**Definitely, Bitcoin's digital scarcity has value, but how big?Authorsarticlequantitatively estimates scarcity using the ratio of stocks to the increase in the total number of assets (stock-to-flow), and also uses this ratio to model the future price of bitcoin.**

## Introduction

Satoshi Nakamoto Published Bitcoin White PaperOctober 31, 2008 [1], created the genesis block on January 3, 2009 and released the Bitcoin software code on January 08, 2009. So the foundation was laid for the 70 billionth bitcoin market today.

Bitcoin is the world's first scarce digital asset. It exists in limited quantities, like silver or gold, and it can be transmitted via the Internet, radio waves, satellites, etc.

"In

As a thought experiment, imagine that there is a non-ferrous metal that is as scarce as gold, but has the following properties: boring gray color, poor conductivityelectricity, which is not particularly durable [..], is not well suited for any practical or decorative purpose ... and with one magical property – it can be transmitted through communication channels." – Satoshi Nakamoto[2]

Definitely, this digital scarcity hasvalue. But how big? In this article, I quantify scarcity using the ratio of held stocks to growth in the total amount of assets (stock-to-flow), and I also use this ratio of stocks to growth to model bitcoin prices.

## Deficit and the ratio of stocks to the increase in the number of assets

In dictionaries, scarcity is usually defined as "a situation in which it is not easy to get something" or "the absence of something."

Nick Sabo gave scarcity a more useful definition: "unadulterated value."

“What do antiques, time and gold have in common?They are expensive, either inherently or because of their incredible history, and this high value is difficult to counterfeit. [..] The realization of non-falsifiable value by computational means, on a computer, is associated with certain difficulties. If we overcome these problems, we can achieve digital gold,” Nick Szabo. [3]

“Precious metals and objectscollectibles have non-falsified scarcity due to the cost of their creation. This property was once also possessed by money, the value of which was, by and large, independent of any third parties. [..] [but] metal money cannot be paid online. So, it would be nice if there was a protocol that allowed you to create falsifiable valuable bits online and with minimal dependence on third parties, and then store, transfer and analyze them with the same minimum need for trust. Digital Gold, ”- Nick Szabo [4].

Bitcoin has non-falsifiable value,because it takes a lot of electricity to create new bitcoins. Bitcoins cannot be easily faked. Please note that fiat money and altcoins with unlimited emission, not using proof-of-work (PoW), with a low hash rate or with a small group of people or companies that can easily affect the volume of their offer do not have this quality.

Sayfiddin Ammus speaks of scarcity from the point ofview of the ratio of stocks to growth (SF). He explains why gold and Bitcoin are different from consumables such as copper, zinc, nickel, brass, oil - because they have a high SF ratio.

«For any consumable goods [..] doubling production will reduce existing stocks, causing prices to fall and hurting holders. For gold, the price hike that would cause annual production to double would be small, increasing inventories by 3% rather than 1.5%.”

“It is this invariably low level of gold supply that is the main reason that it has retained its monetary role throughout human history.”

«The high stock-to-growth ratio makes it the product with the lowest price elasticity of supply».

«Existing Bitcoin stocks in 2017were approximately 25 times greater than the number of new coins created in the same year. This is still half that of gold, but around 2022 Bitcoin's stock-to-growth ratio will exceed that of gold,” Saifiddin Ammus. [5]

Thus, the deficit can be quantified through the ratio of stocks to growth.

**SF = stock / gain**

Stock is the size of existing stocks orreserves. Growth is annual production. Instead of the ratio of stocks to growth, the supply growth rate (growth / stock) is also used. Please note that SF = 1 / supply growth rate.

Let's take a look at the SF values for some assets.

</p>Gold has the highest SF ratio of 62 – forIt takes 62 years of production to obtain the current supply of gold. Silver is in second place with SF 22. Such a high SF value gives them the properties of monetary products.

SF for all other commodity assets, includingat best, palladium and platinum barely exceed 1. Existing stocks are usually equal to or less than the annual production volume, which makes production a very important factor. For commodity assets, it is almost impossible to get a higher SF value, because as soon as someone succeeds, the price rises, as a result, production rises, and the price drops again. It is very difficult to break out of this trap.

