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GWP data points, 1,000,000 BCE → 2019
Two million years of human output looks, from far enough away, like a single smooth curve racing to a singularity. Up close it looks like a flight of straight steps — a new, faster regime with each great invention. We pulled the data and fit both. Here is what it can, and can't, decide.
“Is growth one smooth hyperbola — or just discontinuous straight trend-breaks, one per great invention? And how else could it be modelled?”
That question — a reader's — runs against the grain of the most-shared chart in the genre: Paul Christiano's 2017 plot of world GDP curving up toward a finite-time blow-up. The reader's instinct is that the smooth curve is a trick of scale, and the real object is a sequence of exponential epochs: forager, farmer, industry — each roughly a straight line in log-space, each far steeper than the last. We took both seriously and fit them to the same series.
The staircase is a real, respected model — essentially Robin Hanson's exponential modes — and on this data it fits better in-sample than one smooth hyperbola. But three things decide how far to trust it:
So: the reader is right that the staircase is legitimate and arguably the most descriptive way to see the past. They're wrong that it's uniquely correct, or as clean as one straight line per invention. The rest of this page earns that answer — starting with the chart that started the argument.
Christiano's post shows world GDP two ways — on a linear axis it's a hockey-stick that flatlines for ten millennia then detonates; on a log axis the detonation resolves into a long, steady, upward-bending climb. Here are both, redrawn point-for-point from our own GWP series. Toggle the axis.
Why the log view matters. A plain exponential — constant growth rate — is a straight line on a log axis. Our series isn't straight: it bends upward, meaning the growth rate itself keeps rising. That upward bend is the whole ballgame. It's what makes a finite-time singularity even thinkable, and it's the pattern every model on this page is trying to explain.
Christiano's 2017 post is often cited as having “fit a hyperbola that predicts a 2043 singularity.” It didn't, quite. The post is qualitative: it tabulates shrinking doubling times and shows these two charts. The equation ẋ = xᵃ, a>1 and the ~2040 pole appear only in his extrapolated doubling list and a comment — no exponent is fit, no date is derived in the body. The rigorous fitted version is Michael Kremer's (1993) and David Roodman's (2020). We keep that distinction throughout: where we quote a fitted B or a singularity date, it's ours or Roodman's, never Christiano's.
What the post really leans on is a list — the length of each successive doubling of world output, and the year it finished. Transcribed verbatim, it falls from a millennium to fifteen years and then, tellingly, ticks back up. The blue points below are Christiano's own numbers; the dashed points are his speculative extrapolation to a mid-2043 blow-up (his words: “I have little idea whether it will start in 2025 or 2045 or 2075”).
The staircase, in Christiano's own data. Nearly every doubling is faster than the last — bar a flat 40-year step from 1900 to 1940 — until 1975, where the fastest solid point (15 yr) gives way to a longer one (≈20 yr). That uptick is the post-1970 slowdown, and it's the single most awkward fact for any “singularity is imminent” reading.
Now both models on the same data. The hyperbola is one smooth curve that must reach infinity at a finite date (≈2029 for our fit). The staircase is straight log-lines with a break wherever the data most reward one. Switch to a linear axis, or zoom into an era, to see where each one wins and fails.
Zoom to 1700→now on a log axis and the tell appears: the hyperbola (blue) overshoots — modern data bend below its rush to the pole. That gap is the slowdown again. The data-placed staircase hugs every era, but watch where its breaks land versus the invention-anchored version: the Industrial Revolution smears across 1700–1820, agriculture never surfaces as its own break, and the data instead prefer breaks at 1940 and 1975 — a war recovery and a deceleration, neither a clean invention.
Plotting against time is treacherous when 70% of points are post-1950. A fairer view plots the growth rate against the level. A single power law predicts one straight rising line; clean discrete modes predict flat plateaus with vertical steps. (Illustrative, not a formal test — sparse ancient gaps smear any true step into an averaged point.)
A noisy upward band of slope B≈0.64 — mild support for a smooth power law over literal clean modes — except the modern cloud at top-right, which rolls over instead of climbing. That rollover is the post-1970 stagnation, and it's the single most robust departure from both the pure hyperbola and the “next mode is imminent” story.
Christiano's object is the doubling time of output. Kremer's original (1993) argument was about population — more people make more ideas, which feed more people. Both doubling-time staircases collapse for ten thousand years, then both turn back up within the last 60 years as growth decelerates and fertility falls. We compute both from the full series — including the 264-point population record.
Read the right-hand ends. In our computed staircases both turning points cluster in the third quarter of the 20th century — output's fastest doubling (~13 yr) lands in the mid-1960s, population's (~47 yr) in the mid-1970s — and both have lengthened since. The engine that drove super-exponential growth — ever more people — visibly stalled: world population's growth rate peaked in the late 1960s (~2.1%/yr) and has fallen ever since.
The reader's idea — discontinuous trend-breaks at each invention — is family #2, operationalised with the testing tools of #3. All five agree about the past and diverge about the future, precisely where there's no data.
One feedback law — more output → more ideas → faster growth — makes output a power law that hits infinity in finite time. Roodman's is the rigorous stochastic version.
A sequence of straight log-lines — forager, farmer, industry — each ~100× faster than the last, joined by fast transitions. Each invention = a new mode. (Transitions are smooth blends, not true jumps.)
Not a theory but the test: fit K log-linear segments and ask whether the breaks are real. Markov-switching is the smooth-probabilistic cousin. This is how you'd check the staircase.
A hyperbola decorated with accelerating oscillations — a critical phenomenon. Fits a critical time, but the oscillation fit is notoriously over-flexible.
Today's super-exponential is the lower arm of one big logistic; we're near its inflection. US research productivity has fallen ~41× since the 1930s; US TFP growth fell from 1.89%/yr (1920–70) to 0.57%/yr (1970–94).
We fit the competing models to the series ourselves (code and data in the repo). In descending order of how much weight each result bears:
| Model · full data, log-space | params | RSS | BIC | in-sample fit |
|---|
The residuals are not white noise — the systematic runs are the structure a single smooth curve misses, and the reason a regime model looks appealing in the first place.
This is the crux, and it's a finding, not a hedge. The smooth hyperbola, the log-periodic law and the staircase are observationally near-equivalent on data this coarse — a geometric stack of ever-faster exponentials looks like a power law when you zoom out, and vice-versa.
The reader's staircase is legitimate and arguably the most descriptive way to model the past — provided it's framed as fast smooth regime-changes on historically-anchored dates, held alongside the feedback that explains why the regimes accelerate, and reported with the candour that the data crown neither a single model nor a singularity date.
Literature was gathered by four parallel research agents (hyperbolic baseline; Hanson's modes; structural-break methods; alternatives), each required to source every quantitative claim and tag its confidence C1–C5. The GWP series is David Roodman's compiled Modeling the Human Trajectory dataset (stitching DeLong, Maddison et al.), with a three-point deep-prehistory tail from DeLong 1998; population is OWID's HYDE/Gapminder/UN series. Two years (1900, 1920) are genuinely absent from Roodman's reconciled GWP column and are left as gaps rather than spliced from a different lineage — provenance and a flagged data issue are documented in the repo. Every model was fit in Python (NumPy/SciPy); all charts on this page are drawn live from those fits.
The findings were then hardened with a maker/breaker loop using a second model (codex / GPT-5.5) as an adversarial fact-checker. Earlier rounds caught real errors — a causal claim about the singularity date that was simply backwards, a 10×/100× table typo, and the mis-attribution of a fitted equation to Christiano's qualitative post — all since fixed.