Information Theory & Network Economics
Scale
West, a theoretical physicist and former president of the Santa Fe Institute, shows that organisms, cities and companies follow power laws in size, with one exponent deciding their fate. Sublinear systems grow towards a ceiling; superlinear ones accelerate and need ever faster innovation to avoid collapse. For investors it is a way to ask how a company's output should grow with its size.
The big picture
The book starts from a pattern biologists found early in the 20th century: across mammals from mouse to whale, metabolic rate rises with body mass to the power of about 3/4, known as Kleiber's law. Double the mass and energy use rises by only about 68%, not 100%. West and his co-authors explained the 3/4 with a model of branching supply networks, blood vessels and airways that fill the body and waste as little energy as possible. The same maths, a power law Y = Y0 · N^β in which the exponent β is the whole story, then carries over to cities and companies. In cities, infrastructure such as road surface, cable length and petrol stations scales with population at roughly 0.85 (sublinear: bigger cities need less per head), while wages, GDP, patents and also crime and disease scale at roughly 1.15 (superlinear: each resident produces more in a bigger city). Companies, in West's analysis of US listed firms, behave more like organisms: sales, assets and profits grow a little less than one-for-one with employees.
The exponent has a direct consequence for growth. In West's growth equation, a system takes in resources that scale with N^β and spends a maintenance cost proportional to N. With β below 1, maintenance eventually eats the intake and growth levels off, which is why organisms stop growing and, in his reading, why companies tend to stall and die (the half-life of a US listed company is around ten years). With β above 1, growth runs faster than exponential and reaches infinity in finite time unless an innovation resets the system, and each reset has to come sooner than the last. Why it matters now: in 2026 the question for AI-heavy companies is whether revenue per employee, profit per unit of sales and R&D per unit of sales can break the usual sublinear pattern. The exponent can be measured, on the reader's own data, instead of argued.
The 3 strategic pillars
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One exponent, three worlds
Size alone predicts a large share of how an organism, city or company behaves, and the prediction takes the form of a power law whose exponent falls into one of three classes: below 1, equal to 1, above 1.
On log–log axes a power law is a straight line and its slope is β. Doubling size multiplies output by 2^β: about 1.68 at β = 0.75, 2 at β = 1, about 2.22 at β = 1.15. West's argument is that the exponent comes from the structure of the network that distributes resources: supply networks that serve every cell or household efficiently give economies of scale (sublinear), while social networks, where the number of possible interactions grows faster than the number of people, give increasing returns (superlinear). The biological values are well established; the city values come from Bettencourt, Lobo, Helbing, Kühnert and West (2007); company exponents are less settled and vary by variable.
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Growth: plateau or singularity
The same exponent decides whether a system grows towards a ceiling or accelerates towards a breaking point.
In the growth equation dN/dt = a·N^β − b·N, a is the rate of resource intake and b the maintenance cost per unit of size. For β < 1 the solution is sigmoid-like and levels off at N* = (a/b)^(1/(1−β)). For β = 1 it is plain exponential at rate a − b. For β > 1, starting above an unstable point, it reaches infinity at a finite time t_c. With no maintenance term that time is N0^(1−β) / ((β − 1)·a), so a larger starting size gives a shorter run: each innovation that resets the system buys less time than the one before, and the pace of change keeps rising. The model is stylised, but the closed form makes the three regimes exact.
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Judge against size, not in absolute terms
A big city or a big company is expected to produce more, so raw totals and even per-head ratios mislead; what is informative is how far a unit sits from the line for its size.
West and Bettencourt rank cities by scale-adjusted residuals: the log of actual output minus the log of what the fitted power law predicts at that population. A residual of +0.20 means about 22% above the line (e^0.20 − 1). Per-capita ratios quietly assume β = 1; if the true exponent is 1.15, every big city looks good per head for no reason other than its size. The same holds for companies: sales per employee or margin comparisons across firms of very different size carry a built-in size bias unless the exponent is estimated first.
What a Closelooknet reader does with it
The working use is to measure the exponent instead of assuming it. Take a cross-section of companies in one sector, or one company over a run of years, and fit ln Y against ln N: sales against employees, operating profit against sales, R&D against sales. The slope is β, and its standard error tells whether the data can even tell 0.95 from 1.00; with 20 companies it often cannot. The scale-adjusted residuals then show which firm sits above or below the line for its size, which is a better starting question than a raw ratio. The mistake it prevents is extrapolating a young company's growth as if β were 1 or above when its own history says the cost base grows faster than output. The pack does the fit, the test of β = 1 and the residual ranking on the reader's data, and simulates the growth equation for any β. West's company exponents are reported approximately here; the reader's own fit is what the pack uses.
