Cryptoasset researcher Lanre Ige analyzes Adam Hayes’s cost of production method and Ken Alabi’s Metcalfe’s Law framework as part of his larger paper on cryptoasset valuations.

There is growing interest in cryptoasset valuation techniques and frameworks. I have released a larger piece of research on the topic and over the next few days I will be releasing shorter overviews of the main topics covered in the paper which can be found here:

In this article I will cover Adam Hayes’s cost of production method and Ken Alabi’s Metacalfe’s Law framework. I urge you to first read those two papers before reading this.

Cost of Production method

One interesting and intuitive approach to cryptoasset valuation has been the Cost of Production method by Adam Hayes. He argues that market price of a bitcoin will have its lower bound theoretically set at the marginal cost of bitcoin mining in a competitive market. The marginal product of mining a bitcoin should theoretically, according to Hayes, equal its marginal cost in a competitive market which should, in turn, equal its selling price. Hayes expresses cost per day as $/day and mining production is expressed as BTC/day. Then the lower bound of $/BTC price is simply the ratio of the two. This price, p*, serves as a theoretically lower bound for the market price, below which a miner would operate at a marginal loss.

Therefore, p* can be expressed as follows:

Where E represents $/day. A full breakdown of how this formula was derived can be found in my paper. According to Hayes, if the market price were to drop below p* miners would be operating at a marginal loss and halt production. The below diagrams show the cost of production of a single bitcoin in a selection of countries.

Other considerations

This analysis is an interesting approach to understanding the economic drivers underlying the price of a bitcoin. However, the Hayes model raises several concerns.

1. Transaction fees

The first concern is that Hayes fails to address the effect transaction fees accrued by miners may have on their incentives. Miners are rewarded in bitcoin but also in transaction fees paid by those who use the network. The average Bitcoin transaction fee stands at around 2.673 USD¹, and with the average number of transactions per block currently sitting at around 850, successful miners of a given block can expect to receive around $2,000 in transaction fees. This will only make a small difference to the lower bound of the bitcoin price calculation, given that $2,000 is around 1.5% of the USD price of bitcoin a successful miner currently receives. However, it is not inconceivable for future bitcoin transaction fees to be 20x what they currently are, given that they peaked at an average of $55 in December². At the same time in December, the average number of transactions per block was around 2,450. Here the total transaction fees in USD represented a much higher percentage of the total mining reward (transaction fees + block reward); in such a situation it becomes much more important for the model to consider accounting for transaction fees.

2. Mining centralization

Another concern with Hayes’s model is his assumption that bitcoin mining is a competitive pursuit. The top five bitcoin mining pools currently command 74% of the network’s total hashing power. See the pie chart below:

The current state of the bitcoin mining market is more aptly described as oligopolistic than (perfectly) competitive. Several of the findings in a 2018 paper³ — which studied the distribution of mining power on the Bitcoin network — are useful in understanding bitcoin mining centralization. Moreover, the top four Bitcoin miners have more than 53% of the average mining power. Going further, 90% of the mining power of Bitcoin is controlled by only 16 miners; though this is tempered by the fact that the largest miners are all mining pools whose participants do have the ability to move to competing pools.

The point should be clear that Hayes’ assumption of perfect competition, and therefore his argument for the price of a bitcoin being set where the marginal product of mining is equal to the marginal cost, is shaky at best. Bitcoin miners offer an identical product — electricity — and are price takers, but the economies of scale that are present (due to the high cost of ASIC mining rigs and Bitmain’s monopoly on their production) have led to a degree of centralization. Although there isn’t a single framework to describe oligopolistic markets, further work could be done in this area with some commonly used models like Cournot-Nash, Bertrand, or Kinked Demand.

