Superexponential Growth

In 1960, Heinz Von Foerster graphed world population since the birth of Jesus, fit a line, and predicted infinite population on Friday the 13th of November, 2026. It was tongue-in-cheek, but the mathematics was sound: if you take the best-fitting model of human economic history and project it forward, you hit a singularity — infinite output in finite time. David Roodman of Open Philanthropy spent years refining this analysis with modern data and stochastic calculus, and the paradox only got sharper.¹

The human economy has not just grown exponentially. It has grown superexponentially — the bigger it gets, the faster it doubles. This is qualitatively different from the exponential growth that gets all the headlines. Exponential growth has a fixed doubling time. Superexponential growth has a shrinking doubling time. The global economy doubled from $37 trillion to $74 trillion between 2000 and 2019. Such a fast doubling was unthinkable in the Middle Ages. Our earliest doublings may have taken millennia.¹

The Power Law That Fits Too Well

Roodman assembled GWP (gross world product) data from 10,000 BCE to 2019 — starting at $1.6 billion (4 million people times $400/year subsistence) and ending at $73.6 trillion. On a log-log plot where both the vertical axis (GWP) and horizontal axis (time before a reference year) are logarithmic, the data fall remarkably close to a straight line. The agricultural revolution and the industrial revolution — the two most profound transformations in human history — appear as gentle ripples on this long-term trend.¹

The straight line is a "power law" model, and if you extend it forward, it reaches infinite GWP around 2047 (95% confidence range: plus or minus 16 years). Roodman was careful to point out that this is not a prediction. It's a mathematical consequence of the best-fitting model of the past. The question is what to make of a model that describes history so well yet implies an impossible future.

Stochastic History

To capture the randomness of history — plagues, wars, financial crises, bursts of innovation — Roodman built a stochastic model using the Itô calculus, the same mathematical framework that Black, Scholes, and Merton used for options pricing. The idea is to treat GWP not as a smooth curve but as a random process where the growth rate depends on the current level, with noise folded in.¹

Twenty simulated "rollouts" of this model, all starting from the same initial conditions, produce paths that resemble real history but diverge wildly. Some civilizations explode by 3000 BCE; others don't take off even by 5000 CE. In one imagined history, the wheel was invented a thousand years sooner and the breakthroughs snowballed. The randomness is large enough to change the timing of industrial takeoff by millennia, but the overall tendency toward acceleration is robust.

When Roodman ran 10,000 rollouts and asked what fraction eventually reach infinite GWP, the answer was all of them, essentially. The probability of no eventual explosion was 1 in 10^69 — an atoms-in-the-universe-scale number. Incorporating randomness did not soften the paradox of infinity.¹

Why It Doesn't Explode

The model's failure to match the impossible future tells us something about the present. When Roodman tested whether the industrial revolution "surprised" the model — fitting only to pre-1820 data and projecting forward — the answer was yes: 1820 GWP was in the 95th percentile of simulated paths. The model was surprised again in 1870 and 1913. The industrial revolution genuinely broke with the pre-industrial trajectory.¹

Conversely, the four data points since 1990 are all lower than the model expects. Recent growth has been slower and steadier than the historical pattern predicts. Something is dampening the acceleration — perhaps we're in a transition period between one growth regime and the next, or perhaps the superexponential era is ending.

Roodman traced this tension through the economics literature. Standard macroeconomic models (Solow, Romer) emphasize "steady state" — a constant growth rate. They achieve this by assuming diminishing returns: as you add more capital, each additional unit produces less. But the historical data show increasing returns over long timescales, driven by the non-rival nature of ideas. One person's use of a drill bit excludes others; one person's use of an idea does not. Thomas Jefferson's metaphor: "He who lights his taper at mine, receives light without darkening me."¹

The tension between increasing returns (ideas) and diminishing returns (physical resources) is what makes the human trajectory so uncertain. If ideas win, growth accelerates toward... something. If physical constraints win, growth plateaus. The range of possible futures is enormous, which is Roodman's actual conclusion: "It is our task as citizens and funders, at this moment of potential leverage, to lower the odds of bad paths and raise the odds of good ones."

The Modeling Lesson

What I find most valuable in Roodman's analysis isn't the specific projections — it's the methodology. He built a model that rigorously quantifies its own shortcomings. It captures the long-term acceleration of human economic history better than any previous model, then honestly reports where it fails (the industrial revolution, recent slowdowns). It uses stochastic calculus to express the genuine uncertainty of history rather than pretending the future is determined. And it connects the strange mathematics of superexponential growth to real economics — endogenous growth theory, the non-rival nature of knowledge, the tension between innovation and physical limits.

This connects to the broader theme of ecological modeling: the most useful models aren't the ones that predict correctly, but the ones that tell you clearly what they assume and where they break. World3 was valuable not for predicting collapse but for revealing the structural tendency of coupled-growth systems to overshoot. Roodman's model is valuable not for predicting a 2047 singularity but for demonstrating that the mathematical pattern of human history implies instability — and that the future will be profoundly different from the present in ways we can't determine.

The connection to scaling laws is also worth noting: both biological scaling and economic scaling show power-law relationships that hold over enormous ranges and then break when the system transitions between regimes. Whether the human economy is approaching such a transition — and what's on the other side — is perhaps the most important open question in the world.