Bitcoin Pricing: Scarcity and Network Effects Compared Using Generalized Additive Models

Katey Durham
Investor’s Handbook
8 min readJun 1, 2021


Photo by Dmitry Demidko on Unsplash

Digital Finance has been transforming from a niche space for tech enthusiasts to a mainstream area of investment strategy. There is so much more to this space than just alternatives to fiat currencies, as needed as those are. However, in this article I’ll to take a step back from the larger evolution and explore some theories regarding the drivers of Bitcoin valuation, since its price action is, after all, why many are giving it their attention. There are two commonly-cited drivers, Scarcity and Network Effects, to which Bitcoin’s value is attributed, although frequently each is considered in isolation rather than in tandem. Scarcity can be represented by Stock-to-Flow, a framework that comes from the valuation of other precious commodities. Network Effects can be evaluated by a measure of its size, or simply put the number of people participating in it. Let’s dive in.

Scarcity and Stock-to-Flow

There are a variety of reasons why an object might be valued. It could be essential to human life, like water. It might also be something non-essential but that people like because it’s shiny and they can make jewelry with it, such as gold. To attribute value to this type of material, it should probably be somewhat rare. One way to measure scarcity is with the Stock-to-Flow Ratio, where Stock refers to the amount of the material already in existence, and Flow is the rate at which new amounts of it are discovered or mined. For precious metals, there is a direct relationship between this ratio and the value of the material. For example, the Stock-to-Flow ratios of Gold > Silver > Copper, in alignment with their relative prices. The insightful leap of applying this to Bitcoin valuation was made by PlanB. Many swear by a chart overlapping Bitcoin price with Stock-to-Flow for technical indicators as to when to trade the crypto currency profitably. Something important to point out about Bitcoin mining that leads to the stepped nature of the charts is the Bitcoin halving events. At pre-specified milestones on its way to the cap of 21 million coins, the reward to Bitcoin miners for a new block is reduced by a factor of 2. Below is the schedule so far, with the next halving expected in 2024:

There has been debate as to whether these events are directly related to pricing. Many cite predictable increases in Bitcoin price surrounding these events, while others (famously Ethereum founder Vitalik Buterin) claim this is not the case. Regardless, Stock-to-Flow is arguably the biggest single factor that is often followed by those who wish to predict Bitcoin price movements.

Network Effects and Active Bitcoin Addresses

Another important determinant for Bitcoin valuation may be the extent to which it is held or is in use by the human population. This principle can be referred to as following Metcalfe’s Law. Originally postulated to propose valuation for a telecommunications network based on the number of its users (n), the law states that the value of the network is proportional to n-squared. In words, the value increases in a greater-than-linear way as new users are added. This can be understood in terms of valuing an important type of network that we have today, a social one. If you and your friend are the only ones sending each other videos of your latest dance moves, don’t expect an IPO any time soon. But consider a network with users in the millions or even billions, together with machine/deep learning algorithms suggesting content you might like, and you’ve got something immensely valuable. A measure that approximates the size of Bitcoin’s network is the daily number of active addresses.

Bitcoin Stock to Flow, Active Addresses and Price

Here is a plot of each day’s Bitcoin Price, Stock-to-Flow, and Active Addresses from mid-2010 to the end of May 2021.

Data and Models

Stock-to-Flow and Bitcoin Addresses were explored using daily data obtained from Coin Metrics. Both Bitcoin Scarcity (Stock-to-Flow) and Network Size (Active Addresses) were considered simultaneously by estimating their effect on price for the period from mid-2010 to present (May 29, 2021). Log10 of both of these values, along with centered calendar time and a cyclical term representing month, were included in a Generalized Additive Model (GAM) with log10 Bitcoin price as the response. One way to think of this type of modeling is that is allows more of a conversation between the modeler and the data than in models where the functional form of the relationship between the predictors and response is set beforehand. In a linear model, for example, the effect of the feature on the response is fixed across its range, i.e. there’s just one estimated parameter, no matter what the value the feature has. Small values or big values result in the same per-unit effect on the response. Not so with GAM, where the effect can change depending on the feature’s value. Further, the functional form of this relationship isn’t specified beforehand, but is estimated from the data using smooth terms. More details on the data and modeling process can be found here.

