of different family members could vary greatly for the same data set and difference comparisons between index values for different data sets could vary greatly, depending upon α. The need for some additional constraint on index values would seem to be paramount.

Such variation in index values is also reflected in their differing sensitivities to changes in the distribution P_n = (p₁,…,p_n). In the case of the most important index HHI, as discussed more extensively later in the paper, HHI is more sensitive to changes in P_n for large index values than for small index values. As a numerical illustration, consider the two distributions (0.30, 0.30, 0.20, 0.20) and (0.35, 0.30, 0.20, 0.15) for which HHI = 0.26 and 0.28, respectively, where the first distribution is obtained from the second one by means of a transfer as defined above. Then, consider the same transfer involving the two distributions (0.70, 0.15, 0.10, 0.05) and (0.75, 0.15, 0.10, 0.00) for which HHI = 0.53 and 0.60. The increase in HHI values between the last two more concentrated distributions is about three times that between the first two less concentrated distributions (i.e., 0.07 versus 0.02), with relative increases of 13.2% and 7.7%.

Since all proposed concentration indices take on their extremal values for the two distributions

$$P_n^0=\left(\frac{1}{n},\dots ,\frac{1}{n}\right),{\mathrm{P}}_n^1=\left(1,0,\dots ,0\right),$$ (4)

it may be interesting and informative to consider index values for the following mean of $P_n^0$ and $P_n^1$:

$$P_n^{0.5}=\left(\frac{1}{2}\right)P_n^0+\left(\frac{1}{2}\right)P_n^1=\left(\frac{1}{2}+\frac{1}{2n},\frac{1}{2n},\dots ,\frac{1}{2n}\right).$$ (5)

In terms of this information alone, the only defensible value of a concentration index C is that its value for $P_n^{0.5}$ in (5) should be the mean of its values for $P_n^0$ and $P_n^1$ in (4), i.e.,

$$C\left(P_n^{0.5}\right)=\left(\frac{1}{2}\right)C\left(P_n^0\right)+\left(\frac{1}{2}\right)C\left(P_n^1\right).$$ (6)

For the $P_{10}^{0.5}=(0.55,0.05,\dots ,0.05)$ used in the above example and with $C(P_n^0)=1/n$ and $C(P_n^1)=1$ as for many indices, including HHI, (6) requires that $C(P_{10}^{0.5})=0.55$. However, by comparison, $HHI(P_{10}^{0.5})=0.33$ i.e., 40% less than the corresponding requirement from (6). Other indices fare no better, such as $C_{14}(P_{10}^{0.5})=0.14<<0.55$.

The requirement in (6) can also be expressed in terms of distances or differences as

$$C(P_n^1)-C(P_n^{0.5})=C(P_n^{0.5})-C(P_n^0)$$ (7)

showing the $C(P_n^{0.5})$ is an equal distance from each of the two extreme values $C(P_n^0)$ and $C(P_n^1)$. Furthermore, by considering the distributions $P_n^0,P_n^{0.5},\mathrm{and}{P}_n^1$ as being points or vectors in n-dimensional Euclidean space, the following equality between Euclidean distances can be seen to hold:

$$d\left(P_n^1,P_n^{0.5}\right)=d\left(P_n^{0.5},P_n^0\right).$$ (8)

That is, the distance between $P_n^{0.5}\mathrm{and}{P}_n^1$ is the same as that between $P_n^{0.5}$ and $P_n^0$, supporting the requirement in (7).

2.3.2 Generalized requirement

The development from (4) to (8) can be further generalized by using the lambda distribution

$$P_n^{\lambda }=\left(\lambda +\frac{1-\lambda }{n},\frac{1-\lambda }{n},\dots ,\frac{1-\lambda }{n}\right),0\le \lambda \le 1$$ (9)

introduced by Kvålseth [34] (the 1−λ is used here as a convenient form when dealing with concentration). This particular distribution provides an important and general basis for the value-validity property as first discussed in Kvålseth [34]. The present paper adapts those basic concepts to the measurement of concentration.

The real-valued parameter λ is basically a concentration parameter of which λ = 0, λ = 0.5, and λ = 1 are the particular cases in (4) and (5). In fact, the following linear function or weighted arithmetic mean of $P_n^0$ and $P_n^1$

$$P_n^{\lambda }=(1-\lambda )P_n^0+\lambda P_n^1,0\le \lambda \le 1$$ (10)

includes the mean in (5) as a particular case with λ = 1/2. Furthermore, the condition in (6) would be a particular case of the general formulation

$$C\left(P_n^{\lambda }\right)=\left(1-\lambda \right)C\left(P_n^0\right)+\lambda C\left(P_n^1\right).$$ (11)

Similarly, it is seen that (7) and (8) generalize as follows:

$$C\left(P_n^1\right)-C\left(P_n^{\lambda }\right)=\left(\frac{1-\lambda }{\lambda }\right)\left[C\left(P_n^{\lambda }\right)-C\left(P_n^0\right)\right]$$ (12)

and

$$d\left(P_n^1,P_n^{\lambda }\right)=\left(\frac{1-\lambda }{\lambda }\right)d\left(P_n^{\lambda },P_n^0\right).$$ (13)

Besides considering (11) as a direct extension of (6), although clearly supported by (12)-(13), the general condition for value validity in (11) also follows from the requirement that C should be equally sensitive to small changes in the distribution P_n = (p₁,…,p_n) throughout its range from $C\left(P_n^0\right)$ to $C\left(P_n^1\right)$. This sensitivity requirement can equivalently be expressed in terms of the discriminant ability of C, i.e., the ability of C to discriminate between distributions P_n that are not unduly different. Thus, C should have a constant or uniform ability to detect small changes in any P_n. Such a requirement can be imposed on an index C with Properties (P1)-(P4) because of the following equality:

$$C\left(P_n\right)=C\left(P_n^{\lambda }\right)\mathrm{for}\mathrm{one}\mathrm{unique}\lambda$$ (14)

for any P_n and the $P_n^{\lambda }$ in (10). Consequently, as the p_i‘s become increasingly unequal or uneven so that C(P_n) increases, the value of the concentration parameter λ increases correspondingly for any given n. As a simple illustration involving the index HHI, consider HHI(0.50, 0.30, 0.20) = 0.38 and HHI(0.70, 0.20, 0.10) = 0.54 for which the respective values of λ from (14) is found to be λ = 0.26 and λ = 0.56.

