📈 economics

Universality and Predictability of Technology Diffusion

By analyzing 120 mature technologies, this paper demonstrates that a universal Bertalanffy-Richards process can accurately forecast S-curve technology diffusion decades in advance, revealing that solar and wind energy will likely reach global dominance by 2050 far sooner than current climate scenarios predict.

Original authors: Benjamin Wagenvoort, François Lafond, Joel Dyer, James Doyne Farmer

Published 2026-08-24

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CC BY 4.0

Original authors: Benjamin Wagenvoort, François Lafond, Joel Dyer, James Doyne Farmer

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Every new technology seems to have its own unique story. Some arrive quietly, others explode onto the scene, and most eventually settle into a steady rhythm as they become part of daily life. For decades, scientists and economists have tried to find a single rule that explains this pattern, a universal law that predicts how fast a new invention will spread from a handful of users to the entire world. This question is not just academic curiosity; it is a matter of survival for our planet. As humanity races to replace fossil fuels with renewable energy, knowing exactly how quickly solar panels and wind turbines can be deployed is the difference between meeting climate goals and failing them. Yet, past attempts to forecast these transitions have often been wildly inaccurate, leaving planners to guess whether a technology will take off or fizzle out.

A team of researchers at the University of Oxford has now tackled this uncertainty by looking at history not as a collection of unique stories, but as a massive dataset of patterns. They gathered information on 120 different technologies that have already reached maturity, ranging from ancient canals to modern mobile phones. By studying these completed journeys, they discovered that despite their different origins and purposes, these technologies all follow a remarkably similar shape as they grow. This shape, often called an S-curve, starts with slow growth, accelerates into a rapid boom, and then slows down again as the market becomes saturated. The researchers found that this pattern is so consistent that it can be described by a single mathematical process, which they call the Bertalanffy-Richards model. This model acts like a reliable map, allowing them to predict the future path of a new technology with a known level of accuracy, even when they only have a small amount of early data.

To test if this map works, the team used a method called Bayesian forecasting, which allows them to update their predictions as new information arrives. They ran thousands of simulations, pretending they were in the past and trying to forecast the future based only on the data available at that time. They found that their model could predict the final size of a technology's adoption with surprising precision. Even when looking at a technology that had only reached five percent of its potential market, the model could forecast its future size within a factor of two. This means that if a technology eventually reaches 100 million users, the model would predict a number between 50 million and 200 million, a level of accuracy that is rare in long-term forecasting. The researchers also compared their method to other popular forecasting tools and found that those older methods often produced results that were systematically too low or too confident, missing the true potential of the technology.

The team then applied this new, validated method to the two most critical technologies for our energy future: solar photovoltaic panels and wind turbines. The results challenge the prevailing wisdom found in many major climate reports. While current scenarios from groups like the Intergovernmental Panel on Climate Change and the International Energy Agency often assume a moderate growth for both, the Oxford team's analysis suggests a very different outcome for solar power. Their median forecast indicates that by 2050, solar energy could supply approximately 85 petawatt-hours of useful energy annually. This is a figure that dwarfs current expectations; it is roughly equivalent to the total amount of useful energy consumed by all of human society today. In contrast, their forecast for wind energy is more modest, predicting a supply of about 5 petawatt-hours, which is significantly lower than the optimistic scenarios often presented in policy documents.

The study suggests that the world is likely to see a much faster transition to solar power than widely believed, potentially meeting ambitious climate targets sooner than anticipated. The researchers note that this rapid growth is not guaranteed by policy alone but is driven by the inherent dynamics of how successful technologies spread. They found that solar power is currently in a phase where its growth is still indistinguishable from a rapid exponential rise, making it difficult to know exactly when it will begin to slow down. However, the data strongly suggests that the upper limits of solar adoption are far higher than current models assume. This does not mean that the transition will be easy or that all obstacles will vanish, but it does provide a scientifically grounded reason for optimism. The study concludes that by understanding the universal laws of technology diffusion, we can move beyond guesswork and make investment and planning decisions based on a clearer, more accurate picture of what is possible.

Technical Summary: Universality and Predictability of Technology Diffusion

Problem Statement

Forecasting the growth and diffusion of technologies is critical for planning and investment, particularly in sectors like renewable energy and artificial intelligence. Historically, technology growth has been observed to follow "S-curves," characterized by slow initial growth, rapid acceleration, and eventual saturation. However, the precise functional form governing this growth has remained a subject of debate, with various models (e.g., logistic, Gompertz, Bass) proposed but none achieving consensus.

Previous forecasting attempts have suffered from two primary limitations:

Lack of Ground Truth: Studies often relied on immature technologies where the final saturation level is unknown, making it impossible to validate forecasts against reality.
Unreliable Accuracy: Simple S-curve fitting has historically yielded pessimistic or inaccurate results, and probabilistic methods have lacked rigorous out-of-sample validation against mature technologies.

Consequently, the universality of technology diffusion processes and the ability to make forecasts with known accuracy remained highly uncertain.

Methodology

The authors address these gaps through a comprehensive empirical and statistical approach:

1. Data Assembly

The study constructs a database of 120 mature technologies (ranging from canals to mobile phones), selected because they have reached at least 80% of their estimated asymptotic deployment. This "ground truth" dataset allows for rigorous backtesting. The initial search yielded 1,362 time series, filtered down to ensure robustness against selection bias.

