Circulating Supply Equilibrium for Ethereum and Minimum Viable Issuance under Proof of Stake

Hello. I’m a researcher from another field and find blockchain tech very intriguing. I have been following Ethereum for a while and appreciate the ethos of openness. The effects of a deflationary Ethereum, and the potential equilibrium for the circulating supply, seem to me like relevant topics to explore. Feel free to correct things I may have gotten wrong. In order to approach more elaborate models, I would need to understand more deeply how Ethereum approaches minimum viable issuance in proof of stake, including potential mechanisms for enforcing a minimum viable issuance policy in the future. I would appreciate links to writings about these topics.

Abstract

This text explores various equilibria for the circulating supply of Ethereum under different assumptions, and discusses the effects of enforcing a minimum viable issuance policy and the benefits and drawbacks of allowing Ethereum to be perpetually deflationary. While deflationary pressure will only have a substantial effect on the circulating supply over longer time horizons, my hope is that highlighting low potential equilibria and discussing different philosophies concerning monetary policy could be useful also in the short term.

Introduction

With the introduction of EIP-1559, a deflationary mechanism has been added to Ethereum, burning base fees. This mechanism counteracts the inflation from ETH issued for securing the network. From a “monetary policy” perspective, it could be good to estimate the equilibrium for the circulating supply under current circumstances, and to make some projections for different equilibria under various assumptions (see for example this thread, and other discussions that are connected to, or implicitly depend on, estimates of the future circulating supply).

Before modeling the equilibrium, it can be noted that if the rewards required for securing the network can be kept lower than the ether burned in base fees, Ethereum could stay deflationary for a very long time, perhaps perpetually. From one perspective, this would not be all too surprising. Payment providers are generally able to make profits year after year after they mature (with stock buybacks corresponding to burned base fees). But this could only happen under a minimum viable issuance policy, which to my understanding is not currently enforced for staking. This is discussed towards the end of the text. On the other hand, staking rewards are issued to holders that stake their ether as well, so Ethereum need not necessarily be deflationary to be a viable investment.

The models for the equilibrium serve to build an understanding of the current monetary policy, but they are not strictly needed for the arguments presented by the end of the text concerning deflation and minimum viable issuance; they could be skipped by the casual reader. This was just my first way of approaching and trying to understand the effect of the monetary policy of Ethereum, so I thought other readers could find some insights there as well.

Current statistics

During the last two weeks, more than 10 000 ETH has been burned each day on average, reaching around 8 000 ETH during days of lower demand. If we assume the lower 8 000 burned ETH/day, this corresponds to a yearly burn B of around 2.9 million ETH. Given the current circulating supply S \approx 117.4 million ETH, the burn rate b = B/S \approx 0.025. Thus, around 2.5 % of the circulating supply is burned each year at the current burn rate.

The current deposit size D for the staking contract is around 7.5 million ETH. The yearly issuance I of ether to validators under PoS can be computed from the deposit size and the base reward factor F as

I = cF \sqrt{D}

where c is a constant c \approx 2.6 derived from the number of epochs in each year and compensating for the fact that the protocol denominates ether in gwei. The base reward factor F is controlled by the developers and is set to 64. Since it determines issuance, and thus how high rewards are for staking, it could be used to control staking demand in Ethereum. I include a further discussion about this at the end of the text. Inserting the given numbers into the equation, we find that at the current rate, the yearly issuance is around 455 000 ETH.

Equilibrium with a fixed deposit size and burn rate

First, we will look at the equilibrium for the naïve case where the amount of staked ETH is kept fixed. The burn rate as a proportion of the circulating supply will also be kept fixed. Let us assume a fixed deposit size D of 15 million ETH, which would produce an initial yearly issuance of around 644 000 ETH.

The yearly burned ether B will be modeled as a burn rate b of the circulating supply S,

B = bS,

assuming that fewer ETH will be burned when the supply is reduced. The equilibrium when I = B gives us

cF \sqrt{D} = bS,

which means that the circulating supply at the equilibrium will be

S = \frac{cF \sqrt{D}}{b}.

As a base case, with a 2.5 % burn rate and 15 million ETH deposited, the equilibrium for the circulating supply would thus be 26 million ETH:

S = \frac{2.6\times64\sqrt{15\times10^6}}{0.025} \approx 2.6\times10^7 \textrm{ ETH}.