The requirement or condition that C(P_n) should be equally sensitive to small changes in the form of P_n throughout its range becomes equivalent to requiring the partial derivative $\partial C\left(P_n^{\lambda }\right)/\partial \lambda$ to be constant for all λ and any given n. Consequently, $C\left(P_n^{\lambda }\right)$ has to be a linear function of λ, which leads immediately to (11).

3. Assessment of indices

A concise historical account of concentration indices proposed to date is summarized in Table 1. Some of these are parameterized families of indices such as C₁₀, C₁₁, C₁₂, C₁₈, and C₁₉ since they depend on an arbitrary parameter α. Two of the individual indices, C₃(HHI) and C₁₃, are seen to be members of the C₁₈-family (for α = 1 and α→0). Note that, although all indices in Table 1 have the Symmetry Property (P2), the indices C₁, C₄, C₆, C₇, C₉, and C₁₇ are based on the descending order among the p_i‘s as in (1).

Table 1. Proposed concentration indices based on probabilities (proportions, market shares) p₁≥p₂≥⋯≥p_n and their lacking properties (LP, in Section 2.1).

C i	Formula	Reference	Notes	LP
C1 (CRm)	${\displaystyle \sum _{i=1}^m{p}_i}$	Various authors		P1, P4
C 2	$\sqrt{{\displaystyle \sum _{i=1}^n{p}_i^2}}$	Hirschman [18]		P5
C3 (HHI)	${\displaystyle \sum _{i=1}^n{p}_i^2}$	Herfindahl [17]		P5
C 4	${\left(2{\displaystyle \sum _{i=1}^{n}i{p}_i-1}\right)}^{-1}$	Rosenbluth [25] and Hall & Tideman [29]		P5
C 5	$‐{\displaystyle \sum _{i=1}^n{p}_i\log p_i}$	Theil [28]	a	P5
C 6	$p_1+{\displaystyle \sum _{i=2}^n{p}_i^2(2-p_i)}$	Horvath [35]		P4, P5
C 7	${\displaystyle \sum _{i=1}^m{p}_i(m+1-i)/m},m\ge 1$	Hart [36]	b	P1, P3, P4
C 8	$2{\displaystyle \sum _{i=1}^n{p}_i^2-}{\displaystyle \sum _{i=1}^n{p}_i^3}$	Hart [37]	c	P5
C 9	$\frac{1}{n(n-1)}{\displaystyle \sum _{m=1}^{n-1}\left(\frac{n-m}{m}\right)}\left(\frac{C{R}_m}{1-C{R}_m}\right)$	Linda [38]		P1,P3,P4,P5
C 10	${\displaystyle \sum _{i=1}^n\left[p_i^2+{\left\{p_i\left({\displaystyle \sum _{j=1}^n{p}_j^2-p_i^2}\right)\right\}}^{\alpha }\right]}$, α>1	Hause [39]		P5
C 11	${\left({\displaystyle \sum _{i=1}^n{p}_i^{\alpha }}\right)}^{1/(1-\alpha )}$, α>0	Hannah & Kay [3]	d	P5
C 12	${\left(n{\displaystyle \sum _{i=1}^n{p}_i^2}\right)}^{\alpha }/n,\alpha >0$	Davies [40]		P1, P3, P5
C 13	${\displaystyle \prod _{i=1}^n{p}_i^{p_i}}$	Bruckmann [41], Häni [42]		P5
C 14	${\displaystyle \sum _{i=1}^n{p}_i}{\left[1+n(1-p_i)\right]}^{-1}$	Ginevičius [43]		P1, P3, P5
C 15	${\displaystyle \sum _{i=1}^m{p}_i^2}/{\displaystyle \sum _{i=1}^n{p}_i^2},\mathrm{m}\ge 1$	Anbarci & Katzman [44]		P4, P5
C 16	${\displaystyle \sum _{i=1}^n{\left(\log p_i\right)}^2}/n-{\left({\displaystyle \sum _{i-1}^n\log p_i}\right)}^2/n^2$	Hannah & Kay [3]		P1, P4, P5
C 17	$1-\left(2{\displaystyle \sum _{i=1}^{n}i{p}_i-1}\right)/n$	Marfels [45]		P1, P3
C 18	${\left({\displaystyle \sum _{i=1}^n{p}_i{}^{\alpha +1}}\right)}^{1/\alpha },\alpha >-1$	Bruckmann [41]		P5
C 19	${\alpha }^{-1}\log {\displaystyle \sum _{i=1}^n{p}_i{e}^{\alpha p_i}},\alpha >0$	Bruckmann [41]		P5

Notes: (a) This is the famous entropy by Shannon [46], it is strictly Schur-concave; (b) Hart [36] also suggested replacing m in C₇ with n; (c) The C₈ was proposed as an alternative to C₆; (d) This family of indices ranges from $C_{11}(P_n^0)=n\mathrm{to}{C}_{11}(P_n^1)=1$ for the market-share distributions in (4) and measures deconcentration; it is strictly Schur-concave.

As indicated in the right column of Table 1, the various indices lack some of the Properties (P1)-(P5) as can easily be verified. A number of the potential indices lack the strict Schur-convexity (Property (P4)) while several others lack the zero-indifference Property (P3). Only three of the indices, C₁(CR_m), C₇, and C₁₇, meet the condition in (11) required by the value-validity Property (P5). The other indices fail to meet even the weaker condition in (6). However, each of those three indices lack other properties as indentified in Table 1. In particular, CR_m lacks Property (P1) and the strict Schur-convexity (Property (P4)), although it can be shown to be Schur-convex.

In the case of C₃(HHI) and the distributions of $P_n^0\mathrm{and}{P}_n^1$ in (4) and $P_n^{\lambda }$ in (9), it is seen that

$$HHI(P_n^{\lambda })={\lambda }^{2}HHI(P_n^1)+(1-{\lambda }^2)HHI(P_n^0)=(1-\frac{1}{n}){\lambda }^2+\frac{1}{n}$$ (15)

with $HHI(P_n^0)=1/n$ and $HHI(P_n^1)=1$. When compared with (1−1/n)λ+1/n as required by (11), it is clear that HHI understates the true extent of the concentration. This negative bias occurs throughout the range of values of HHI, but is seen to be most pronounced when λ = 1/2. From the partial derivative $\partial HHI\left(P_n^{\lambda }\right)/\lambda =2(1-1/n)\lambda$ and the equality in (14), the implication is that for any given n, HHI(P_n) is not equally sensitive to small changes in P_n throughout its range. Rather, HHI(P_n) has the bias of being increasingly sensitive to changes in P_n = (p₁,…,p_n) as the p_i‘s become increasingly uneven or variable, i.e., with increasing HHI(P_n)-values. The important point is this: while a concentration index should clearly be sensitive to changes in P_n, such sensitivity should not be biased in a particular direction such as toward uneven (skewed) distributions as in the case of HHI.

