A Buyer-Based Measure of Seller Concentration

Abstract

In most markets, buyers differ in their ability or willingness to switch supplier. This note proposes a novel industry concentration measure that takes this heterogeneity into account. The index increases in the share of captive sales, coincides with the Hirschman–Herfindahl Index when none of the buyers are captive, and takes the “pure monopoly” value of 1 when all are captive.

1. Introduction

“The purpose of a concentration ratio presumably is to show the likelihood of monopolistic policies in an industry.”

([1] Herfindahl, 1950, p. 15)

Imagine a market with eight equally sized firms that face demand from consumers, none of whom is captive. To describe this market’s structure, one may employ the popular seller concentration measures, the “4-firm Concentration Ratio” (CR₄) and the “Hirschman–Herfindahl Index”(HHI). The first is 50%, which tells us that the four largest sellers have a combined market share of 50%. The second is 0.125, which characterizes this market as competitive ([2] p. 163). Now, imagine the same market, but with one important difference: all buyers are captive. If one employs the same measures to describe this market’s structure, then one would find precisely the same numbers: 50% and 0.125, respectively.

The CR₄ and the HHI are structural measures of seller concentration, both of which are frequently employed to proxy the intensity of competition in a market (See [3] for a recent, detailed discussion). As the above stylized example illustrates, however, they do not take account of the robustness (or “contestability”) of market shares. In the first case, even a marginal change in the strategy of a firm (e.g., a modest price cut) may significantly affect the size distribution of firms. Yet, this effect is negligible in the second case. Given, then, that such concentration indices are intended to measure the intensity of competition in a market, one would like them to somehow take account of the (in)stability of market shares.

The purpose of this note is to propose a seller concentration measure that does precisely that. Specifically, we present a simple market concentration index (coined the $\mathcal{C}$-index) that captures the degree of contestability of market shares. This index is ceteris paribus increasing in firms’ captive customer bases. It coincides with the standard HHI when market shares are fully contestable (i.e., when there are no captive customers) and equals 1 when all buyers are captive. The latter case is characterized as a “pure monopoly”, since firms, although operating in one and the same market, are not actually competing.

As an illustrative example, consider the market for luxury cars. In this market, the German car brands BMW and Mercedes are notable competitors. If, say, Mercedes offers a significant discount on its most popular model, then it may well attract part of BMW’s clientele. Yet, a share of BMW’s buyers would not switch (i.e., BMW’s captive consumers). For example, BMW has an existing fan base that simply refuses to drive a Mercedes (or any other car brand) no matter what the circumstances. Furthermore, some consumers may have signed long-term contracts with BMW (e.g., taxi companies, driving schools, etc). The classic concentration measures CR_n and HHI are, however, based on total market shares (captive and non-captive together). Ideally, then, a market concentration measure takes account of the ratio captive/non-captive if it is to properly describe the intensity of competition.

The question of how best to measure market concentration has a long tradition, and many concentration indices have been proposed over the last century. The most prominent ones take account of both the number of firms and their size (-distribution). Notable examples include the aforementioned CR_n and HHI, as well as different entropy and Lorentz-based measures of industry concentration (See, amongst many others [4,5,6]). Ref. [7] provides a synthesis of different concentration ratios that are based on both the number of firms and their size distribution. More recently, several authors suggested measures of competition that focus directly on firm performance rather than industry structure. The most well-known ones assess how changes in factor prices affect a firm’s profits, which sheds light on how competitive the market under consideration is ([8,9,10]). However, none of these concentration indices is “buyer-based” in the sense of incorporating the (non-)captive status of consumers.

The next section presents the $\mathcal{C}$-index. Section 3 relates this measure to market structure and the degree of competition. Section 4 concludes.

2. The $\mathcal{C}$-Index

We propose a new “buyer-based” measure of competition that takes into account the robustness of market shares (which we coin the $\mathcal{C}$-index). Specifically, this concentration index takes a higher value the larger the share of captive consumers, all else equal.

Consider an industry comprising a fixed and finite set of firms $N=\left\{1,\dots ,n\right\}$. At given prices, firm i$\in N$ has a market share $s_i=\frac{q_i}{Q}$, where $q_i\ge 0$ is its level of supply and $Q={\sum }_{i=1}^n{q}_i$ is the total industry output within a given period. It is assumed that part of firm i’s market share is contestable, denoted by $c_i=\frac{y_i}{Q}$, whereas the complementary part is non-contestable, indicated with $b_i=\frac{x_i}{Q}$. Consumers may be (temporarily) captive because they are brand-loyal or have signed long-term contracts with their supplier, for example. Hence, assuming that the market clears, $s_i=c_i+b_i$ and $q_i=y_i+x_i$ or $q_i=c_{i}Q+b_{i}Q$. Thus, $y_i=c_{i}Q$ is firm i’s contestable demand and $x_i=b_{i}Q$ is its non-contestable demand. Finally, it holds that $Q={\sum }_{i=1}^n{q}_i={\sum }_{i=1}^n{y}_i+{\sum }_{i=1}^n{x}_i$ and ${\sum }_{i=1}^n{s}_i=1$.

