Herfindahl–Hirschman index

Based on Wikipedia: Herfindahl–Hirschman index

The Number That Decides Whether Mergers Happen

Somewhere in Washington, a team of lawyers and economists is running calculations. Two massive corporations want to combine, and billions of dollars hang on a single number. If it comes out too high, the deal dies. Too low, and the merger sails through with barely a second glance.

That number is the Herfindahl-Hirschman Index, or HHI for short. It's one of the most consequential metrics in modern capitalism—a deceptively simple formula that serves as the first line of defense against monopoly power.

A Tale of Two Economists

The index carries the names of two economists who developed it independently: Orris Herfindahl and Albert Hirschman. Hirschman, in particular, led a fascinating life. Born in Berlin in 1915, he fled Nazi Germany, fought against Franco in the Spanish Civil War, and helped rescue thousands of refugees from Vichy France before eventually becoming one of the twentieth century's most influential development economists.

The tool he helped create has become standard equipment for antitrust regulators worldwide. Every time the Department of Justice or the Federal Trade Commission evaluates whether a corporate merger might harm consumers, the HHI is among the first things they calculate.

The Math Is Surprisingly Simple

Here's how it works. Take every company in a market and calculate its market share as a decimal. Square each share. Add them all together.

That's it.

Consider a market with five companies, each controlling twenty percent. You'd calculate it as: 0.20 squared plus 0.20 squared plus 0.20 squared plus 0.20 squared plus 0.20 squared. That gives you 0.04 five times over, for a total HHI of 0.20, or 2,000 points when expressed on the ten-thousand-point scale that regulators often prefer.

Why square the market shares rather than just adding them up? The squaring is the crucial insight. It gives more weight to larger players. A company with forty percent of the market contributes far more to the index than two companies with twenty percent each, even though the raw numbers add to the same total. This captures something intuitive about market power: dominance by a few giants is different from fragmentation among many smaller players.

What the Numbers Mean

The HHI ranges from nearly zero to one (or from zero to ten thousand on the points scale). At the extremes, the interpretation is clear.

An HHI below 0.01—that's under 100 points—indicates a fiercely competitive industry. Think of something like local restaurants or small retail shops. Hundreds or thousands of businesses, none with meaningful market power.

An HHI of 1.0, or ten thousand points, means pure monopoly. One firm controls everything.

The interesting territory lies between. American antitrust authorities consider markets with an HHI below 0.15 (1,500 points) to be "unconcentrated." Mergers in these markets rarely raise concerns. Between 0.15 and 0.25 (1,500 to 2,500 points), regulators start paying attention—this is "moderately concentrated" territory. Above 0.25 (2,500 points), the market is considered "highly concentrated," and proposed mergers face intense scrutiny.

But these aren't rigid cutoffs. The change in HHI matters as much as the absolute level. A merger that increases the index by more than 0.01 (100 points) generally triggers a closer look, especially in already-concentrated markets.

A Merger in Action

Let's watch the HHI at work. Imagine a market with three firms. Two of them each control forty percent, and the third holds twenty percent.

Before any merger, the HHI is 0.40 squared plus 0.40 squared plus 0.20 squared. That's 0.16 plus 0.16 plus 0.04, giving us 0.36, or 3,600 points. High, but not catastrophically so.

Now suppose the two larger firms decide to merge. Suddenly the calculation becomes 0.80 squared plus 0.20 squared. That's 0.64 plus 0.04, for a total of 0.68, or 6,800 points.

The HHI nearly doubled. One company now controls eighty percent of the market. This is approaching monopoly territory, and any antitrust regulator would likely move to block such a deal.

The Equivalent Number of Firms

There's an elegant mathematical property hidden in the HHI. Take its reciprocal—that is, divide one by the HHI—and you get the "equivalent number of firms."

This tells you how many equal-sized companies would produce the same level of concentration. A market with an HHI of 0.20 has an equivalent number of five firms. An HHI of 0.25 corresponds to four equivalent firms. An HHI of 0.50 means the market behaves as if it had only two equal competitors.

This interpretation makes the HHI more intuitive. When someone says a market has an HHI of 0.33, you can immediately think "that's like having three equal-sized companies competing."

The Problem of Defining Markets

The HHI's power depends entirely on one thorny question: what counts as a "market"?

Get the definition wrong, and the index becomes meaningless or misleading.

