The Dirichlet Model — The Math Behind How Brands Grow

Definition (≤25 words). The Dirichlet model is a statistical model of repeat buying that predicts how penetration, purchase frequency, and market share relate in stable consumer markets.

TL;DR. Most marketing theory is anecdote dressed up. The Dirichlet model is the rare exception: a piece of actual science, built by Goodhardt, Ehrenberg and Chatfield in 1984, that predicts how buyers distribute their purchases across competing brands in a stable market. It is the theoretical engine behind the patterns every modern evidence-led marketer leans on — double jeopardy, the duplication of purchase, and the primacy of market penetration. You don't need to do the maths to use it. But you should know it exists, because it is the reason "how brands grow" is a settled, predictive science rather than a collection of opinions.

What problem was the Dirichlet model built to solve?

Start with what empiricists like Andrew Ehrenberg noticed about buyer panel data from the 1960s onward: across wildly different categories — baked beans, gasoline, toothpaste, banking — buying behaviour kept producing the same regular patterns. The same statistical shapes turned up again and again, regardless of product, country, or decade. That regularity begged for an explanation, and ideally a predictive model.

The behaviours begging to be explained were:

• Penetration and purchase frequency move together. Brands with more buyers also have those buyers buying slightly more often — the pattern we now call double jeopardy. Small brands suffer twice: fewer buyers, and the buyers they have buy a bit less.
• Brands share customers in proportion to their size. A brand's customers overlap with another brand's customers roughly in line with that other brand's market share — the duplication of purchase law. There is little evidence of tight, exclusive "tribes" of loyalists.
• Buying is irregular but predictable in aggregate. Individual households buy erratically, but the distribution of purchase counts across households is stable and follows a known statistical shape (the negative binomial distribution).

The challenge Ehrenberg's group set itself was ambitious: build one model that generates all of these patterns at once, from a small number of plausible assumptions. The result was the Dirichlet model.

What are the model's core assumptions?

"Dirichlet" here is the Dirichlet distribution — a mathematical object for describing how a whole is split into shares. The model makes a small set of assumptions about how buyers and markets behave, and then derives the rest as consequences. The key assumptions are:

1. Buyers differ in how much they buy the category. Some households are heavy category buyers; most are light. This variation is modelled with a well-known distribution (the gamma), which produces the negative-binomial shape of purchase counts.
2. Within the category, each buyer has a stable probability of choosing each brand, and those choice probabilities vary across households according to a Dirichlet distribution. In plain terms: buyers have a personal "mix" of brands they buy, and that mix differs from household to household.
3. The market is approximately stationary — without strong trend, and without rigid sub-market structure. Brands are, in the model's language, broad substitutes competing "without structure."

That third assumption is the load-bearing one. It says: in a normal, stable consumer market, brands are close substitutes and buyers pick among them with probabilities shaped by availability and mental presence, not by deep, partitioned loyalty. When the model fits the data well — which it does, remarkably often — that assumption is vindicated.

What does the model predict, and how well?

This is the startling part. From those few assumptions, the Dirichlet model predicts, for any brand in a category:

• its penetration (share of category buyers who buy it),
• its average purchase frequency (how often those buyers buy it),
• its share of category requirements (the share of a buyer's category purchases it captures),
• and the overlap of its customer base with every other brand.

And it does so using little more than each brand's market share as input. The predicted values land very close to the observed panel data across a wide range of real categories. Patterns that look like free-floating empirical "laws" — double jeopardy, the duplication of purchase, the relationship between penetration and loyalty — all fall out of the model as necessary consequences of the assumptions, not separate discoveries.

The practical upshot is that the Dirichlet model gives marketers a benchmark for what "normal" looks like. If your brand's penetration and loyalty look like the model predicts for its size, you're behaving like a normal brand in a normal market — and growth must come from getting bigger (more penetration), not from impossible loyalty gains. If the data deviates from the model in a structured way, that deviation is a signal: maybe there's a genuine sub-market partition, a real quality difference, or an availability problem worth investigating.

Why is this the backbone of "how brands grow"?

Byron Sharp's How Brands Grow (2010) is, in essence, the Dirichlet model translated for practitioners. Nearly every counter-intuitive claim in the book — that loyalty is hard to grow, that penetration drives share, that brands compete as near-substitutes, that double jeopardy is a law not a theory — is a direct consequence of the model and the decades of panel data that validated it.