Note that the industry effectively consists of $n+1$ distinct market segments: n captive segments and 1 non-captive segment. To describe the degree of competition in such a market, we propose the following measure:

$$\mathcal{C}=\sum _{i=1}^n\left(\frac{c_i^2}{{\displaystyle \sum _{j\in N}}{c}_j}+b_i\right).$$

(1)

For each firm $i\in N$, the $\mathcal{C}$-index takes the sum of the contestable and non-contestable market share weighted by firm i’s shares in the respective segments ($y_i/{\displaystyle \sum _{j\in N}}{y}_j$ and 1, respectively). To see this more clearly, one can rewrite (1) as:

$$\mathcal{C}=\sum _{i=1}^n\left(c_i·\frac{y_i}{{\displaystyle \sum _{j\in N}}{y}_j}+b_i·1\right).$$

It is easy to see that the $\mathcal{C}$-index can be expressed exclusively in terms of contestable shares as follows.

$$\begin{array}{ccc}\hfill \mathcal{C}& =& \sum _{i=1}^n\left(\frac{c_i^2}{{\displaystyle \sum _{j\in N}}{c}_j}+b_i\right)=\sum _{i=1}^n\left(\frac{c_i^2}{{\displaystyle \sum _{j\in N}}{c}_j}\right)+\sum _{i=1}^n{b}_i\hfill \\ & =& \sum _{i=1}^n\left(\frac{c_i^2}{{\displaystyle \sum _{j\in N}}{c}_j}\right)+\sum _{i=1}^n\left(s_i-c_i\right)=1-\sum _{i=1}^n{c}_i·\left(1-\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}\right).\hfill \end{array}$$

Thus, if all customers are captive $\left(c_i=0,\forall i\in N\right)$, then $\mathcal{C}=1$. Such a situation effectively mimics a pure monopoly. The other extreme has only non-captive consumers $\left(c_i=s_i,\forall i\in N\right)$ so that:

$$\mathcal{C}=1-\sum _{i=1}^n{c}_i·\left(1-\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}\right)=1-\sum _{i=1}^n{s}_i·\left(1-\frac{s_i}{{\displaystyle \sum _{j\in N}}{s}_j}\right)=\sum _{i=1}^n{s}_i^2,$$

which is equivalent to the standard Hirschman–Herfindahl Index (HHI) based on total market shares. This means that $\mathcal{C}\in \left[HHI,1\right]$. That is, the $\mathcal{C}$-index is between the traditional HHI (maximal competition) and 1 (no competition). Moreover, for a given market share allocation, it is monotonically increasing in a firm i’s captive customer base $b_i$. To see this, fix $s_i$, $\forall i\in N$. Taking the first-derivative with respect to $b_i$ gives:

$$\frac{\partial \mathcal{C}}{\partial b_i}={\left(\frac{-\left({\displaystyle \sum _{j\in N\backslash \left\{i\right\}}}{c}_j\right)}{{\displaystyle \sum _{j\in N}}{c}_j}\right)}^2>0.$$

In conclusion, the proposed concentration measure takes into account the possibility that sellers may effectively compete for a subset of buyers only. The $\mathcal{C}$-index takes this non-captive segment as a basis. The resulting HHI value is then corrected upward to incorporate the monopolistic (captive) part of the market.

3. Market Structure and Competition

Let us now relate the $\mathcal{C}$-index to market structure and the degree of competition. To that end, we consider a price-setting oligopoly in which suppliers choose prices simultaneously to maximize their profits.