2. Model Selection and Stochastic Formulation

The authors test several functional forms for the growth rate $r(t)$ within the differential equation $dY(t)/dt = r(t)Y(t)$. They identify the Bertalanffy-Richards (B-R) process as the superior model, defined by:
$r(t) = k(1 - (Y(t)/L)^\beta)$
where $L$ is the asymptotic deployment, $k$ is the initial growth rate, and $\beta$ is a shape parameter. The authors find $\beta = 2/3$ to be a robust universal value.

To account for real-world data characteristics (non-stationarity, autocorrelation, and heteroscedasticity), the deterministic model is recast as a stochastic process:
$Y(t) = \frac{L}{[1 + \exp(-\beta k(t - t_0) + \beta\sigma(\epsilon_t + \rho\epsilon_{t-1}))]^{1/\beta}}$
where $\sigma$ represents noise amplitude and $\rho$ is an autocorrelation parameter. The noise is multiplicative, resulting in a roughly log-normal distribution for $Y(t)$ .

3. Bayesian Forecasting and Validation

The authors employ Empirical Bayesian inference to generate probabilistic forecasts.

Priors: Priors for growth rate $k$ and noise volatility $\sigma$ are derived from the full-sample analysis of the 120 technologies.
Universal Parameters: To improve forecasting stability, the authors assume universal values for the shape parameter ( $\beta = 2/3$ ) and autocorrelation ( $\rho = 0.8$ ), rather than estimating them individually for every technology.
Backtesting: Extensive out-of-sample backtesting is performed. Forecasts are generated at various diffusion origins (5%, 10%, 25%, 50%) and compared against future realized data.
Metrics: Accuracy is measured using the Median Absolute Log Error (MedALE). Calibration is assessed via Quantile-Quantile (Q-Q) plots and Probability Integral Transform (PIT) to ensure forecast distributions are well-calibrated.
Surrogate Data: A surrogate ensemble method is used to test statistical significance, comparing empirical results against an idealized model where each technology strictly follows the B-R process.

Key Results

1. Universality of S-Curve Shapes

When the 120 technologies are rescaled into non-dimensional coordinates (normalizing time, scale, and initial growth), their growth trajectories collapse onto a single curve.

The Bertalanffy-Richards model with $\beta = 2/3$ explains 93% of the variance in the data.
This suggests that while idiosyncratic factors affect specific parameters (like $k$ or $L$ ), the underlying shape of the growth path is remarkably universal across diverse technologies.

2. Forecasting Accuracy

Point Forecasts: The B-R model outperforms the Logistic ( $\beta=1$ ) and Gompertz models. Point forecasts made from a 5% diffusion origin are typically accurate to within a factor of two (MedALE < 100%), even for horizons extending decades into the future.
Model Comparison: The Gompertz model exhibits strong bias specifically at the 5% diffusion origin, meaning its forecasts are systematically too low, and it is strongly overconfident in this context. While it shows somewhat lower forecasting errors for short horizons generally, the B-R model is substantially better for horizons beyond 50%. The Logistic model performs better than Gompertz but is generally outperformed by the B-R model, particularly at longer horizons.
Calibration: The probabilistic forecasts are well-calibrated. After a minor adjustment to widen the predictive distribution to account for model mis-specification, the empirical coverage falls within the expected confidence intervals.

3. Forecasts for Solar PV and Wind Energy

Applying the validated method to solar photovoltaic (PV) and wind energy (measured in useful energy, PWh/yr):

Solar PV: The median forecast for 2050 is approximately 85 PWh/yr (90% interval: 18–290 PWh/yr). This is comparable to the total useful energy consumed globally today. The authors note that solar is currently in a noisy exponential phase, making the upper bound of the forecast highly uncertain.
Wind: The median forecast for 2050 is approximately 5 PWh/yr (90% interval: 4–17 PWh/yr).
Comparison to Scenarios: The authors' median solar forecast is more than 10 times higher than the median of IPCC AR6, IEA, and NGFS scenarios (which project ~8 PWh/yr). Conversely, the wind forecast is roughly half the typical IPCC scenario. The study concludes that current scenarios are likely too pessimistic regarding solar and too optimistic regarding wind.

Significance and Claims

The paper claims to resolve long-standing questions regarding the nature of technology diffusion by demonstrating that:

Universality Exists: Technology growth follows a universal S-curve shape well-described by the Bertalanffy-Richards process with $\beta \approx 2/3$ .
Predictability with Known Accuracy: It is possible to forecast future technology adoption with quantifiable accuracy, even from early stages (5% diffusion), provided the process follows an S-curve.
Methodological Advancement: The study provides a validated, statistically rigorous framework for forecasting that outperforms existing popular models (Logistic, Gompertz, Bass).
Implications for Energy Transition: The results suggest that ambitious climate targets involving solar energy may be met faster than widely believed by current policy scenarios, potentially reducing reliance on negative emission technologies.

The authors acknowledge limitations, including potential selection bias toward successful technologies, the exclusion of auxiliary variables (cost, policy) in the unconditional forecasts, and the assumption of a constant asymptote. However, they assert that the methods developed offer a robust foundation for guiding investment and planning for emerging technologies, extending beyond energy to fields such as AI.