4. The new index

4.1 Index formulation

Several of the indices defined in Table 1 can be considered as members of the following class of weighted sums:

$$C={\displaystyle \sum _{i=1}^n{w}_i{p}_i}$$ (16)

where w₁,…,w_n are positive weights with each w_i being a function of (or depending upon) p_i and possibly other components of the distribution P_n = (p₁,…,p_n). Other indices in Table 1 can be viewed as strictly increasing functions of the C in (16). The most obvious members of (16) are perhaps HHI with w_i = p_i for i = 1,…,n and CR_m with w_i = 1 for i = 1,…,m and 0 otherwise (with the p_i’s being ordered as in (1)).

A family of concentration indices as in (16) can also be derived from theoretical models. For example, Encaoua and Jacquemin [27] arrived at (16) by means of an axiomatic characterization. In their derivation, w_i is a nondecreasing function of p_i such that w_ip_i is convex (see also Tirole [5, pp. 221–223]). Dickson [47] showed that a concentration index can be expressed as in (16) with w_i (i = 1,…,n) being the firms’ so-called conjectural variation elasticities, with w_i∈[0, 1] being the economically meaningful or valid interval for each w_i.

Two of the members of (16) defined in Table 1 (C₇ and C₁₇) use weights w_i (i = 1,…,n) based on the ranks from the distributions P_n = (p₁,…,p_n) when ordered as in (1). The C₄ is also based on ranks, but this index belongs to the reciprocal of the family in (16). With rank 1 for p₁ (the largest p_i), rank 2 for p₂ (the second largest p_i), etc., each of these three indices are decreasing functions of the weighted mean rank ${\displaystyle \sum _{i=1}^n{p}_{i}i}$.

Alternatively, instead of linearly decreasing weighted mean rank or the reciprocal weighted mean rank, consider the following weighted mean reciprocal rank:

$$C_K={\displaystyle \sum _{i=1}^n{p}_i/i}$$ (17)

with the p_i‘s ordered as in (1). This index is proposed as a new concentration index with desirable properties. In the case of market shares p₁,…,p_n, C_K is simply the market share of the largest firm (divided by 1), plus the market share of the second largest firm divided by 2,…., plus the market share of the nth largest (i.e., smallest) firm divided by n. The expression in (17) applies whether some or all of the p_i‘s are equal. In the extreme case of $P_n^0$ in (4), $C_K(P_n^0)={\displaystyle \sum _{i=1}^n(1/n)(1/i)}$.

When there are ties among the p_i‘s so that p_i = p_i+1 = ⋯ = p_i+k = p for any i and k, it may appear as if different terms of C_K in (17) contribute differently to the value of C_K because their set of weights {1/i} differ even though the p_i‘s are equal. However, the contribution toward C_K by such tied p_i‘s can be expressed equivalently in terms of the arithmetic mean $\left({\displaystyle \sum _{j=i}^{i+k}1/j}\right)/k$ of their reciprocal ranks as their common weight such that

$$p/i+p/(i+1)+\cdots +p/(i+k)=\left[\left({\displaystyle \sum _{j=i}^{i+k}1/j}\right)/k\right](p+p+\cdots +p).$$

This expression shows clearly how the terms with equal p_i‘s can all be considered as contributing equally toward the overall value of C_K. Of course, $C_K\left(P_n^0\right)$ represents the extreme case when p₁ = ⋯ = p_n = 1/n.

Besides being a weighted mean of reciprocal ranks or a weighted sum of the p_i‘s (ordered as in (1)), it may be of interest to note that C_K is also the reciprocal of the weighted harmonic mean of the ranks 1,2,…,n. The C_K can also be given a geometric interpretation in terms of the cumulative p_i‘s and 1/i as explained and illustrated in Fig 1.

Fig 1. The value of C_K in (17) corresponds to the area above the step function formed by the cumulative p_i‘s and the reciprocal rank 1/i where the cumulative proportions (CP) = 0 for 0<1/i<1/n; CP = p_n for 1/n≤1/i<1/(n−1); CP = p_n+p_n−1 for 1/(n−1)≤1/i<1/(n−2),…; CP = p_n+p_n−1+⋯+p₂ for 1/2≤1/i<1; and CP = 1 for 1/i = 1.

This exemplary graph is for the distribution P₅ = (0.40, 0.30, 0.15, 0.10, 0.05) with C_K = 0.635.

The range of values of C_K based on the extreme distributions in (4) is given by

$$C_K(P_n^0)=\left(\frac{1}{n}\right){\displaystyle \sum _{i=1}^{n}1/i}\le C_K(P_n)\le C_K(P_n^1)=1.$$ (18)

The lower bound on C_K is simply the reciprocal of the harmonic mean of the first n positive integers (or the ranks 1,2,…,n). This is also by itself an interesting mathematical quantity. In particular, $H_n={\displaystyle \sum _{i=1}^{n}1/i}$ is the partial sum of the harmonic series, also referred to as the n-th harmonic number. A recurrence relation for $C_K(P_n^0)$ follows immediately from H_n+1 = H_n+1/(n+1). Although tabulated H_n values are readily available, a quick way to obtain values of $C_K(P_n^0)$ is by means of the following formula:

$$C_K(P_n^0)=\left[\log (n+1/2)+0.5772\right]/n$$ (19)

where the 0.5772 is the Euler-Mascheroni constant (to 4 decimal places). This formula is found to be correct to at least 3 decimal places for n>5.

4.2 Properties of C_K

It is readily apparent from the expression in (17) that C_K has each of the Properties (P1)-(P3). The strict Schur-convexity of C_K (Property (P4)) is rather apparent from the form of (17) and the Schur-convexity definition in (2)–(3). It also follows from the fact that (a) C_K has the symmetry Property (P2) and the partial derivative ∂C_K/∂p_i = 1/i is strictly decreasing in i = 1,…,n [32, p. 84].

What sets C_K apart from other concentration indices is the fact that C_K has the value-validity property (P5). From (10) and (17),

$$C_K(P_n^{\lambda })=\lambda +(1-\lambda )C_K(P_n^0)$$ (20)

so that, with $C_K(P_n^1)=1,C_K(P_n^{\lambda })$ complies with the value-validity condition in (11). The $C_K(P_n^0)$ is the lower bound on C_K defined in (18).