Suppose that each firm selects a single price. The profits of firm $i\in N$ are then given by:

$${\pi }_i\left(p_i,p_{-i}\right)=p_i\left(y_i\left(p_i,p_{-i}\right)+x_i\left(p_i\right)\right)-C_i\left(q_i\left(p_i,p_{-i}\right)\right),$$

(2)

where $p_i\in {\mathbb{R}}_{+}$ is its own price and $p_{-i}\equiv \left(p_{1,}\dots ,p_{i-1},p_{i+1},\dots ,p_n\right)$ is the vector of its competitors’ prices. Note that $x_i$, since it is firm i’s non-contestable output, is not sensitive to its rivals’ prices. In what follows, it is assumed that both $y_i$ and $x_i$ are continuous, differentiable, and decreasing functions of the own price $p_i$. Let the absolute value of the price elasticity of demand for captive and non-captive customers be given by ${\eta }_i^b\left(p_i\right)=\left|x_i^{\prime }·\frac{p_i}{x_i}\right|$ and ${\eta }_i^c\left(p_i\right)=\left|y_i^{\prime }·\frac{p_i}{y_i}\right|$, respectively. Since captive customers are ceteris paribus less price sensitive than non-captive customers, it holds that ${\eta }_i^b\left(p_i\right)<{\eta }_i^c\left(p_i\right)$ at all $p_i$.

We make the following assumptions on demand and cost. In stating these assumptions, let $m_i$ be the firm’s marginal cost at zero output.

For all $i\in N$ and $p_{-i}$:

A1 

$C_i\left(0\right)=0$ and $C_i\left(q_i\right)$ is twice continuously differentiable with $\frac{d{C}_i\left(q_i\right)}{d{q}_i}\ge 0$ and $\frac{d^2{C}_i\left(q_i\right)}{d{q}_i^2}\ge 0$.

A2 

$q_i(p_i,p_{-i})$ is twice continuously differentiable with $\frac{\partial q_i(p_i,p_{-i})}{\partial p_i}<0$ and $\frac{{\partial }^2{q}_i(p_i,p_{-i})}{\partial p_i^2}\le 0$.

A3 

$q_i\left(m_i\right)>0$.

By A1, all fixed costs are taken to be sunk. This assumption is made for analytical convenience. Alternatively, one could assume part of the fixed costs to be avoidable (i.e., not sunk) and sufficiently scale up all values to ensure that being productive is profitable. Furthermore, firms’ production technologies exhibit constant or decreasing returns. A2 implies that products are less-than-perfect substitutes. Finally, A3 (in combination with A1 and A2) ensures firms have an incentive to be productive.

Notice that, under these assumptions, the objective function is strictly concave within the relevant range (i.e., the range of outputs for which the price weakly exceeds marginal costs).

$$\frac{{\partial }^2{\pi }_i\left(p_i,p_{-i}\right)}{\partial p_i^2}{\mid }_{p_i\ge \frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}}=$$

$$2\frac{\partial q_i\left(p_i,p_{-i}\right)}{\partial p_i}+\frac{{\partial }^2{q}_i\left(p_i,p_{-i}\right)}{\partial p_i^2}·\left(p_i-\frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}\right)-\frac{d\left(\frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}\right)}{d{q}_i}·{\left(\frac{\partial q_i\left(p_i,p_{-i}\right)}{\partial p_i}\right)}^2<0.$$

Therefore, the first-order conditions suffice to define the Bertrand price equilibrium:

$$\frac{\partial {\pi }_i\left(p_i,p_{-i}\right)}{\partial p_i}=q_i\left(p_i,p_{-i}\right)+\frac{\partial q_i\left(p_i,p_{-i}\right)}{\partial p_i}·\left(p_i-\frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}\right)=0,\forall i\in N.$$

Next, let $L_i$ be firm i’s Lerner index:

$$L_i=\frac{p_i-\frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}}{p_i},\forall i\in N.$$

Akin to the classic Lerner index, $L_i$ is a measure of market power on firm i’s contingent demand curve. The following result relates the individual Lerner index to the price elasticity of (non-)captive demand.

Proposition 1.

Assume the Bertrand price equilibrium. For all $i\in N$, it holds that:

$$L_i=\frac{s_i}{{\eta }_i^c{c}_i+{\eta }_i^b{b}_i}.$$

Proof. 

Consider the first-order condition for some firm $i\in N$:

$$q_i\left(p_i,p_{-i}\right)+\frac{\partial q_i\left(p_i,p_{-i}\right)}{\partial p_i}·\left(p_i-\frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}\right)=0.$$