3. Drivers of mining difficulty

One final, especially interesting variable highlighted by Hayes is that of mining difficulty, δ. To keep the rate of bitcoin creation relatively consistent, mining difficulty adjusts every 2016 blocks according to the overall hash power of current miners. Therefore, mining difficulty can be said to be a function of the total hash power of bitcoin mining, Ρ, such that:

where Z is white noise representing the mining difficulty deviation (from ideal difficulty for the given hash power) from the 2016-block delay.

Interestingly, the total hash power of mining is a function of the (1) bitcoin price, (2) macroeconomic circumstance (a nation-state banning cryptocurrencies or positive legal change, for example), and (3) technological advancements (improvements in ASIC quality, for example). With regards to (1), one may notice the slightly circular (or at the very least recursive) nature of Hayes’s bitcoin price framework. The lower bound of the bitcoin price depends partially on mining difficulty, which in turn depends on total mining hash power, which in turn depends on the bitcoin price. Bitcoin mining displays a reflexive relationship with its price. This is not necessarily a problem, but it does leave us wondering whether there is a more fundamental way (based on exogenous factors) to measure cryptoassets like bitcoin.

4. Application to other cryptoassets

As a final note on this framework, it is unclear how well that this can be applied to other cryptoassets. Some cryptoassets (e.g., Delegated Proof of Stake projects, future Proof of Stake implementations for Ethereum) do not use electricity-based mining; while there isn’t a specific timeline on a Proof of Stake consensus algorithm implementation for Ethereum, the ultimate aim is to transition to Proof of Stake.

Metcalfe’s Law

Ken Alabi⁴ argues that the value of certain cryptoasset networks (such as Bitcoin, Ethereum, and Dash) can be modelled using Metcalfe’s Law. Metcalfe’s Law says that the value of a network is proportional to the number of its nodes or end users. More specifically, the value of the network is proportional to the square of the nodes of the network (V ∝ N²) where V is the network value and N is the number of nodes. The relationship between network value and size is known as the network effect.

Metcalfe’s Law can be formalized as:

Alabi’s study was subject to the following parameters:

The network value is modelled by the price of the network’s digital currency. Price and market capitalization (network value) have a direct relationship and are used interchangeably. The number of end users is the number of unique addresses participating in the network per day. The price curve on the network will contain bubbles and bursts — noise deviating from the mean — that should be filtered out to ascertain the ‘true model of growth and value of the network’. Network growth (under Metcalfe’s Law) begins once critical mass is reached. Critical mass is defined as the threshold number of users from which the network becomes viral.

Eventually, Alabi’s value function eventually becomes:

Where V(N) is the value of a cryptoasset network, N is the number of people holding the asset at a given time, C is a constant and, γ is the network’s intrinsic scaling factor. I advise you to read Alabi’s article, as well, to fully understand the model and his empirical findings⁵.

Is Metcalfe’s Law valid?

A common criticism⁶ of valuation methodologies that use Metcalfe’s Law is the claim that Metcalfe’s Law is not a good estimate for the value of networks. To get an understanding of the grounds for such an argument it is useful to compare the other most common network models⁷:

Metcalfe’s Law:

As more individuals join a network, each adds to the overall network value in a non-linearly fashion. The mathematics behind the law is based on pair-wise connections apparent in systems like telephone networks; for example, if there are 5 people with telephones, there can be a maximum of 10 connections (4 + 3 + 2 + 1) — assuming equality among the members’ network connections. The value of the network is derived from the sum of all possible pairings between users and is therefore generalized for n users as:

Sarnoff’s Law:

The value of a (broadcast) network is directly proportional to the number of viewers. Value is created through the network’s one-to-many relationship and not peer-to-peer.

Reed’s Law:

The utility of large networks scales exponentially with the size of the network. This is because the number of possible sub-groups of network participants is generalized as:

Odlyzko’s Law

The growth rate of the network decreases as new members join because the most valuable links are likely to be formed early on.