Bitcoin currently has a margin of 17.5million koins, and an issue volume of 0.7 million / year = SF 25. This puts Bitcoin in the category of cash products such as silver and gold. Bitcoin's market capitalization at the current price is about $ 70 billion.

The volume of Bitcoin issuance is limited.New bitcoins are created with each new block. Blocks are generated (on average) every 10 minutes when a miner finds a hash that satisfies the PoW required for a valid block. The first transaction in each block, called coinbase, contains the block reward for the miner who found the correct hash. The block reward consists of the fees that users pay for transactions and the newly created coins (called the subsidy). The subsidy amount was initially 50 bitcoins and is halved every 210,000 blocks (about 4 years). This is why this subsidy reduction is so important for the money supply and Bitcoin SF. This also means that the change in supply growth rate (commonly referred to as “monetary inflation” in the context of Bitcoin) will be incremental and uneven.

## The ratio of stocks to growth and value

The hypothesis of this study is thatasset scarcity, measured in SF, has a direct impact on its price. The data in the table above confirm that the market value is usually higher, the higher the SF asset. The next step is to collect data and create a statistical model.

### Data

I calculated monthly SF and Bitcoin value withDecember 2009 to March 2018 (a total of 111 data points). The number of blocks per month can be obtained directly from the Bitcoin blockchain using Python / RPC / bitcoind. The actual number of blocks differs significantly from the theoretical one, since blocks are not formed strictly every 10 minutes (for example, in 2009, in the first year of Bitcoin's existence, there were significantly fewer blocks). Knowing the number of blocks and the size of the subsidy, you can calculate the growth and stock. When calculating SF, I arbitrarily ignored the first million coins (7 months) as an adjustment for the number of lost coins. A more accurate adjustment to the number of lost coins is a subject for further research.

Bitcoin price data is available in differentsources, but begin in July 2010. I added the first known Bitcoin prices ($ 1 for 1309 BTC in October 2009, the first BitcoinMarket quote of $ 0.003 / BTC in March 2010, two pizza at $ 41 for 10,000 BTC in May 2010) and interpolated them to the missing data points. Data archeology is also the subject of further research.

And we already have data points for gold (SF 62, market capitalization of $ 8.5 trillion) and silver (SF 22, market capitalization of $ 308 billion), which I used as a guideline.

### Model

First scatter plot (scatter plot)shows that for market value it is better to use logarithmic values or an axis, since it covers 8 orders (from 10 thousand to 10 billion dollars). When using the logarithmic values or axis and for SF, a linear relationship is found between the natural logarithms (ln) of SF and market value. Note that I am using the natural (ln with base e), not the decimal logarithm (log with base 10), which would give similar results.

Establish linear regression to dataconfirms what is visible to the naked eye: a statistically significant relationship between SF and market value (95% R2, significance F 2.3E-17, p-value of slope 2.3E-17). The likelihood that the relationship between SF and market value is caused by chance is close to zero. Of course, other factors also influence the price - regulation, hacks and other news reasons - that’s why R2 is not 100% (and not all points are located on a black straight line). However, the main driver seems to be scarcity, or SF.

What is interesting is that gold and silver,completely different markets also fit into the SF values simulated for bitcoin. This gives additional confidence in the viability of the model. Note that at the peak of the bull market in December 2017, the bitcoin SF was 22, and the market capitalization was $ 230 billion, which is very close to silver.

Given how big value is on SFhalving the rewards for mining (halving), I added a color layer to the schedule with the number of months until the next halving. Dark blue indicates the month of halving, and red - the month following it. The next halving will occur in May 2020. The current value of SF 25 will double to 50, which is already very close to gold (SF 62). After each halving, the price of bitcoin increases by 8 times (pay attention to this constant factor).

Bitcoin Predicted Market Capitalizationafter halving 2020, it will be $ 1 trillion, which corresponds to a price of $ 55,000 per bitcoin. That sounds impressive. Time will tell whether this hypothesis is true. I think we will probably find out about this a year or two after the halving, in the year 2020 or 2021. This will be an excellent out-of-sample test for these hypotheses and models.

They ask me where the money can come from,necessary for Bitcoin to reach a $1 trillion market cap? My answer is: silver, gold, countries with negative interest rates (Europe, Japan, soon US), countries with predatory government regimes (Venezuela, China, Iran, Turkey, etc.), billionaires and millionaires hedging risk &# 171;quantitative easing» monetary policy, and institutional investors discovering the best performing asset of the last 10 years.

