One formula, several different objects
Gabaix’s review considers empirical power laws in economics and finance and different mechanisms that may produce them, including random growth and superstar effects. It also identifies open explanatory questions. This is a useful source of research hypotheses, not proof that every group of large companies or every price series follows one unchanging law.[1]
A constructed tail with an explicit threshold
Set an educational model: for sizes no smaller than ten, the probability of exceeding x is the square of ten divided by x. The probability of exceeding twenty is 25%, forty is 6.25% and eighty is 1.5625%. Doubling the threshold reduces the probability fourfold. This follows from the deliberately selected exponent two and is not an estimate of real companies or markets.
The formula applies only above the lower boundary of ten. It cannot automatically be extended to every positive size. Likewise, an exceedance probability should not be labelled density at a point. If a chart counts objects in intervals of different widths, disclose the normalisation; otherwise the visible shape may reflect binning rather than the underlying distribution.
What a rank chart means
After sorting companies by size, plot size against list position. If the first positions greatly exceed the rest, concentration appears. But a ranking describes a sample on a particular date. It does not prove that today’s leader remains the leader or that buying the largest object produces higher returns. Business size, the price of its security and investment performance are different variables.
A simple fictional set of sizes is 50, 25, 15 and 10. The sum is 100 and the largest share is one half. A concentration index computed as the sum of squared shares is 0.345. This describes the specified set, but does not establish a power law: four observations are plainly insufficient, and a concentration index does not select a distributional form.
Why a straight line on a log scale does not finish the test
A logarithmic display is convenient for large ranges and ratios. It can make some differences less visible, especially with few large observations. First check the raw data, sample boundary, missing values and range over which the model is proposed. Then compare alternatives, such as another heavy-tailed distribution or a mixture of groups. Form selection should not depend only on visual impression.
When estimating, show parameter uncertainty, sensitivity to the lower threshold and the composition of observations left beyond the threshold. If the conclusion changes after removing one largest observation, that is an important result, not a reason to delete it silently. Large observations may define the tail. A narrow interval chosen after viewing the graph can create an illusion of stable scaling.
How extrapolation goes wrong
One error is transferring a cross-sectional relationship between different companies’ sizes into the time path of one company. Even if large and small objects form a particular cross-sectional distribution, it does not define an individual trajectory. Likewise, industry leadership does not establish expected share returns: the price may already reflect optimistic expectations, while future competition changes outcomes.
A second error is treating a power-law tail as permission to ignore risk measurement. Rare large observations make scale and scenario checks more important. The connection with [heavy tails and stress testing](/guides/fat-tails-stress/) concerns the distribution of possible outcomes, not a precise date forecast for an extreme event. For a portfolio, separately check [diversification](/guides/diversification/), not only the number of names.
A testable conclusion
Name the variable, whether the data are cross-sectional or time-series, the lower threshold, the estimation method and alternative models. Separate observed concentration from a proposed mechanism and from an investment decision. A sound formulation may be narrow: “On this range, this model approximately describes these observations.” It is more useful than the grand claim that a “law explains the market,” because its boundaries and possible refutation can be tested.
Do not mix frequency and size
Clarify whether the chart shows the number of objects of a given size or the share of total capital represented by them. These distributions answer different questions and use different weights. A few largest objects may be a small fraction by count but a large fraction by value. Therefore, [power-law distribution](/glossary/stepennoe-raspredelenie/) requires defining the observation unit before choosing a logarithmic scale.
Sources and verification limits
[1] Power Laws in Economics and Finance, NBER working paper 14299. Review of empirical power-law distributions and possible mechanisms.
Zipf: an explicit size-to-rank relationship
In a rank model, the size at position r is S(r)=C/r^β. The special case β=1 is commonly called Zipf’s law; equality is a model approximation over a stated range, not an identity for every real ranking. Gabaix’s review distinguishes power-law patterns from the mechanisms that generate them; naming the law does not establish the growth mechanism of an individual company.[RPL1]
Independent example: C=120 and β=1 produce 120, 60, 40 and 30 for four ranks. Their sum is 250 and the first share is 48%. The second rank is half the first, and the fourth is half the second. On logarithmic coordinates, log S=log C−β log r, so the slope is −1. Adding a fifth rank of 24 changes the concentration denominator; the size law does not fix the leader’s share without the number of objects and ranking boundaries.
The connection between rank and tail also requires care. If, for many objects, the share with size at least x is approximately (x_min/x)^α, then rank r is roughly proportional to x^−α. Solving for x gives size proportional to r^−1/α; β≈1/α. For the continuous density, the exponent is already α+1. Therefore, an exponent of 1 on a rank plot cannot be inserted as the density exponent. Ranks depend on the sample, and the approximation is especially rough near the top rank. Parameter estimation and alternative checks remain necessary.
Biological allometry and a resource pyramid are different models
The work of West, Brown and Enquist derives biological scaling from a distribution-network model with specific constraints: space filling, conditions on terminal vessels and transport costs. This is a separate construction, not a consequence of sorting objects by rank.[RPL2]
Consider a limited calculation without presenting it as proof of a biological law. Suppose resource use by one organism is B(M)=aM^3/4, where M is mass and a is given. Increasing mass sixteenfold increases use eightfold. Use per unit mass B/M=aM^−1/4 falls by one half. With a fixed available resource flow E, the maximum supported number of identical organisms N=E/B(M) falls eightfold. The equality requires complete use of one resource and the same environment; in nature, available flow, temperature, age and competition need not match.
A trophic pyramid uses another relationship. If 10,000 notional energy units pass to the next level at 10% in our model, levels receive 10,000, 1,000 and 100. At a need of 1, 10 and 100 units per organism, calculated counts are 10,000, 100 and 1. The percentage and needs are invented, not measured for elephants, ants or firms. Reduced resource and higher need explain the direction only within this construction; they do not derive a specific rank-distribution exponent.
Economic niche: a testable boundary for transfer
A business niche is a combination of customer need, technology, access channel and resources under which serving demand covers costs. This is more useful than the image of a “large predator eating smaller ones”: a customer may pay for several services, information can be copied and technology can change marginal costs. A biological exponent of 3/4 cannot be transferred to revenue without separate measurement.
In an author-built model for a narrow service, price p=10, variable cost v=6 and fixed cost F=120. At volume q, profit is (p−v)q−F and break-even volume is q=30. If all available demand is 40, one supplier earns 40. If two identical suppliers split it equally, each earns −40. This explains a possible limit on the number of profitable participants under the stated costs, but does not establish a monopoly: prices, technology, differentiation or demand may change.
When new technology lowers F or expands available demand, the niche and sustainable scale change. Verification requires real data on service units, utilisation, customer retention and costs, followed by comparison of alternative models. Rank size does not answer these questions. Economic niche, allometry and Zipf may suggest different research; none proves that a share price is fair or that its future return will be positive.