Equilibrium with a fixed deposit ratio and a variable active validator cap

It can be noted that the deposit ratio

d = \frac{D}{S}

is above 50 % (d \approx 0.58) at this equilibrium. Ethereum has had a policy of minimum viable issuance during the PoW-era, stipulating that the issuance of new tokens should be high enough to secure the blockchain, but not higher. Under proof of stake, the deposit ratio will be an important factor determining the security of the chain. If the deposit ratio is too low, Ethereum becomes less secure, and if it is too high, economic activity will be reduced in favor of locking up ether for staking. The latter case reduces the “velocity of money” and adds an inflationary pressure both due to increased issuance and decreased burn rate. There is thus a range within which the deposit ratio should be. I invite people more knowledgeable about Ethereum than me to give some insights into these matters.

A previous proposal suggests a Simplified Active Validator Cap at 2^{19} validators and 16.8 million ETH, corresponding to d \approx 0.14 at the current circulating supply. The purpose of the cap is to increase “confidence that a given level of hardware will always be sufficient to validate the beacon chain,” and it is therefore not variable. It however implies that such a deposit ratio should ensure sufficient security according to Buterin (hope this interpretation is correct). What happens to the equilibrium if we set d = 0.14? The assumption is that there would be some mechanism to enforce it, such as a variable active validator cap at 14 % staked ether.

By expressing the deposit size as a function of the circulating supply and deposit ratio,

D = Sd,

we can combine the Eqs. for the circulating supply in the base case S = \frac{cF \sqrt{D}}{b} and for the deposit size, while substituting D

S = \frac{cF \sqrt{Sd}}{b}.

This gives

\frac{Sb}{cF} = \sqrt{Sd},

leading to

\frac{S^2b^2}{c^2F^2} = Sd,

\frac{Sb^2}{c^2F^2} = d,

which provides an equation of the circulating supply as a function of the deposit ratio

S = \frac{dc^2F^2}{b^2}.

The equation stipulates that this equilibrium is independent of the current circulating supply and that the circulating supply with a variable active validator cap will fall according to the deposit ratio multiplied by a constant divided by the power of the burn rate. Using the numbers provided as a base case, we find that the equilibrium would be

S = \frac{0.14 \times 2.6^2\times 64^2}{0.025^2} \approx 6.2\times10^6 \textrm{ ETH}.

Thus, a deposit ratio kept at 14 % and a burn rate of 2.5 % would eventually push the circulating supply down to 6.2 million ETH. The figure illustrates how the equilibrium varies with burn rate and deposit ratio, with the maximum circulating supply of Bitcoin included as a reference (white line).

Equilibrium.pdf (285.3 KB)

Discussion of models enforcing minimum viable issuance

It can be noted from the example in the previous section that the staking rewards would be unrealistically high at the equilibrium, 17.9 % + tips. Thus, even with a variable active validator cap where rewards fall linearly with additional validators, a significant proportion of the circulating supply would be locked up to capture the yield. However, this would lower the velocity of money and push the burn rate lower. Such an equilibrium may thus not exist and may not be desirable from a minimum viable issuance perspective (the network could entice a sufficient number of staking validators also with a lower staking yield).

We have now reached the point where I can expand on my comment in the original thread that got me interested in the issue in the first place. As the circulating supply shrinks, the yield at the same deposit ratio becomes higher and higher. I would like to consider the effect of instead keeping yields at minimum viable issuance levels, so that Ethereum may become perpetually deflationary. In conjunction, it would be suitable to reduce the number of ether per validator in phase with the reduction of the circulating supply. It seems to me that such a policy would be best for a majority of stakeholders in the Ethereum ecosystem, with a higher velocity of money, and passive value accrual also to people who are not staking their ether. Formulated a little differently:

If Ethereum attracts a higher deposit ratio than what is strictly needed for security, would it serve the ecosystem better to slowly reduce yields (e.g., adjust the base reward factor), equally rewarding all holders and participants in the ecosystem in the form of deflation?

Here it should be mentioned that we would only see substantial effects on the circulating supply for different issuance policies when regarding longer time horizons: if we assume that a 5 % yield (+ tip) will be sufficient over the long run to keep the deposit ratio at 0.14, and use a burn rate of 2.5 %, it would take \frac{ln(21\times10^6/S)}{ln(1-0.025+0.14\times0.05)}\approx95 years for the circulating supply to reach 21 million ETH; with the second equilibrium model depicted in the figure, it would take much longer (or perhaps more realistically never, if the burn rate falls as people choose to stake their ether). So these issues are not pressing but could be interesting to expand on over the coming years.