This condition can be rearranged as follows:

$$\begin{array}{ccc}\hfill y_i\left(p_i,p_{-i}\right)+x_i\left(p_i\right)+\left(\frac{\partial y_i\left(p_i,p_{-i}\right)}{\partial p_i}+\frac{\partial x_i\left(p_i\right)}{\partial p_i}\right)·p_i& =& \frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}·\frac{\partial q_i\left(p_i,p_{-i}\right)}{\partial p_i}\hfill \\ & ⟺& \\ \hfill -\left(\frac{\partial y_i\left(p_i,p_{-i}\right)}{\partial p_i}+\frac{\partial x_i\left(p_i\right)}{\partial p_i}\right)·\left(p_i-\frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}\right)& =& y_i\left(p_i,p_{-i}\right)+x_i\left(p_i\right)\hfill \\ & ⟺& \\ \hfill \frac{p_i-\frac{d{C}_i\left(q_i\left(p_i,p_{-i}\right)\right)}{d{q}_i}}{p_i}& =& \frac{y_i\left(p_i,p_{-i}\right)+x_i\left(p_i\right)}{-\left(\frac{\partial y_i\left(p_i,p_{-i}\right)}{\partial p_i}+\frac{\partial x_i\left(p_i\right)}{\partial p_i}\right)p_i}\hfill \\ & ⟺& \\ \hfill L_i& =& \frac{y_i\left(p_i,p_{-i}\right)+x_i\left(p_i\right)}{{\eta }_i^c{y}_i+{\eta }_i^b{x}_i}\hfill \\ & ⟺& \\ \hfill L_i& =& \frac{s_i}{{\eta }_i^c{c}_i+{\eta }_i^b{b}_i},\hfill \end{array}$$

which is the equilibrium value of the individual Lerner index. □

Observe that without the captive segment (i.e., $b_i=0$), the measure reduces to $L_i=\frac{1}{{\eta }_i}$, where ${\eta }_i=-\frac{\partial q_i\left(p_i,p_{-i}\right)}{\partial p_i}·\frac{p_i}{q_i}$ is the absolute value of the standard price elasticity of demand. Note further that $L_i$ is monotonically increasing in the size of the captive customer base (for a given market share $s_i$) since ${\eta }_i^c>{\eta }_i^b$. Moreover, $L_i$ reaches its maximum at $\frac{1}{{\eta }_i^b}$ when all customers are captive (i.e., $s_i=b_i$). In sum, and all else equal, firm i’s equilibrium price positively depends on its market share $s_i$ and, for a given market share, on its captive market share $b_i$.

Finally, the next result relates the $\mathcal{C}$-index to market structure in equilibrium.

Proposition 2.

Assume the Bertrand price equilibrium. The $\mathcal{C}$-index is given by:

$$\mathcal{C}=\sum _{i=1}^n\left(\left({\eta }_i^c{c}_i+{\eta }_i^b{b}_i\right)·\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}·L_i+b_i·\left(1-\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}\right)\right).$$

Proof. 

By Proposition 1, it holds that:

$$L_i=\frac{s_i}{{\eta }_i^c{c}_i+{\eta }_i^b{b}_i},$$

which can be written as:

$$L_i=\frac{\frac{c_i^2}{{\displaystyle \sum _{j\in N}}{c}_j}+b_i-b_i·\left(1-\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}\right)}{\left({\eta }_i^c{c}_i+{\eta }_i^b{b}_i\right)·\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}},$$

which is equivalent to:

$$\left({\eta }_i^c{c}_i+{\eta }_i^b{b}_i\right)·\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}·L_i+b_i·\left(1-\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}\right)=\frac{c_i^2}{{\displaystyle \sum _{j\in N}}{c}_j}+b_i.$$

Hence, the $\mathcal{C}$-index can be written as:

$$\mathcal{C}=\sum _{i=1}^n\left(\left({\eta }_i^c{c}_i+{\eta }_i^b{b}_i\right)·\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}·L_i+b_i·\left(1-\frac{c_i}{{\displaystyle \sum _{j\in N}}{c}_j}\right)\right),$$

which gives the equilibrium relation. □

As before, one can easily verify that $\mathcal{C}={\sum }_{i=1}^n{b}_i=1$ when $c_i=0$ and $\mathcal{C}={\sum }_{i=1}^n{c}_i^2={\sum }_{i=1}^n{s}_i^2$ when $b_i=0$, for all $i\in N$.

4. Conclusions

In a recent report on methodologies to measure market competition, the OECD [3] (2021, p. 7) states that:

“Individually, each measure of competition provides only limited information, but together they can provide useful information to help build a better understanding of the competition dynamics at play. Therefore, the use of a plurality of measures is necessary when analysing the intensity of competition.”

The proposed $\mathcal{C}$-index is no exception and by itself incapable of providing a definite conclusion about the nature of competition in a particular market. Yet, by explicitly taking into account the eagerness of buyers to switch seller, it arguably gives a more accurate picture of competition intensity in an industry than traditional concentration indices. And that, to paraphrase Herfindahl (1950, p. 15), is presumably the purpose of any market concentration measure.