This is why the evidence-led camp talks with such confidence (and why it annoys the positioning-and-differentiation school). When you have a model that predicts market structure from first principles and fits the data across categories and countries, "it depends" and "every brand is unique" start to sound like evasions. The science says: in stable markets, brands grow by the same mechanisms. The job is to understand and exploit them.

What are the model's limits?

The Dirichlet model is powerful precisely because its assumptions are restrictive, so it's honest about where it breaks:

• It describes stable, mature markets. New categories, explosive growth phases, or markets in disruption don't sit still long enough for the stationary assumption to hold.
• It assumes near-substitutability. Markets with genuine, hard partitions (a budget tier vs. a luxury tier that barely share customers) will deviate — and that deviation is itself informative.
• It is descriptive and predictive of buying patterns, not a causal model of why a specific brand wins. It tells you the shape competition takes, not which advertising will break you out of it.
• It abstracts away marketing activity. The model explains the structure; mental availability, distinctive assets, reach, and pricing are the levers marketers pull to shift a brand's position within that structure.

None of these limits undermines the model's value. They define it. The Dirichlet model is the map of the terrain; strategy is how you move across it.

Where people get it wrong

• Dismissing it as "just a model." Yes, it's a model — one that has fit decades of real panel data across categories and countries. That's what good science looks like.
• Using it to argue that marketing "doesn't matter." The model describes the pattern of buying; it doesn't say you can't influence which brands buyers pick. Distinctiveness and availability are exactly how you shift your brand within the structure.
• Expecting it to predict a single brand's advertising ROI. It doesn't. It predicts market-level buying patterns, not campaign-level sales lift.
• Applying it to non-stationary markets as if it were gospel. In a brand-new or rapidly changing category, the stationary assumption fails; use the model as a rough reference, not a precision tool.
• Treating deviations as bad data. A deviation from the model is a finding — often the most interesting thing in the analysis, pointing to real structure or a real problem.

In practice — what to actually do

1. Benchmark your brand against the Dirichlet predictions for its market share: are your penetration, frequency, and loyalty where the model expects? If yes, stop chasing impossible loyalty and focus on penetration.
2. Investigate deviations seriously. If your loyalty is lower than predicted, ask whether it's a real availability, quality, or mental-availability problem — not a fluke.
3. Use it to set realistic goals. If the model says your loyalty ceiling is near where you are, reframe growth as a penetration target rather than a retention one.
4. Expect competitors to share your customers in proportion to their size. Plan reach and category entry points accordingly — you're fishing in a shared, porous pond.
5. Treat it as the scientific foundation, and layer your mental availability and reach strategy on top — the model explains the terrain; your job is to move across it.

So what?

The Dirichlet model is the unsung reason evidence-led marketing is a discipline rather than a collection of slogans. It is rare in this field — a predictive scientific model, repeatedly validated, that explains why brands compete the way they do. You don't need to derive it. You do need to respect what it implies: that growth has a structure, that loyalty is bounded, and that penetration is the lever. Fundamentals over fashion, with the maths to back it up.

Evidence & sources

• Goodhardt, G.J., Ehrenberg, A.S.C. & Chatfield, C. (1984). "The Dirichlet: A Comprehensive Model of Buying Behaviour." Journal of the Royal Statistical Society, Series A, 147(5), 621–655. — the foundational paper.
• Ehrenberg, A.S.C. (1988). Repeat-Buying: Facts, Theory and Applications. — the broader repeat-buying tradition the model rests on.
• Uncles, M.D., Ehrenberg, A.S.C. & Hammond, K. (1995). "Patterns in buyer behaviour: just how regular are they?" — empirical validation of the model's regularities.
• Sharp, B. (2010). How Brands Grow. Oxford University Press — the practitioner translation of the Dirichlet model's implications.
• Romaniuk, J. & Sharp, B. (2016). How Brands Grow 2. Oxford University Press — mental availability and distinctive assets as the marketing levers that move brands within the model's structure.

Related terms

• Double Jeopardy Law
• Market Penetration
• Mental Availability
• Distinctive Assets

Definition type: DefinedTerm. Reviewed: 2026-07-09. Written by Andrei Tarnovski for Boringly Effective. Fundamentals over fashion.