Market concentration may be viewed as consisting of two different components: the size of a market or industry (n) and the inequality between the market shares. The index C_K can be formulated so as to reflect those two components separately by expressing C_K(P_n) as follows:

$$C_K(P_n)=\left[C_K\left(P_n^1\right)-C_K\left(P_n^0\right)\right]C_K^{*}(P_n)+C_K(P_n^0),C_K^{*}(P_n)=\frac{C_K\left(P_n\right)-C_K\left(P_n^0\right)}{C_K\left(P_n^1\right)-C_K\left(P_n^0\right)}$$ (21)

with $C_K\left(P_n^1\right)=1$ and $C_K^{*}\left(P_n\right)\in \left[0,1\right]$. The $C_K\left(P_n^0\right)$ is a function of n only and the normalized form $C_K^{*}\left(P_n\right)$, which controls or adjusts for n, is a measure of market-share inequality. It is clear from (21) that for any fixed $C_K^{*}(P_n)$, C_K(P_n) is a strictly increasing function of $C_K(P_n^0)$ and strictly decreasing function of n. Similarly, for any fixed n, C_K(P_n) is strictly increasing in $C_K^{*}(P_n)$. That is, C_K(P_n) increases as the size of the market decreases and as the inequality between the market shares increases.

The sensitivity of C_K(P_n) to changes in the inequality $C_K^{*}(P_n)$ and to changes in n via $C_K(P_n^0)$ can be compared by using partial derivatives and treating $C_K(P_n^0)$ as a continuous variable for mathematical purpose. Then, it follows from (21) that

$$\frac{\partial C_K(P_n)}{\partial C_K^{*}(P_n)}=\left(\frac{1-C_K(P_n^0)}{1-C_K^{*}(P_n)}\right)\frac{\partial C_K(P_n)}{\partial C_K(P_n^0)}$$ (22)

It appears from (22) that whether C_K(P_n) is more sensitive to changes in $C_K^{*}(P_n)$ than to changes in $C_K(P_n^0)$ or vice versa depends on whether $C_K^{*}(P_n)>C_K(P_n^0)$ or $C_K^{*}(P_n)<C_K(P_n^0)$.

As an extension of the weighted mean of the two extremal distributions $P_n^0$ and $P_n^1$ in (10), consider the weighted mean of any two distributions P_n = (p₁,…,p_n) and Q_n = (q₁,…,q_n), both arranged in descending order as in (1). It then follows immediately from the definition of C_K in (17) that the value of C_K for the mixture distribution with weights w and 1-w is given by

$$C_K\left[w{P}_n+(1-w)Q_n\right]=w{C}_K(P_n)+(1-w)C_K(Q_n),0\le w\le 1.$$ (23)

This property of C_K, which would seem to be an intuitively reasonable one, is basically an extension of the value-validity requirement in (11). Thus, if an index complies with (23), it will necessarily comply with (11) since $P_n^0$ and $P_n^1$ are simply special forms of Q_n and P_n, respectively.

The C_K does not have the replication property, although, as discussed in Section 2.1, this property cannot be considered essential and may rather be a questionable one. In fact, the replication property and the value-validity property (P5) are inconsistent or mutually exclusive. This inconsistency can be proved by first noting that for the distribution $P_n^{\lambda }$ in (9) and analogously to (21), the value-validity condition in (11) can also be expressed as

$$C^{*}\left(P_n^{\lambda }\right)=\frac{C\left(P_n^{\lambda }\right)-C\left(P_n^0\right)}{C\left(P_n^1\right)-C\left(P_n^0\right)}=\lambda =\left(\frac{n}{\sqrt{n-1}}\right)s_n=s_n^{*}\left(P_n^{\lambda }\right)$$ (24)

where s_n is the standard deviation (with devisor n) of the n components of $P_n^{\lambda }$ and $s_n^{*}\left(P_n^{\lambda }\right)$ is its normalized form. If C were to have the replication property, then $C\left(P_n^{\lambda }/k\right)=C\left(P_n^{\lambda }\right)/k,C\left(P_n^0/k\right)=C\left(P_n^o\right)/k$, and $C\left(P_n^1/k\right)=C\left(P_n^1\right)/k$ so that $C^{*}\left(P_n^{\lambda }/k\right)=C^{*}\left(P_n^{\lambda }\right)$. However, since $s_{kn}\left(P_n^{\lambda }/k\right)=s_n\left(P_n^{\lambda }\right)/k$ and hence $s_{kn}^{*}\left(P_n^{\lambda }/k\right)\ne s_n^{*}\left(P_n^{\lambda }\right)$, C cannot simultaneously have the replication property and meet the condition in (24) for value validity.

Some prefer to use a concentration index that has the property of being a so-called numbers equivalent index such as the index family C₁₁ by Hannah and Kay [3] defined in Table 1. The numbers equivalent NC_K of C_K in (17) can be obtained by simply setting $C_K\left(P_n\right)=C_K\left(P_{N{C}_K}^0\right)=\left({\displaystyle \sum _{i=1}^{N{C}_K}1/i}\right)/N{C}_K$ and solving for the nearest integer NC_K for any given P_n. Thus, for the concentration C_K(P_n) of the market-share distribution P_n = (p₁,…,p_n), NC_K becomes the number of firms in an equivalent market with equal market shares and whose market concentration equals the given C_K(P_n). However, while NC_K does provide an alternative interpretation for any index value C_K(P_n), its utility is limited by its lack of the value-validity property (P5) since NC_K is not a linear function of C_K(P_n) as would be required for compliance with (11).

Another aid for interpreting or visualizing values of C_K may be the use of the equivalent lambda distribution $P_n^{{\lambda }_e}$ obtained from (10) and (14) and by setting $C_K\left(P_n\right)=C_K\left(P_n^{\lambda }\right)$ for the $C_K\left(P_n^{\lambda }\right)$ in (20) and solving for λ = λ_e and for any distribution P_n = (p₁,…,p_n). This solution is seen to equal the normalized $C_K^{*}\left(P_n\right)$ in (21), i.e., ${\lambda }_e=C_K^{*}\left(P_n\right).$ Thus, for example, consider P₅ = (0.40, 0.30, 0.15, 0.10, 0.05) for which C_K(P₅) = 0.64 and $C_K\left(P_{{}_5}^0\right)=0.46$ so that ${\lambda }_e=C_{{}_K}^{*}\left(P_5\right)=0.33$. This gives $P_5^{0.33}=(0.48,0.13,\dots ,0.13)$ as being the equivalent lambda distribution having (approximately) the same C_K-value as C_K(P₅) for the given P₅.