The bridge to the Closelooknet approach
Closelooknet tracks several places where scaling decides the outcome. The Power Constraint is Kleiber's problem at data-centre size: compute grows only as fast as the energy supply that feeds it, and Tokens per Watt measures output per unit of that scarcest input. The CapEx Cliff asks when the build-out stops growing, the plateau question in capital-spending form, and The Digital Labor Economy covers the case where agents let output grow without headcount, a test of whether sales per employee can scale above 1. The Hurst Exponent uses the same log–log regression on a time series. The glossary defines z-scores, used for the residuals, and operating margin and sales efficiency, two ratios that assume β = 1 without saying so. In the same cluster, Modern Monopolies covers network value growing faster than users, Information Rules the cost structure behind increasing returns, and The Information the theory of the data these networks carry. The pack fits the reader's own data; its output is a research note, not a signal.
Action-Kit — from theory to practice
Tooling & data
| What you need | Where to get it | Cost |
|---|---|---|
| Company filings Employees, sales, operating profit and R&D per fiscal year, for the size and output columns of the fit | SEC EDGAR full-text search Headcount is disclosed once a year in the 10-K; use the same fiscal year for every company in a cross-section. | Free |
| Standardised multi-year financials Ten or more years of revenue, profit, R&D and employees for one company, for a time-series fit of its own exponent | stockanalysis.com The free tier shows a limited history; longer histories need the paid plan. | Freemium |
| Structured XBRL data Machine-readable financial statement values across many filers for building a sector cross-section | SEC EDGAR XBRL frames API Employee counts are not always tagged in XBRL; fill them from the 10-K text where missing. | Free |
| City scaling data for comparison Metro-area GDP and population to reproduce a superlinear city exponent as a sanity check of the method | US Bureau of Economic Analysis (GDP by county and metro area) | Free |
The formulas
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Scaling exponent by OLS on logs
ln Y = ln Y0 + β · ln N; β = Σ(xᵢ − x̄)(yᵢ − ȳ) / Σ(xᵢ − x̄)², x = ln N, y = ln Y- N: size (employees, sales or assets), positive
- Y: output (sales, operating profit, R&D), positive
β < 1 sublinear, β = 1 linear, β > 1 superlinear. Doubling N multiplies Y by 2^β.
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Standard error, confidence interval and test of β = 1
SE(β) = s / √Σ(xᵢ − x̄)², s = √(SSE / (n − 2)); CI = β ± t(1 − α/2, n − 2) · SE; t = (β − 1) / SE- n: number of usable rows
- α: two-tailed level, e.g. 0.05 for a 95% interval
Call a sample sublinear only if the whole interval lies below 1, superlinear only if it lies above 1.
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Scale-adjusted residual
rᵢ = ln Yᵢ − (ln Y0 + β · ln Nᵢ); % vs line = e^rᵢ − 1; zᵢ = rᵢ / s- Fitted ln Y0 and β from the regression
- s: residual standard deviation
Positive rᵢ: output above what size alone predicts. The company analogue of Bettencourt and West's scale-adjusted city indicators.
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Growth equation and its two limits
dN/dt = a·N^β − b·N; β < 1: N* = (a/b)^(1/(1−β)); β > 1: t_c = ln[(a/b) / (a/b − N0^(1−β))] / ((β − 1)·b)- a: resource intake coefficient
- b: maintenance cost per unit of size
- N0: starting size (above the unstable point (a/b)^(1/(1−β)) for β > 1)
Exact solution: with k = 1 − β, N^k = a/b + (N0^k − a/b)·e^(−k·b·t). Example a = 1, b = 0.5, N0 = 1: N* = 16 at β = 0.75, t_c ≈ 9.24 at β = 1.15.
Applied Pack · free members
West Applied Pack
A sub/superlinear scaling model: fit the exponent β of your own companies' output against size with standard error and R², rank each firm by its scale-adjusted residual, and simulate how growth plateaus below β = 1 and races to a finite-time singularity above it.
- West_Scaling_Model.xlsx — READ ME; Fit (up to 40 rows of size and output, amber inputs, EXAMPLE_ rows → live β, intercept, Y0, R², standard error, confidence interval, t-test and p-value for β = 1, reading, fitted value, residual, z, % vs line and rank); Simulator (exact closed-form paths of dN/dt = a·N^β − b·N for a sublinear, linear and superlinear β, plateau N*, time to 90% of N*, singularity time t_c, log-scale chart, shrinking cycle-length table); Reference (exponents reported by West and co-authors with confidence notes)
- west_scaling.py — stdlib-only Python: reads a CSV of your own data and prints the fit with confidence interval and the ranked residual table; --simulate prints growth paths for any set of exponents
- scaling_sample.csv — 20 synthetic EXAMPLE_ rows with employees, sales, operating profit and R&D
- README.txt — the maths, what comes from the book and what is the pack's choice, caveats and the educational-use disclaimer
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Educational templates — a research diary companion, not investment advice.
Closelooknet publishes a market diary, not investment advice. This condensed read restates the book's ideas in our own words for education — for the author's full argument, go to the source.