The Logic Behind Bitcoin’s Power Law

A few years back, a European researcher released a power law model to fit Bitcoin’s price behavior over time. The basic idea is that power laws are useful for describing a wide variety of phenomena in nature and science. Formerly, the power law fitted to Bitcoin is a simple equation

Price = A(t – t_0)^n

Here t is time, t_0 is initial time, A is a scaling factor, and n is an exponent. Put simply, the price of Bitcoin is a function of the amount of time that has passed since its inception. This relationship can be seen most visually by taking the logarithm of both sides. If you do that, you will arrive at

log⁡Price = log⁡A + n log⁡(t – t_0)

Then, you can fit a linear regression to log price against log time.

My data runs from 2011, and you’ll see that there is a very clear linear trend in the log-log space. This differs from my previous analysis, where I had used and plotted log price against time. Here, I’m plotting the log price against log time.

The full article on the power law theory by its author is a deeper exposition of why power laws work in nature. Most of that essay is worth reading, though I will disagree on one point. The author claims that the power law has nothing to do with scarcity, since it is a statistical fit to a price chart. However, scarcity is the foundation of Bitcoin’s value proposition and is exactly why new buyers continue to enter the market.

But why?

My explanation for why the power law theory fits is the cycle of FOMO. Because Bitcoin is provably scarce, new adopters will always know that the limited supply will keep the price higher in the future and will not lead to a collapse in price, as can happen in other assets like gold. Gold miners can increase their production if the price of gold rises. Therefore, new gold miners shift the supply curve of gold production forward, leading to a decrease in the price of gold since more gold could be mined.

Bitcoin miners have no ability to change the issuance of new bitcoins, since that is determined by the protocol. Bitcoin miners can only contribute new hash power to the network, which increases their own chance of earning the block reward. However, they have no market-wide impact on price, unlike gold miners.

New buyers of Bitcoin, therefore, know that there will always be other buyers coming in after them who will face the same decision problem, just later in time. Except that those buyers will have even fewer coins available. So, it is optimal to enter the market once you learn of this provable scarcity. At any point in time, there will always be someone before you who bought earlier when there were more bitcoins available at a lower price, and also that there will always also be someone after you who will buy when there are fewer Bitcoins available at a higher price. This cycle is ultimately what drives the upward trajectory in price. No other cryptocurrency has this unique feature.

There are, of course, shocks to the system, like the FTX collapse, Mount Gox, the Silk Road, and the China crackdown on Bitcoin mining. Those are the deviations around the trend line. But the chief principle here is the continuing spread of knowledge about Bitcoin across the population.

Will this ever stop? Once Bitcoin adoption penetrates the entire world then there is no guarantee that the FOMO cycle will continue. At that time, price will stabilize and volatility will decrease, as befits a widely held asset. But we are far from that point, and so I do believe there is some truth behind the power law graph.

It would be intellectually dishonest to admit that price has no impact on Bitcoin adoption. An honest assessment of human behavior is that price is all that matters. The provable scarcity of Bitcoin ensures that new buyers will enter the market. I’m not even sure that Satoshi knew about this fundamental feature of Bitcoin’s scarcity, as the supply cap is not even mentioned in the white paper. But as the power law shows, something is driving this behavior. Provable scarcity.