5. Comparison with other indices

The most significant difference between the new index C_K in (17) and other indices proposed to date is due to the fact that C_K has the value-validity property (P5) whereas most other indices lack this property. However, the most interesting comparison is between C_K and the two indices CR_m and HHI since those are by far the most popular ones (e.g., [13, pp.1216-118], [14, pp. 97–101], [6, Ch. 8]). Among the CR_m members, the most commonly used member is the four-firm concentration ratio (CR₄) (e.g., [16, p. 255]). Comparisons between HHI and CR₄ have been extensively discussed [15].

In order to compare the values of C_K with those of HHI and CR₄ for a wide variety of market-share distributions P_n = (p₁,…,p_n), a computer algorithm was used to randomly generate P_n as follows. First, n was generated as a random integer between n = 5 and n = 100 (inclusive). The lower limit n = 5 was chosen since the four-firm CR₄ was used and to avoid CR₄ = 1 values simply because of values n<5 being generated. Then, for each such generated n, each p_i (ordered as in (1)) was generated as a random number (to the desired decimals) within the following intervals:

$$\frac{1}{n}\le p_1\le 1$$

$$\frac{1-{\displaystyle \sum _{j=1}^{i-1}{p}_j}}{n-(i-1)}\le p_i\le \min \left\{p_{i-1},1-{\displaystyle \sum _{j=1}^{i-1}{p}_j}\right\}\mathrm{for}i=2,\dots ,n-1$$

$$p_n=1-{\displaystyle \sum _{j=1}^{n-1}{p}_j}.$$

A total of 1000 such distributions were generated and the corresponding values of C_K, CR₄, and HHI were computed. The results for the C_K−CR₄ comparison is illustrated in Fig 2.

Fig 2. Comparison of values of CR₄ and C_K for 1000 randomly generated market-share distributions P_n = (p₁,…,p_n) with the number of firms n varying as a random integer between 5 and 100.

It is clear from their differing expressions that there cannot be any precise functional relationship between CR₄ and C_K as is also demonstrated in the result in Fig 2. The rather systematic result is that the variation in CR₄-values for any C_K-value tends to increase quite dramatically with increasing C_K. More specifically, the change in the variation of CR₄-values tends to increase up to about C_K = 0.5 beyond which it decreases. The same type of general relationship is also found between CR₄ and HHI [15].

From the simulation results for HHI versus C_K in Fig 3, it is rather strikingly clear that for any given value of either C_K or HHI, the variation in the other index is quite limited. In fact, this scatter diagram supports the proposition that a functional relationship exists between the two indices.

Fig 3. Comparison of values of HHI and C_K for 1000 randomly generated market-share distributions P_n = (p₁,…,p_n) with the number of firms n varying as a random integer between 5 and 100.

In order to explore this potential relationship, three additional sets of distributions P_n have been analyzed: (1) the lambda distribution in (10), (2) randomly generated P_n as described above, and (3) real market-share distributions. The real data consist of the market shares of the firms within 20 different markets and were chosen so as to be readily available, include a wide variety of markets, and cover a reasonably wide range of concentration values. Those 20 data sets were simply obtained from internet searches with the source of each being identified. The purpose of using those real data sets was for exploring the potential relationship between C_K and HHI rather than providing realiable and representative market concentration results for specific markets. The same data sources were also used in Kvålseth [15] for comparing CR₄ and HHI.

Those three different types of market-share data were then used to explore potential functions and transformations that would provide a reasonably accurate description of the apparent relationship between C_K and HHI. The following result has emerged from this analysis:

$$C_K\approx C_{KH}=\left(\frac{1}{3}\right)\log \left[(e^3-1)HHI+1\right]$$ (25)

for any distribution P_n = (p₁,…,p_n) where the logarithm is to base e (natural logarithm) and the exponential term e³−1 = 19.0855. The approximation between C_K and the transformation C_KH in (25) results from regression analysis. Specifically, for the regression (through the origin) model C_K = αC_KH, the following estimates are obtained: $\widehat{\alpha }=0.98$ for the data in Table 2 involving $P_n^{\lambda }$ in (10) and different values of n and λ, $\widehat{\alpha }=1.00$ for the data in Table 3 involving randomly generated P_n, and $\widehat{\alpha }=1.02$ for the real market results in Table 4.

Table 2. Values of C_K in (17), HHI defined in Table 1, and C_KH in (25) for ${\mathit{P}}_{\mathit{n}}^{\mathit{\lambda }}$ in (9) with varying λ and n.

λ	n	C K	HHI	C KH
0.10	2	0.78	0.51	0.79
0.30	2	0.83	0.55	0.81
0.50	2	0.88	0.63	0.86
0.70	2	0.93	0.75	0.91
0.90	2	0.98	0.91	0.97
0.10	5	0.51	0.21	0.54
0.30	5	0.62	0.27	0.61
0.50	5	0.73	0.40	0.72
0.70	5	0.84	0.59	0.84
0.90	5	0.95	0.85	0.95
0.10	10	0.36	0.11	0.38
0.30	10	0.51	0.18	0.50
0.50	10	0.65	0.33	0.66
0.70	10	0.79	0.54	0.81
0.90	10	0.93	0.83	0.94
0.10	20	0.26	0.06	0.25
0.30	20	0.43	0.14	0.43
0.50	20	0.59	0.29	0.63
0.70	20	0.75	0.52	0.80
0.90	20	0.92	0.82	0.94
0.10	30	0.22	0.04	0.19
0.30	30	0.39	0.12	0.40
0.50	30	0.57	0.28	0.62
0.70	30	0.74	0.51	0.79
0.90	30	0.91	0.82	0.94
0.10	50	0.18	0.03	0.15
0.30	50	0.36	0.11	0.38
0.50	50	0.54	0.27	0.61
0.70	50	0.73	0.50	0.79
0.90	50	0.91	0.81	0.93

Table 3. Values of C_K in (17), ${\mathit{C}}_{\mathit{K}}^{*}$ in (21) and (26), d* in (26), HHI and CR₄ defined in Table 1, and C_KH in (25) for randomly generated P_n = (p₁,…,p_n) and 2≤n≤30.

