The Ethereum power law

The idea

Plot the logarithm of Ether’s price against the logarithm of its age in days, and the points fall roughly along a straight line. A straight line on log–log axes is a power law: price grows like age raised to a fixed power, here n = 2.24.

Networks tend to grow fast early and then slow down. A power law captures that: the price keeps rising, but each year by a smaller percentage. Today the line rises about +21% a year; doubling fair value from here takes about 5 years. Unlike exponential growth, the yearly rate keeps falling — a modest baseline rather than a promise of moonshots.

The idea comes from physicist Giovanni Santostasi’s work on Bitcoin, whose power law has an exponent of almost 6 (counted from Bitcoin’s genesis block). Ether’s is much smaller and its fit looser, so treat the line as a description of the past rather than a law of nature.

Robustness checks

The exponent depends on the data window. Refitting with data only up to each date gives exponents between 1.82 and 2.44. Long-range forecasts are therefore much less certain than the clean lines suggest.

No separate cycle curve. Adding a repeating market cycle (best fit about 3.4 years) to fair value raises the in-sample fit from R² 0.78 to 0.89. But when trained on older data and tested on later years, it predicts worse than the plain trend, so fair value stays the trend line:

• Trained until 1 Jan 2022: error 0.19 for the plain trend vs. 0.36 with a cycle.
• Trained until 1 Jan 2023: error 0.21 for the plain trend vs. 0.40 with a cycle.

Error = root-mean-square difference in log₁₀ price on the held-out years.

Limitations

• Ether’s history is short (about ten years), and the fit explains 78% of the variation in log price.
• Ether has changed a lot along the way: fee burning in 2021, proof of stake in 2022, and the move of activity to layer-2 networks. Past behaviour may not carry over.
• The bottom line is defined by the single lowest day since 2017. A new low would push it down at the next refit.
• Daily data starts in May 2016 (Coinbase). The first ten months of trading are not included.

Questions

What is the Ethereum power law?

A straight-line fit of log price against log age: price = 10^-4.487 · t^2.241, with t in days since 30 July 2015.

How well does it fit?

R² is 0.78: the line explains 78% of the variation in log price. Bitcoin’s power law fits more tightly.

Can I use it to predict the price?

Only as a rough baseline. See the price predictions for the yearly values and their uncertainty, and the outlook for what happened after similar days.