GAM Model Fit

In the final GAM model, all of the terms appear to be highly significant in relation to Bitcoin price, and appear to explain virtually all of its variation (over 99%). Below are the smooth model coefficients and what they indicate for each parameter.

Active Addresses

The smoothed effect of the number of Active Addresses on price is less complex and requires fewer smoothing terms than for time and Stock-to-Flow, so it is easier to interpret. The coefficient is always positive and appears to be increasing as the number of active addresses increases. This is consistent with Metcalfe’s Law, which indicates a larger increase in value as the network size increases. What’s more, using the smoothed parameter estimate, we could estimate the critical mass at which the network growth really begins to accelerate its effect on price at just over 3000 addresses. This might be useful for evaluating the price movements of other digital currencies. Another observation is that, although theory would suggest that a network would eventually exhaust itself, there are no indications of having reached saturation at this point in the smooth term.

Stock to Flow

The Stock-to-Flow coefficient requires a larger basis dimension to represent its shape, resulting in significantly wider confidence bounds. It may indicate an increasing effect on price as the Stock-to-Flow approaches 10 (original units) and a decreasing effect size (although possibly still positive) as the Stock-to- flow increases beyond 10. This indicates that the Scarcity effect on price would eventually level off. However, there are some additional twists and turns in the smooth relationship, and really any curve that could be drawn within the confidence bounds should be considered a possibility. The smooth term for Stock-to-Flow should be interpreted with caution.

Calendar Time

The smooth term for centered calendar time captures the time effect on the overall increasing Bitcoin price over the period. After the mid-point (23 Dec 2015) there appears to be an increasing effect that settles at a new, higher level. This term may be capturing time-related features not included yet in the model.


Although still significant, the month effect is not at the same magnitude as the other smooth terms in the model. It may be indicating a modest seasonal effect toward higher pricing in the spring to summer months that drops sightly in the fall.

Conclusions and Thoughts about Prediction

TL;DR: These are the major take-aways from the GAM model for Bitcoin price fit using Stock-to-Flow and Active Addresses.

  1. Both Stock-to-Flow and Active Addresses have highly significant relationships to Bitcoin price, and together with time effects explain almost all of the observed variation over the period from mid 2010 to May 2021 (over 99%).
  2. Active Addresses displays an increasing, non-linear relationship to Bitcoin price and the relationship appears to be in line with Metcalfe’s Law, with no signs of network saturation at this point. The smoothed function may indicate an inflection point at which increasing network size may really start to impact value (over 3000 active addresses).
  3. Stock-to-Flow displays a more complex relationship to Bitcoin price, with an effect that may be exhibiting diminishing increases on price as the Stock-to-Flow ratio increases. However this effect requires many more smoothing terms to describe, so has a greater range of possible values.

And finally, here are some thoughts regarding the use of this model to predict future Bitcoin price. By its nature, projecting future values from any model will always be out of sample, and thus more risky than when predictions involve the observed range of data. However, models can give some guidance and structure to future prediction, especially if the alternative is a crystal ball or hunch. With GAMs, however, what is useful in terms of letting the data tell their story is a liability when out of the observed range of information. The model will likely simply extend the most recent trend, with rapidly increasing error bars as the time horizon increases. The best way to use the insights from GAM modeling to predict the future would be to use the information from the smooth parameters to develop a more parametric representation of the effects, and then use this more structured model to estimate a best guess for future values.



Katey Durham
Investor’s Handbook

Predictive Analyst who uses models and intuition ~ NFT jams ~ Mashing it all up at