n	C K	${\mathit{C}}_{\mathit{K}}^{*}$	d*	HHI	CR 4	C KH
27	0.84	0.81	0.76	0.59	0.94	0.84
29	0.16	0.03	0.03	0.04	0.18	0.17
19	0.75	0.69	0.68	0.49	0.79	0.78
8	0.50	0.24	0.23	0.17	0.63	0.49
12	0.45	0.26	0.24	0.14	0.55	0.43
9	0.71	0.58	0.53	0.36	0.96	0.69
15	0.48	0.33	0.32	0.16	0.54	0.47
3	0.79	0.46	0.47	0.48	1.00	0.77
14	0.24	0.01	0.01	0.07	0.30	0.29
4	0.72	0.42	0.40	0.37	1.00	0.70
21	0.35	0.22	0.20	0.09	0.38	0.33
24	0.36	0.24	0.21	0.08	0.42	0.32
13	0.82	0.76	0.72	0.56	0.89	0.82
12	0.91	0.88	0.85	0.75	0.95	0.91
15	0.24	0.02	0.02	0.07	0.29	0.28
29	0.60	0.54	0.48	0.26	0.78	0.60
20	0.26	0.10	0.09	0.06	0.31	0.25
8	0.77	0.65	0.60	0.44	1.00	0.75
13	0.40	0.21	0.19	0.11	0.53	0.37
17	0.92	0.90	0.88	0.78	0.95	0.92
19	0.38	0.24	0.22	0.10	0.43	0.36
14	0.28	0.06	0.05	0.07	0.36	0.29
28	0.29	0.17	0.16	0.06	0.35	0.25
17	0.65	0.56	0.52	0.32	0.73	0.65
25	0.49	0.40	0.39	0.19	0.50	0.50
17	0.91	0.89	0.85	0.74	0.97	0.90
25	0.36	0.24	0.22	0.08	0.44	0.32
21	0.18	0.01	0.01	0.05	0.20	0.22
29	0.34	0.24	0.22	0.08	0.37	0.31
23	0.78	0.74	0.68	0.49	0.98	0.78

Table 4. Values of C_K in (17), ${\mathit{C}}_{\mathit{K}}^{*}$ in (21) and (26), d* in (26), HHI defined in Table 1, and C_KH in (25) for a sample of real market-share data.

n	C K	${\mathit{C}}_{\mathit{K}}^{*}$	d*	HHI	CR 4	C KH	Source	(Market type)
16	0.37	0.20	0.20	0.10	0.50	0.36	[48]	(Airline travel)
16	0.42	0.27	0.25	0.12	0.60	0.40	[48]	(Airline travel)
8	0.47	0.20	0.20	0.16	0.75	0.47	[49]	(U.S. distilled liquor)
10	0.45	0.22	0.21	0.14	0.64	0.43	[50]	(Paints, coatings)
10	0.38	0.12	0.11	0.11	0.52	0.38	[51]	(Pharmaceuticals)
15	0.39	0.22	0.19	0.10	0.54	0.36	[52]	(Insurances companies)
12	0.52	0.35	0.32	0.18	0.69	0.50	[53]	(Weapons exporters)
30	0.26	0.15	0.13	0.05	0.34	0.22	[54]	(Car sales, Britain)
12	0.39	0.18	0.20	0.12	0.60	0.40	[55]	(Auto Mnf., US)
8	0.49	0.23	0.25	0.18	0.77	0.50	[50]	(Craft beer, US)
9	0.46	0.21	0.23	0.16	0.75	0.47	[50]	(Running shoe sales)
5	0.97	0.94	0.94	0.91	1.00	0.97	[56]	(Search eng., Norway)
4	0.71	0.39	0.38	0.36	1.00	0.69	[57]	(Comm. water heaters)
3	0.90	0.74	0.74	0.70	1.00	0.89	[58]	(Microprocessors)
10	0.51	0.31	0.28	0.17	0.72	0.48	[50]	(Top charter airlines)
5	0.58	0.23	0.22	0.24	0.79	0.57	[59]	(Global cigarettes, 2019)
10	0.52	0.32	0.30	0.18	0.68	0.50	[50]	(Farm mach., equip.)
20	0.32	0.17	0.18	0.08	0.48	0.31	[60]	(Global car sales)
10	0.40	0.15	0.15	0.12	0.60	0.40	[50]	(Top airlines, world)
8	0.70	0.55	0.51	0.35	0.83	0.68	[61]	(Consumer products)

In fact, for $\widehat{\alpha }=1.00$, the fitted model ${\widehat{C}}_K=C_{KH}$ provides an excellent fit to all data sets. The coefficient of determination, when properly computed [62], is found to be $R^2=1-{{\displaystyle \sum {\left(C_K-C_{KH}\right)}^2/{\displaystyle \sum \left(C_K-{\overline{C}}_K\right)}}}^2=0.985,0.990,\mathrm{and}0.990$ for the data in Tables 2–4, respectively. That is, 99% of the variation of C_K (about its mean) is explained (accounted for) by the fitted model.

While the excellent fit of the relationship C_K≈C_KH in (25) is apparent from the data in Tables 2–4, it is more conveniently depicted in Fig 4. It is clear from this scatter diagram that (25) is indeed an appropriate relationship supported by the data. When looking at the residuals (C_K−C_KH) in Fig 4, they tend to fall within a rather uniform band along each side of the curve. There appears to be no other systematic pattern among the residuals that would indicate the need for any alternative expression.

Fig 4. Scatter diagram of HHI versus C_K from the data in Table 2 (dots), Table 3 (crosses) and Table 4 (circles).

The curve represents the fitted model in (25).

Tables 3 and 4 also give the results for the following normalized forms of C_K and of the Euclidean distance $d\left(P_n,P_n^0\right)$ (generalized from (8) and (13)):

$$C_K^{*}(P_n)=\frac{C_K\left(P_n\right)-C_K\left(P_n^0\right)}{1-C_K\left(P_n^0\right)},d^{*}\left(P_n,P_n^0\right)=\frac{d\left(P_n,P_n^0\right)}{d\left(P_n^1,P_n^0\right)}.$$ (26)

It is clear from these results that the approximation $C_K^{*}\left(P_n\right)\approx d^{*}\left(P_n,P_n^0\right)$ holds to a highly respectable degree of accuracy. In fact, the properly computed R²-values for the fitted model ${\widehat{C}}_K^{*}\left(P_n\right)=d^{*}\left(P_n,P_n^0\right)$ are found to be $R^2=1-{{\displaystyle \sum {\left(C_K^{*}-d^{*}\right)}^2}/{\displaystyle \sum \left(C_K^{*}-{\overline{C}}_K^{*}\right)}}^2=0.990$ for all the data in each of Tables 3 and 4.