Critics of Metcalfe’s Law suggest that one of the other laws may more accurately reflect the Bitcoin networks. Reed’s Law, however, seems undoubtedly to be a worse fit than Metcalfe’s Law given that the concept of ‘sub-groups’ isn’t coherent in the Bitcoin financial transaction network. In addition, Sarnoff’s Law seems too conservative since it seems to imply that the sum of individual disconnected notes in a cryptoasset are equal to a single network constituted of each node. This argument seems nonsensical, since there are undoubtedly some network effects present on cryptoasset networks and they do influence the value (especially the fundamental financial value) of said network.

Having eliminated the other models, this leaves Metcalfe’s Law and Odlyzko’s Law. The crucial difference between the two is that the former assumes homogeneity between the value added for each new node introduced to the network whilst the latter assumes diminishing returns to value for newer nodes. Odlyzko and Tilly (2005) suggest that Metcalfe’s Law “provides irresistible incentives for all networks relying on the same technology to merge” and this conclusion is divorced from the reality of modern networks, thus untenable.

We can apply similar logic to cryptoasset network forks. Consider a single network with n members. The network’s participants decide to fork, such that the first network has 9n/10 users and the second n/10 users. Therefore:

Their combined value would be:

Compared to the value of a single, unforked network:

When we solve forking transaction costs, ð, (where ð < 0) and assume that the cumulative value of the forked networks (accounting for transaction costs) are greater than that of the original network, we see that |ð|>1.82k · n². This implies that the transaction costs of forking a cryptoasset are likely quite since k is likely much greater than 1/n² for large values of n. Given the open source nature of most cryptoasset networks and the ease with which a developer can fork a given network, such a claim seems unlikely. Under Odlyzko’s law, the transaction costs of forking would be much smaller and scale logarithmically with the cryptoasset network’s number of users.

This implies that early cryptoasset networks may face large transaction costs to fork (relative to their overall value) but more mature ones would not. Based off the ‘digital gold’ investment thesis of a cryptoasset like Bitcoin, Odlyzko’s Law seems like a much more appropriate model for network value. Perhaps, Metcalfe’s Law will be more appropriate for a strictly medium-of-exchange cryptoasset where there are likely to be greater network effects and the value added by each node in the network is more evenly distributed.

An observation made by Alabi is that his proposed model does not result in a ‘pre-ordained exponent’ in the same way the N² formulation of Metcalfe’s Law does. Exponential growth is not always guaranteed with N under his model as γ can be altered to match the fundamentals of a given cryptoasset network.

Conclusion

This article has covered some of the main points made in my larger research paper. Over the next few days I will be releasing the next article in this series which will cover ‘Equation of Exchange’ valuation attempts. If you have enjoyed this article please have a read of the full paper here and leave any comments or criticisms below.

Note: Any views contained in this article represent those of the author and do not reflect the views of the Mosaic.

The full paper can be found here. Click here to visit our website.

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[1] https://bitinfocharts.com/comparison/bitcoin-transactionfees.html as of 26/02/18 15:18 UTC

[2] https://bitinfocharts.com/comparison/bitcoin-transactionfees.html

[3] Adem Efe Gencer, Soumya Basu, Ittay Eyal, Robbert van Renesse, and Emin Gün Sirer “Decentralized in Bitcoin and Ethereum Networks” (2018) https://arxiv.org/pdf/1801.03998.pdf

[4] Hayes, A. S. (2017). Cryptocurrency value formation: An empirical study leading to a cost of production model for valuing bitcoin. Telematics and Informatics, 34(7), 1308–1321.

[5] https://medium.com/@alabs.ken/a-macro-mathematical-model-for-the-observed-value-of-digital-blockchain-networks-23cc8e0dc7ea

[6] See Odlyzko and Tilly (2005) A refutation of Metcalfe’s Law and a better estimate for the value of networks and network interconnections. (Unpublished manuscript.) http://www.dtc.umn.edu/~odlyzko/doc/metcalfe.pdf

[7] This section borrows from Peterson (2017) — Metcalfe’s Law as a Model for Bitcoin’s Value