We can also simulate the price of bitcoin.directly from SF. The parameters of the formula, of course, are different, but the result is the same - 95% R2 and the predicted price of $ 55,000 / BTC at SF 50 after a halving in May 2020.

I compared the bitcoin price chart modeled on the basis of SF with the actual price chart and overlaid them with a colored layer of the number of blocks.

Pay attention to the degree of compliance,especially to the almost immediate reaction to halving in November 2012. The price adjustment after halving in June 2016 was much slower, possibly due to competition from Ethereum and the DAO hack. In addition, you can see fewer blocks per month (blue) in the first (2009) year and during downward adjustment of difficulty at the end of 2011, mid-2015 and at the end of 2018. The advent of GPU miners in 2010–2011 and ASIC miners in 2013 led to an increase in the number of blocks per month (red).

## Power Laws and Fractals

It is also very interesting that there are signs of a relationship with the law of power dependence.

Linear Regression Function: ln (market capitalization) = 3.3 * ln (SF) + 14.6

can be written as a function of the power dependence: market capitalization = exp (14.6) * SF ^ 3.3

Power dependence is rare. The probability of a power law dependence with 95% R2 of more than 8 orders of magnitude adds confidence that SF is indeed the main driver of the bitcoin price.

The law of power dependence is a relation inwhere a relative change in one quantity leads to a proportional relative change in another quantity, regardless of the initial size of these quantities. [6] In the appendix at the end of the article you can find several famous examples of the manifestation of the law of power dependence.

Power laws are interesting becausereveal the basic regularity in the properties of seemingly random complex systems. Complex systems usually have properties in which changes between processes at different scales are independent of the scales we are looking at. Thus, the picture that we observe on one scale is somewhat similar to the picture that can be seen on another scale. This property of self-similarity underlies the relations of the law of power dependence. We see this in Bitcoin: the collapses of 2011, 2014 and 2018. look very similar (in all cases it was a drop of 80%), but on completely different scales ($ 10, $ 1,000 and $ 10,000, respectively). This can be seen on a logarithmic scale. Scale invariance and self-similarity are associated with fractals. In fact, parameter 3.3 in the function of the power law above is the “fractal dimension”. For more information on fractals, see the famous study of the length of coastlines. [7] The laws of power dependence and fractals in Bitcoin are a subject for further research.

## Conclusion

Bitcoin is the first scarce digital asset in the world. It is scarce like silver and gold and can be transmitted over the Internet, radio waves, satellites, etc.

Definitely, this digital scarcity hasvalue. But how big? In this article, I quantify scarcity using the ratio of held stocks to growth in total assets, and use the ratio of stocks to growth to model the value of bitcoin.

There is a statistically significant relationshipbetween the ratio of stocks to growth and the market value of the asset. The probability that the relationship between the ratio of stocks to growth and market value is caused by chance is close to zero.

The following factors add confidence in the viability of the model:

- Gold and silver, completely different markets, also fit into the SF values simulated for bitcoin.
- There are signs of compliance with the law of power dependence.

According to the model, the projected market capitalization of Bitcoin after a halving in May 2020 will be $ 1 trillion, which corresponds to a price of $ 55,000 / BTC.

## Quoted sources

[1] https://bitcoin.org/files/bitcoin-paper/bitcoin_ru.pdf - Satoshi Nakamoto, 2008

[2] https://bitcointalk.org/index.php?topic=583.msg11405#msg11405 - Satoshi Nakamoto, 2010

[3] https://unenumerated.blogspot.com/2005/10/antiques-time-gold-and-bit-gold.html - Nick Szabo, 2008

[4] https://unenumerated.blogspot.com/2005/12/bit-gold.html - Nick Szabo, 2008

[5] The Bitcoin Standard: The Decentralized Alternative to Central Banking - Saifedean Ammous, 2018

[6] https://necsi.edu/power-law

[7] http://fractalfoundation.org/OFC/OFC-10-4.html

## Appendix: examples of the manifestation of the law of power dependence

#### Kepler (planets)

#### Richter (earthquake)

#### Kleiber (animals)

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