In many practical situations involving potential concerns about mergers and market structure, the number of firms n within a market or industry may be quite small. Although the data in Tables 2–4 include a number of data sets with n≤10, it may be worthwhile to look at some additional data as in Table 5 obtained for market-share distributions P_n = (p₁,…,p_n) computer generated from the algorithm described above for 2≤n≤10.

Table 5. Values of C_K in (17), ${\mathit{C}}_{\mathit{K}}^{*}$in (21), d* in (26), HHI defined in Table 1, and C_KH in (25) for randomly generated P_n = (p₁,…,p_n) and 2≤n≤10.

n	C K	${\mathit{C}}_{\mathit{K}}^{*}$	d*	HHI	C KH
2	0.99	0.96	0.94	0.94	0.98
9	0.79	0.70	0.65	0.49	0.78
4	0.87	0.73	0.72	0.64	0.86
5	0.57	0.21	0.22	0.24	0.57
6	0.65	0.41	0.40	0.30	0.64
5	0.68	0.41	0.43	0.35	0.68
7	0.61	0.38	0.37	0.26	0.60
2	0.76	0.04	0.00	0.50	0.79
5	0.56	0.19	0.22	0.24	0.57
4	0.67	0.31	0.33	0.33	0.66
6	0.92	0.86	0.83	0.74	0.91
8	0.70	0.55	0.56	0.40	0.72
8	0.50	0.24	0.23	0.17	0.48
3	0.71	0.26	0.29	0.39	0.71
5	0.63	0.32	0.35	0.30	0.64
9	0.96	0.94	0.94	0.90	0.97
5	0.46	0.01	0.00	0.20	0.52
8	0.59	0.38	0.36	0.24	0.57
5	0.64	0.34	0.32	0.28	0.62
8	0.65	0.47	0.42	0.28	0.62
3	0.82	0.54	0.51	0.51	0.79
7	0.70	0.52	0.51	0.37	0.70
10	0.61	0.45	0.45	0.28	0.62
8	0.97	0.95	0.95	0.92	0.97
10	0.58	0.41	0.38	0.23	0.56
6	0.57	0.27	0.28	0.23	0.56
10	0.93	0.90	0.86	0.77	0.92
7	0.43	0.10	0.09	0.15	0.45
3	0.69	0.21	0.23	0.37	0.70
4	0.97	0.94	0.93	0.90	0.97

These results clearly support those in Table 2 for 2≤n≤50, Table 3 for 3≤n≤29, and Table 4 for 3≤n≤30. Again, the relationship between C_K and HHI is quite accurately described by (25), with R² = 0.986 for the fitted model ${\widehat{C}}_K=C_{KH}$ and the data in Table 5. Similarly, the normalized $C_K^{*}\in \left[0,1\right]$ and the normalized Euclidean distance d*∈[0, 1] show an impressive agreement, with R² = 0.993 for the fitted model ${\widehat{C}}_K^{*}=d^{*}$ and the data in Table 5.

This close approximation between $C_K^{*}\left(P_n\right)$ and $d^{*}\left(P_n,P_n^0\right)$ has important implications. First, it provides $C_K^{*}\left(P_n\right)$ with an interesting mathematical property: $C_K^{*}\left(P_n\right)$ becomes approximately a normalized distance metric (see, e.g., Chen et al. [63] for such metrics). This is of significance since the metric property is a strong one with far-reaching mathematical consequences. Second, this close approximation between $C_K^{*}\left(P_n\right)$ and $d^{*}\left(P_n,P_n^0\right)$ also means that C_K(P_n) is an approximately linear function of the standard deviation s_n of p₁,…,p_n (with devisor n). Specifically, it readily follows that

$$C_K\left(P_n\right)\approx \left[1-C_K\left(P_n^0\right)\right]\left(\frac{n}{\sqrt{n-1}}\right)s_n+C_K\left(P_n^0\right).$$ (27)

In the case of $P_n=P_n^{\lambda }$ in (10) and since C_K satisfies (11)–(13), the relationship in (27) becomes an equality as is also implied by (24). Therefore, because of the equality in (14) and from (24), the approximate relationship in (27) is not an unexpected one. It is important because it shows that, similarly to (21), C_K(P_n) is a function of both the number of firms in a market and their market-share variation as measured by the well-known standard deviation.

For the sake of completeness, the results for CR₄ are also given in Tables 3 and 4. Even though there is a substantial correlation between C_K and CR₄ (Pearson’s r = 0.97 and 0.92 and Spearman’s r_s = 0.89 and 0.95 for Tables 3 and 4, respectively), there are also substantial individual differences. Such differences between the values of C_K and CR₄ for individual data points seem to increase with increasing values of the two indices as in Fig 2. The HHI and CR₄ are also highly correlated with r = 0.89 and r_s = 0.91 and r = 0.81 and r_s = 0.96 for Tables 3 and 4, respectively. In spite of such high correlations, the different indices can certainly produce very conflicting results.

6. Discussion

6.1 Comments on (25)

The approximate, although quite accurate, relationship between C_K and HHI in (25) has important implications. First, with HHI becoming the increasingly dominant concentration index [7], reported HHI values can be converted into the corresponding C_K values for valid comparisons and interpretations of market concentrations.

Second, while the varying sensitivity of HHI to small changes in market-share distributions was discussed above based on (15) and on the particular distribution $P_n^{\lambda }$ in (10), this analysis can be done for any distribution P_n by using (25). With C_K and HHI being functions of P_n and taking the derivatives of both sides of (25) with respect to P_n gives the following approximate relationship:

$$\frac{dHHI\left(P_n\right)}{d{P}_n}\approx A\frac{d{C}_K\left(P_n\right)}{d{P}_n},A=3\left[HHI\left(P_n\right)+{\left(e^3-1\right)}^{-1}\right].$$ (28)

Compared with C_K, which has constant sensitivity or discriminant power because of its value-validity property (P5), that of HHI is seen from (28) to increase with increasing HHI-values. Since A = 1 for HHI = 0.32 in (28) and since C_K behaves appropriately, the implication is that HHI lacks adequate sensitivity when HHI<0.32 and has excessive and rapidly increasing sensitivity for HHI>0.32. This result is not inconsistent with the more general representation by the data in Fig 4.

6.2 C_K and economic theory

One of the major reasons for the popularity of HHI is its solid theoretical relationship with market power. Cowling and Waterson [64], for example, showed that for an oligopolistic market (industry) with profit-maximizing firms, the average price-cost margin is directly related to the HHI index. See also Martin [65, pp. 150–151, 337–338] and Carlton and Perloff [16, p. 283].