Effects of price and market cap

I have purposefully left out price and market cap discussions from this text. It seems to me that a policy favoring staking above minimum viable issuance would not necessarily drive the market cap higher than a model where non-staked ether is equally favored. While it is true that favoring staking above minimum viable issuance will result in more locked up ether, it will also reduce (or more accurately, not to the same extent induce) demand for non-staked ether since the value accrual due to deflation will be slightly lower for non-staked ether than at minimum viable issuance. It can be mentioned here also that an increase in market cap will serve to push down the burn rate, because participants in the ecosystem will be willing to pay a lower fee in relation to the dollar-denominated market cap. But counteracting these forces is the scaling of Ethereum, which will contribute to a higher tolerance for high L1-fees when transactions move to L2; although this could initially temporarily reduce L1 demand a bit.

Undesirable long-term effect of deflation

There is an additional long-term effect of a deflationary Ethereum I would like to mention. As the circulating supply shrinks, the proportion of lost ether in relation to the ether that still can be moved will rise. Importantly, this proportion will be unknown. Therefore, at the very long time frame, determining minimum viable issuance may become harder, or such attempts even undesirable.

Conclusion

This text explores potential equilibria for the circulating supply of Ethereum. If the burn rate stays high, the circulating supply will fall substantially over the long term, approaching the circulating supply of Bitcoin in certain scenarios. The persistency of the deflation will depend on how Ethereum approaches minimum viable issuance under proof of stake. Such a policy is not currently enforced to my understanding. Enforcing minimum viable issuance could make Ethereum perpetually deflationary and will positively affect stakeholders in the Ethereum ecosystem that do not stake their ether. But monetary policy and attempts to automate it is a complex issue, and long-term effects are hard to predict currently. Enforcing minimum viable issuance would also likely come with its own issues and undesirable side effects.

Please note that I have likely overlooked certain aspects, and that this text as an attempt to learn more about the ecosystem. Hopefully, it can lead to a fruitful discussion.

Perhaps I missed it, but the burn rate isn’t a fixed value. It is a function of how much people are willing to pay for inclusion. To illustrate why this needs to be taken into consideration, you can imagine there is 1 ETH left in the supply (the rest burned), it is very unlikely that people would be willing to burn 0.1 ETH in transaction fees (10% of total supply).

If the supply decreases, the demand remains constant, and cows are spheres then one would expect the burn rate to be a function of total supply at transaction time, not an absolute/fixed value.

Thanks for reading my text. The burn rate is not modeled as a fixed value. It is modeled as the proportion of the circulating supply burned each year, so perhaps think of it as a yearly burn ratio. In these models, the circulating supply S is not a fixed constant, but the equilibrium at the point the yearly issuance I equals the yearly burned ether B.

Why is it modeled as a percentage of circulated supply burned rather than as a percentage of total supply?

The variable b corresponds to the proportion of the circulating supply that is burned each year. The “circulating supply” and “total supply” are the same thing for Ethereum, in the sense that there is not vesting. Excluding or including the ether temporarily locked in staking until they can be withdrawn does not affect the equilibrium. I however mention towards the end of the text that permanently lost ether will start to affect potential heuristics for minimum viable issuance:

As the circulating supply shrinks, the proportion of lost ether in relation to the ether that still can be moved will rise. Importantly, this proportion will be unknown. Therefore, at the very long time frame, determining minimum viable issuance may become harder, or such attempts even undesirable.

What I mean by this is that over very long time frames, under a minimum viable issuance policy, uncertainty regarding the true circulating supply (excluding permanently lost ether) will lead to uncertainty regarding the deposit ratio. At that point, enforcing minimum viable issuance would need to come with certain heuristics and safe guards (e.g., in some safe way account for ether that did not move in the last 50 years).

I am not so worried about the basics being incorrect (though of course there can always be errors). My main questions are regarding the overall philosophy concerning minimum viable issuance in the PoS era, and security considerations at different deposit ratios when looking at longer time scales.

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The concern that “lost/stuck” ETH will eventually dominate the supply is valid I think. We could do something like a 10 year turnstile at some point to help address this if it becomes a serious threat/problem.

I might have missed it. Can the burn of tokens surpass the issuance rate? meaning that it would actually be deflationary instead of just decrease the inflation rate.

Thanks

Yes. The section Current statistics specifies the situation if we were to move to PoS today with the current rates. Ethereum would burn 2 922 000 ETH (at 8000 ETH/day) and issue around 455 000 ETH yearly. This corresponds to around 2 % deflation.

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