Decomposing Herfindahl–Hirschman (HHI) Index
Herfindahl–Hirschman (HHI) Index is a well-known market concentration measure determined by two factors:
- the size distribution (variance) of firms, and
- the number of firms.
Intuitively, having a hundred similar-sized gas stations in town means a far less concentrated environment than just one or two available, and when the number of firms is constant, their size distribution or variance determines the magnitude of market concentration.
Since these two properties jointly determine the HHI measure of concentration, naturally we want a decomposition of HHI that can reflects these two dimensions respectively. This is particularly useful when two distinct markets have the same level of HHI measure, but the concentration may result from different sources. Note that here these two markets do not necessarily have to be industry A versus industry B, but can be the same industry niche in two geographical areas, for example.
Thus, we can think of HHI as the sum of the actual market state’s deviation from 1) all firms having the same size, and the deviation from 2) a fully competitive environment with infinite number of firms in the market. Some simple math can solve our problem.
Some math
Let’s say in a market ther are
In the first scenario where all firms’ sizes are equal, we can describe it with:
where
The Euclidean distance between the point
For the ease of discussion, let’s consider the other spectrum of the second scenario where there’s only one firm in the market instead of infinite firms, assuming its size is the sum of all firms in the first scenario (i.e. its size is
Hence, the distance of any market state
Thus we can derive a relative index of concentration (when
Now, given the definition of Herfindahl-Hirschman Index
we can get:
Here comes the important implications. Recall that
When we observe a market state
Further,
The graph below shows that
where
We mentioned before that
This decomposition is appealing also in that
Thus, it’s safe to say this decomposition produces two components explaining the observed market concentration, 1)
Another finding from the graph is that with higher market concentration measured by
When
is small, most of the concentration is resulted from as highlighted below, which means the number of firms has a greater impact on market concentration.When
is larger, on the other hand, contributes more to , which means the firm size inequality plays a bigger role in market concentration.
A potential implication for regulators who are concerned about market concentration, I think, is to 1) focus more on reducing the entry barrier if the current concentration level is moderate, and to 2) focus more on antitrust if the concentration level is already high.
Another implication for researchers is that even though
Acknowledgement
This post is largely a replicate of the paper “A Decomposition of the Herfindahl Index of Concentration” by Giacomo de Gioia in 2017.