More specifically, the price-cost margin of Firm i is defined as PCM_i = (P_i−MC_i)/P_i where P_i is the market price set by the firm and MC_i is the firm’s marginal cost. The PCM_i, which is also the well-known Lerner index, is considered as a measure of a firm’s market power. Then, as shown by Cowling and Waterson [64] and under certain economic assumptions (including the assumption that P_i = P for all i), the weighted mean (PCM) of all the PCM_i‘s within a market or an industry, using the market shares as the weights $(\mathrm{i}.\mathrm{e}.,PCM={\displaystyle \sum _{i=1}^n{p}_{i}PC{M}_i})$, is related to HHI as follows:

$$PCM=HHI/\left|\eta \right|$$ (29)

where η is the market (industry) price elasticity of demand. The price elasticity of demand, which measures the variation in demand in response to a variation in price, is generally negative so that the right side in (29) may also be expressed as −HHI/η.

From (29), and by inverting the expression in (25), it follows that

$$PCM\approx \left(\frac{1}{\left|\eta \right|}\right)\left(\frac{e^{3C_K}-1}{e^3-1}\right).$$ (30)

That is, while the relationship between PCM and HHI is linear, the PCM is related to C_K in terms of an approximate exponential function. Since the close approximation in (25) has been empirically established, the approximation in (30) can be expected to be similarly close. Consequently, with PCM being a well-known measure of market power, the relation in (30) shows that increasing market concentration as measured by C_K results in increasing market power and hence decreasing competition and efficiency.

6.3 C_K and merger implications

The HHI is often computed from market shares as being percentages (rather than probabilities or proportions), which is the most typical form for reporting such data. Thus, the potential values of HHI range from 0 to 10,000 as is also used in the DOJ and FTC Merger Guidelines. For any HHI∈(0, 10,000], the corresponding values of C_K in percentage points can be obtained from (25) as

$$C_K\approx C_{KH}=\left(\frac{100}{3}\right)\log \left[\left(e^3-1\right)\left(\frac{HHI}{10,000}\right)+1\right]\in (0,100].$$ (31)

Based on (31), the most recent (2010) Horizontal Merger Guidelines may then be summarized in terms of C_K-values and changes ΔC_K as follows:

• Small Change in Concentration: Mergers involving an increase ΔC_K<2 percentage points are unlikely to have adverse competitive effects and ordinarily require no further analysis.

• Unconcentrated Markets (C_K<45): Mergers resulting in unconcentrated markets are unlikely to have adverse competitive effects and ordinarily require no further analysis.

• Moderately Concentrated Markets (45<C_K<58): Mergers resulting in moderately concentrated markets that involve ΔC_K>2 potentially raise significant competitive concerns and often warrant scrutiny.

• Highly Concentrated Markets (C_K>58): Mergers resulting in highly concentrated markets that involve 2<ΔC_K<3 potentially raise significant competitive concerns and often warrant scrutiny. Mergers resulting in highly concentrated markets that involve ΔC_K>3 will be assumed to likely enhance market power. The presumption may be rebutted by persuasive evidence showing that the merger is unlikely to enhance market power.

Similarly, the European Commission (EC) Merger Guidelines (Regulation) also uses the HHI index [13, p. 120] [66]. The EC guidelines may be expressed in terms of C_K in percentage points as follows:

• It is unlikely that a merger will cause horizontal competition concerns if (a) the post-merger C_K<36, (b) the post-merger C_K is between 36 and 52 and ΔC_K<4,or (c) the post-merger C_K>52 and ΔC_K<2, except in case of special given circumstances.

While these guidelines are directly transferred from those using HHI, it is important to note that since HHI lacks the value-validity property (P5), changes ΔHHI in HHI-values do not truly reflect changes in the extent of the concentration characteristic, but rather reflect changes in HHI values. As discussed above, the sensitivity of HHI to small changes in the market-share distribution P_n, rather than being constant as in the case of C_K, increases substantially with increasing HHI values. Thus, for example, an increase of ΔHHI = 100 at HHI = 1,500 and at HHI = 2,500, as used by the Merger Guidelines, cannot be considered as equivalent changes in concentration. In fact, the corresponding changes in C_K from (25), rather than being equal, are found to be 1.7 and 1.1, respectively.

Such invalid assumptions or interpretations of changes in HHI are of increasing importance, both theoretically and empirically, from studies emphasizing changes in concentration over its actual level. For example, Nocke and Whinston [8] show “that there is both a theoretical and empirical basis for focusing solely on the change in the Herfindahl index, and ignoring its level, in screening mergers for whether their unilateral effect will harm consumers (p.1).” Similarly, Miller et al. [67] used a Monte Carlo simulation study to evaluate price effects of mergers, finding a strong correlation between ΔHHI and the price change.

Whatever the purpose is of an investigation involving market concentration, whether theoretical or empirical, one proposition seems clear: changes in the values of a concentration index may be as important, if not more so, than its individual index values. Results that truly represent reality require an index with the value-validity property (P5). The C_K is claimed to be such an index.

7. Conclusion

Value validity is introduced as a property required of a concentration index in order for its numerical values and their changes to provide reliable and true representations of the market or industry concentration characteristic. Since other indices proposed to date, including the most popular HHI, lack this value-validity property (P5), the new concentration index C_K is introduced. Besides having this additional property, C_K shares properties (P1-P4) with other indices, including HHI.

One desirable property of any summary measure is its simplicity and meaningfulness. The C_K is certainly simple to compute and, as discussed, has meaningful interpretations. Computationally, C_K simply equals the largest market share (divided by 1), plus the second largest market share divided by 2, plus the third divided by 3, and so forth for all the n firms within the market or industry. The fact that the computation of C_K requires the market shares to be ordered from the largest to the smallest as in (1) is not a real disadvantage since market-share data are typically reported in that format.

From the definition of C_K in (17), it is clear that when the number of firms n is large, the weights (1/i) assigned to the different market shares (p_i) become sufficiently small such that the very small p_i‘s can effectively be ignored from the computation of C_K. This is really a practical advantage of C_K (as it also is with HHI) since market-share data are typically reported for the larger firms while the very small firms are grouped into an “others” category_.

Reported results from research involving market concentration seem to be increasingly based on the HHI. The empirically determined relationship in (25) can then conveniently be used to derive the corresponding values of C_K to a high degree of accuracy. Thus, valid concentration comparisons can be based on C_K.

Data Availability

All data underlying the findings of this manuscript are included in the manuscript.

Funding Statement

The author received no specific funding for this work.