> Credit where credit is due: after writing the bulk of this article I
found out
> that Monero developer [smooth_xmr](https://www.reddit.com/user/smooth_xmr/
)
> also observed that tail emission results in a stable coin supply
> [a few years ago](
https://www.reddit.com/r/Monero/comments/4z0azk/maam_28_monero_ask_anything_monday/d6sixyi/
).
> There's probably others too: it's a pretty obvious result.

Fwiw, Joe Lubin, April 2014:  "The expected rate of annual loss and
destruction of ETH will balance the rate of issuance.  Under this dynamic,
a quasi-steady state is reached and the amount of extant ETH no longer
grows."
https://blog.ethereum.org/2014/04/10/the-issuance-model-in-ethereum/

As you say, probably an observation various people have made.  (Ethereum
has had some updates to its issuance model since 2014, in particular
EIP-1559 and the block reward reduction coming with PoS.  But they've had a
fixed rather than halving block subsidy since launch so the question of
whether it implied infinite supply often came up.)


On Sat, Jul 9, 2022, 7:47 AM Peter Todd via bitcoin-dev <
bitcoin-dev@lists.linuxfoundation.org> wrote:

> New blog post:
>
> https://petertodd.org/2022/surprisingly-tail-emission-is-not-inflationary
>
> tl;dr: Due to lost coins, a tail emission/fixed reward actually results in
> a
> stable money supply. Not an (monetarily) inflationary supply.
>
> ...and for the purposes of reply/discussion, attached is the article
> itself in
> markdown format:
>
> ---
> layout: post
> title:  "Surprisingly, Tail Emission Is Not Inflationary"
> date:   2022-07-09
> tags:
> - bitcoin
> - monero
> ---
>
> 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
> reward[^fee-in-reward], with the exception of Ethereum, from June 2020 to
> Aug 2021.
> To date no proof-of-work currency has ever operated solely on transaction
> fees[^pow-tweet], and academic analysis has found that in this condition
> block
> generation is unstable.[^instability-without-block-reward] To paraphrase
> Andrew
> Poelstra, it's a scary phase change that no other coin has gone
> through.[^apoelstra-quote]
>
> [^pow-tweet]: [I asked on Twitter](
> https://twitter.com/peterktodd/status/1543231264597090304) and no-one
> replied with counter-examples.
>
> [^fee-in-reward]: [Average Fee Percentage in Total Block Reward](
> https://bitinfocharts.com/comparison/fee_to_reward-btc-eth-bch-ltc-doge-xmr-bsv-dash-zec.html#alltime
> )
>
> [^instability-without-block-reward]: [On the Instability of Bitcoin
> Without the Block Reward](
> https://www.cs.princeton.edu/~arvindn/publications/mining_CCS.pdf)
>
> [^apoelstra-quote]: [From a panel at TABConf 2021](
> https://twitter.com/peterktodd/status/1457066946898317316)
>
> Monero has chosen to implement what they call [tail
> emission](
> https://www.getmonero.org/resources/moneropedia/tail-emission.html):
> a fixed reward per block that continues indefinitely. Dogecoin also has a
> fixed
> reward, which they widely - and incorrectly - refer to as an "abundant"
> supply[^dogecoin-abundant].
>
> [^dogecoin-abundant]: Googling "dogecoin abundant" returns dozens of hits.
>
> 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](https://www.reddit.com/user/smooth_xmr/
> )
> also observed that tail emission results in a stable coin supply
> [a few years ago](
> https://www.reddit.com/r/Monero/comments/4z0azk/maam_28_monero_ask_anything_monday/d6sixyi/
> ).
> There's probably others too: it's a pretty obvious result.
>
>
> <div markdown="1" class="post-toc">
> # Contents
> {:.no_toc}
> 0. TOC
> {:toc}
> </div>
>
> ## 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](https://en.wikipedia.org/wiki/Exponential_decay):
>
> $$
> \frac{dN(t)}{dt} = k - \lambda N(t)
> $$
>
> 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 rate[^p2pool-tail]. Since
> the number of
> previously lost coins, and the current rate of coin loss, is
> unknown[^unknowable] 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.
>
> [^unknowable]: Being a privacy coin with [shielded amounts](
> https://localmonero.co/blocks/richlist), it's not even possible to get an
> estimate of the total amount of XMR in active circulation.
>
> [^p2pool-tail]: P2Pool operates [a page with real-time date figures](
> https://p2pool.io/tail.html).
>
> 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 larger[^btc-vs-xmr-market-cap] Bitcoin. Additionally, Monero has a
> culture of frequent hard forks that simply does not exist in Bitcoin.
>
> [^btc-vs-xmr-market-cap]: [As of writing](
> https://web.archive.org/web/20220708143920/https://www.coingecko.com/),
> the apparent market cap of Bitcoin is $409 billion, almost 200x larger than
> Monero's $2.3 billion.
>
> 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
> _______________________________________________
> bitcoin-dev mailing list
> bitcoin-dev@lists.linuxfoundation.org
> https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
>