At present, all notable proof-of-work currencies reward miners with both a block reward, and transaction fees. With most currencies (including Bitcoin) phasing out block rewards over time. However in no currency have transaction fees consistently been more than 5% to 10% of the total mining reward1, with the exception of Ethereum, from June 2020 to Aug 2021. To date no proof-of-work currency has ever operated solely on transaction fees2, and academic analysis has found that in this condition block generation is unstable.3 To paraphrase Andrew Poelstra, it’s a scary phase change that no other coin has gone through.4

Monero has chosen to implement what they call tail emission: a fixed reward per block that continues indefinitely. Dogecoin also has a fixed reward, which they widely - and incorrectly - refer to as an “abundant” supply5.

This article will show that a fixed block reward does not lead to an abundant supply. In fact, due to the inevitability of lost coins, a fixed reward converges to a stable monetary supply that is neither inflationary nor deflationary, with the total supply proportional to rate of tail emission and probability of coin loss.

Credit where credit is due: after writing the bulk of this article I found out that Monero developer smooth_xmr also observed that tail emission results in a stable coin supply a few years ago. There’s probably others too: it’s a pretty obvious result.

# Contents

## Modeling the Fixed-Reward Monetary Supply

Since the number of blocks is large, we can model the monetary supply as a continuous function $$N(t)$$, where $$t$$ is a given moment in time. If the block reward is fixed we can model the reward as a slope $$k$$ added to an initial supply $$N_0$$:

$N(t) = N_0 + kt$

Of course, this isn’t realistic as coins are constantly being lost due to deaths, forgotten passphrases, boating accidents, etc. These losses are independent: I’m not any more or less likely to forget my passphrase because you recently lost your coins in a boating accident — an accident I probably don’t even know happened. Since the number of individual coins (and their owners) is large — as with the number of blocks — we can model this loss as though it happens continuously.

Since coins can only be lost once, the rate of coin loss at time $$t$$ is proportional to the total supply at that moment in time. So let’s look at the first derivative of our fixed-reward coin supply:

$\frac{dN(t)}{dt} = k$

…and subtract from it the lost coins, using $$\lambda$$ as our coin loss constant:

$\frac{dN(t)}{dt} = k - \lambda N(t)$

That’s a first-order differential equation, which can be easily solved with separation of variables to get:

$N(t) = \frac{k}{\lambda} - Ce^{-\lambda t}$

To remove the integration constant $$C$$, let’s look at $$t = 0$$, where the coin supply is $$N_0$$:

\begin{align} N_0 &= \frac{k}{\lambda} - Ce^{-\lambda 0} = \frac{k}{\lambda} - C \\ C &= \frac{k}{\lambda} - N_0 \end{align}

Thus:

\begin{align} N(t) &= \frac{k}{\lambda} - \left(\frac{k}{\lambda} - N_0 \right)e^{-\lambda t} \\ &= \frac{k}{\lambda} + \left(N_0 - \frac{k}{\lambda} \right)e^{-\lambda t} \end{align}

## Long Term Coin Supply

It’s easy to see that in the long run, the second half of the coin supply equation goes to zero because $$\lim_{t \to \infty} e^{-\lambda t} = 0$$:

\begin{align} \lim_{t \to \infty} N(t) &= \lim_{t \to \infty} \left[ \frac{k}{\lambda} + \left(N_0 - \frac{k}{\lambda} \right)e^{-\lambda t} \right ] = \frac{k}{\lambda} \\ N(\infty) &= \frac{k}{\lambda} \end{align}

An intuitive explanation for this result is that in the long run, the initial supply $$N_0$$ doesn’t matter, because approximately all of those coins will eventually be lost. Thus in the long run, the coin supply will converge towards $$\frac{k}{\lambda}$$, the point where coins are created just as fast as they are lost.

## Short Term Dynamics and Economic Considerations

Of course, the intuitive explanation for why supply converges to $$\frac{k}{\lambda}$$, also tells us that supply must converge fairly slowly: if 1% of something is lost per year, after 100 years 37% of the initial supply remains. It’s not clear what the rate of lost coins actually is in a mature, valuable, coin. But 1%/year is likely to be a good guess — quite possibly less.

In the case of Monero, they’ve introduced tail emission at a point where it represents a 0.9% apparent monetary inflation rate6. Since the number of previously lost coins, and the current rate of coin loss, is unknown7 it’s not possible to know exactly what the true monetary inflation rate is right now. But regardless, the rate will only converge towards zero going forward.

If an existing coin decides to implement tail emission as a means to fund security, choosing an appropriate emission rate is simple: decide on the maximum amount of inflation you are willing to have in the worst case, and set the tail emission accordingly. In reality monetary inflation will be even lower on day zero due to lost coins, and in the long run, it will converge towards zero.

The fact is, economic volatility dwarfs the effect of small amounts of inflation. Even a 0.5% inflation rate over 50 years only leads to a 22% drop. Meanwhile at the time of writing, Bitcoin has dropped 36% in the past year, and gained 993% over the past 5 years. While this discussion is a nice excuse to use some mildly interesting math, in the end it’s totally pedantic.

## Could Bitcoin Add Tail Emission?

…and why could Monero?

Adding tail emission to Bitcoin would be a hard fork: a incompatible rule change that existing Bitcoin nodes would reject as invalid. While Monero was able to get sufficiently broad consensus in the community to implement tail emission, it’s unclear at best if it would ever be possible to achieve that for the much larger8 Bitcoin. Additionally, Monero has a culture of frequent hard forks that simply does not exist in Bitcoin.

Ultimately, as long as a substantial fraction of the Bitcoin community continue to run full nodes, the only way tail emission could ever be added to Bitcoin is by convincing that same community that it is a good idea.

## Footnotes

5. As of writing, the apparent market cap of Bitcoin is $409 billion, almost 200x larger than Monero’s$2.3 billion.