Consider financial services. Suppose you calculate the HHI for the entire industry and find six major firms with roughly fifteen percent each. Looks competitive. But what if one of those firms controls ninety percent of all checking accounts and physical bank branches, while the others focus on investment banking and commercial lending? Regular consumers face a near-monopoly for their daily banking needs, even though the industry-wide number suggests healthy competition.

The problem cuts both ways. A single movie theater might dominate its local market with ninety percent of ticket sales. But if consumers see video streaming, bars, concerts, and home entertainment as alternatives to going to the movies, that theater's apparent dominance overstates its actual market power.

Geography adds another layer of complexity. Five companies might each hold twenty percent of a national market, but if each one operates as a regional monopoly—one dominates the Northeast, another the Southwest, and so on—consumers in any given area face no real choice. The HHI would miss this entirely.

The European Approach

American and European regulators use the HHI somewhat differently. United States authorities focus heavily on absolute thresholds: below 1,500 points is fine, above 2,500 is concerning.

European regulators pay more attention to the change in concentration. They worry when a merger increases the HHI by more than 0.025 (250 points) in a market that's already concentrated above 0.10 (1,000 points). This approach catches mergers that might push a moderately concentrated market toward dangerous territory, even if the final number doesn't cross American thresholds.

A Normalized Version

Economists sometimes use a "normalized" version of the HHI that always runs from zero to one, regardless of how many firms exist in the market. The regular HHI's minimum value depends on the number of competitors—with ten firms, the lowest possible HHI is 0.10, not zero.

The normalized version adjusts for this, making it easier to compare concentration levels across markets with different numbers of players. But this convenience comes at a cost: you lose information about how many competitors actually exist. A normalized HHI of zero could mean two equal firms or twenty equal firms—very different competitive situations.

The Same Formula Appears Everywhere

One of the most remarkable things about the HHI is that the identical mathematical formula shows up across completely unrelated fields, discovered independently each time.

Ecologists use it as the Simpson diversity index to measure species diversity in ecosystems. A forest with many equally common species has a low index; one dominated by a few abundant species has a high index. The math is exactly the same—substitute "species abundance" for "market share."

Physicists encounter it as the "inverse participation ratio," which measures how localized a quantum state is. Political scientists use the reciprocal to calculate the "effective number of parties" in a legislature.

This convergence isn't coincidence. All these situations involve measuring how concentrated or dispersed something is—whether market power, species abundance, quantum probability, or political representation. The squaring-and-summing formula turns out to be a natural way to capture this concept mathematically.

Beyond the Numbers

The HHI is a screening tool, not a verdict. A high score triggers investigation, but it doesn't automatically doom a merger.

Regulators consider many other factors. Will the merger create efficiencies that benefit consumers? Are there barriers preventing new competitors from entering? Does one of the merging companies appear likely to fail anyway? Are there other forces, like powerful buyers or rapid technological change, that limit market power?

The index also can't capture everything important about competition. It says nothing about whether firms compete aggressively on price or tacitly collude to keep prices high. It ignores the threat of potential competitors waiting in the wings. It doesn't account for how quickly market dynamics might change.

The Connection to Economic Theory

The HHI isn't just an arbitrary metric that happened to catch on. Economists have shown that it emerges naturally from theoretical models of how firms compete.

In what's called Cournot competition—named after nineteenth-century French mathematician Antoine Augustin Cournot—firms choose how much to produce, and the market price adjusts based on total output. When you work through the mathematics of this model, the HHI appears as a key term determining how much profit firms can extract and how much consumers suffer from reduced competition.

This theoretical foundation gives the HHI credibility beyond its empirical usefulness. It's not just a convenient summary statistic; it captures something fundamental about market structure.

A Guardrail for Capitalism

Competition is often described as the engine of capitalism—the force that drives innovation, keeps prices low, and ensures that companies serve consumers rather than exploiting them. But competition tends to erode itself. Successful firms grow larger. Larger firms acquire competitors. Industries consolidate.

The HHI serves as a quantitative guardrail against this tendency. It gives regulators a systematic way to identify when concentration becomes concerning, transforming subjective judgments about market power into concrete, comparable numbers.

It's far from perfect. The difficulties of defining markets, the inability to capture dynamic competition, the risk of gaming through creative market definitions—all these limitations are real. But having an imperfect measure is better than having none at all.

When two corporations announce a major merger and antitrust authorities respond within days with concerns about market concentration, the HHI is almost certainly involved. That quick calculation—square the shares, sum the squares—provides the first glimpse of whether competition is about to get a little less competitive.