From mboxrd@z Thu Jan 1 00:00:00 1970 Return-Path: Received: from smtp4.osuosl.org (smtp4.osuosl.org [IPv6:2605:bc80:3010::137]) by lists.linuxfoundation.org (Postfix) with ESMTP id 18F59C002D for ; Sat, 9 Jul 2022 14:26:29 +0000 (UTC) Received: from localhost (localhost [127.0.0.1]) by smtp4.osuosl.org (Postfix) with ESMTP id D0B8B419F5 for ; Sat, 9 Jul 2022 14:26:28 +0000 (UTC) DKIM-Filter: OpenDKIM Filter v2.11.0 smtp4.osuosl.org D0B8B419F5 Authentication-Results: smtp4.osuosl.org; dkim=pass (2048-bit key) header.d=voskuil-org.20210112.gappssmtp.com header.i=@voskuil-org.20210112.gappssmtp.com header.a=rsa-sha256 header.s=20210112 header.b=mhn2irBd X-Virus-Scanned: amavisd-new at osuosl.org X-Spam-Flag: NO X-Spam-Score: -0.896 X-Spam-Level: X-Spam-Status: No, score=-0.896 tagged_above=-999 required=5 tests=[BAYES_00=-1.9, DKIM_SIGNED=0.1, DKIM_VALID=-0.1, HTML_MESSAGE=0.001, MIME_QP_LONG_LINE=0.001, RCVD_IN_DNSWL_NONE=-0.0001, SPF_HELO_NONE=0.001, SPF_NONE=0.001, URI_DOTEDU_ENTITY=1] autolearn=no autolearn_force=no Received: from smtp4.osuosl.org ([127.0.0.1]) by localhost (smtp4.osuosl.org [127.0.0.1]) (amavisd-new, port 10024) with ESMTP id 7CTvDl8Xkw3U for ; Sat, 9 Jul 2022 14:26:26 +0000 (UTC) X-Greylist: whitelisted by SQLgrey-1.8.0 DKIM-Filter: OpenDKIM Filter v2.11.0 smtp4.osuosl.org 60FC0418AD Received: from mail-pj1-x102b.google.com (mail-pj1-x102b.google.com [IPv6:2607:f8b0:4864:20::102b]) by smtp4.osuosl.org (Postfix) with ESMTPS id 60FC0418AD for ; Sat, 9 Jul 2022 14:26:26 +0000 (UTC) Received: by mail-pj1-x102b.google.com with SMTP id 89-20020a17090a09e200b001ef7638e536so4422274pjo.3 for ; Sat, 09 Jul 2022 07:26:26 -0700 (PDT) DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=voskuil-org.20210112.gappssmtp.com; s=20210112; h=content-transfer-encoding:from:mime-version:subject:date:message-id :references:in-reply-to:to; bh=RiWPs0TKFlZeUz7KQvaFSs3YgPLzvDyuU6nXOXV1TX4=; b=mhn2irBdsQxN+2RBYHGbf0IpbrhNP+3Im/hQCwutIantXQ1kCMcmlhQ1WBv8EnmbPz qqdDOojaGQ9QV2HRZCfajt71NYbuO8JH/Fd9lZEEWZvpqb5zxEfK2lQjdCVNzS+iuuqm dJnCwKN4eMisyE8W/I7satV7gJZ3RWENinu5fg1xP/b6RdbegE7uYXTBZHDvRP/uHrmg Uk61dGa+RAb6Rzp9kh1S+CWA4q1QuUj7x+U9Rx3r1ui7E0EkbzDYEp4R/H1iH/diNJt8 iteBnUL8D9cS73SxLWnIRxms2tGpUpF/SS568Yy97xwdic5GtvDpmRA+XWhIhiXS/Xqu z1sQ== X-Google-DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=1e100.net; s=20210112; h=x-gm-message-state:content-transfer-encoding:from:mime-version :subject:date:message-id:references:in-reply-to:to; bh=RiWPs0TKFlZeUz7KQvaFSs3YgPLzvDyuU6nXOXV1TX4=; b=YeYx+z40o94Yxr5glc88ceqNBjGBk82iWskslhfiXHWcK+h3dMi+YWwaUMq3QaIaH0 r9ygivmK3yOZd87v84Fl5OE7KuuFmJWb4F8OyB0eBj/12bR4QEaeUIH2zD1Jk8O6OLUy s2OLPWGfp+TEun47hXex2UFlWeU4YnA4JvlXNveN5fZIdiUTelXP1KMyA4HQu/Ie5t2a kFT75kLAohQfqZ80e5J6eWlotWDHJj1Q+e4UP9oBI0ctVSZrjKscZxe21PQeKDWSMLyw K2ToD4bokSasjiSfYfq6tiP7Ny2VakrANd2EzyjWPprfuyomM82bYhUf7khD4na5Xnvp Jzzg== X-Gm-Message-State: AJIora9WpBBo/mhD4DleU6TzPXJvCRcM+KLMIQYLHS/sXmmigHKdS9DE TGYm8aQ0FqxN1qHCvWhIGlUh8IYeI8k/IA== X-Google-Smtp-Source: AGRyM1sxLH18CHs7kp+Pj6BgeMZVvNkn6simJvq5F56qGGBwqna9WyjIYZjMVsowrnuMzJ8aZfGQlw== X-Received: by 2002:a17:903:183:b0:16a:5c43:9a9c with SMTP id z3-20020a170903018300b0016a5c439a9cmr8954584plg.153.1657376785496; Sat, 09 Jul 2022 07:26:25 -0700 (PDT) Received: from smtpclient.apple ([50.35.67.197]) by smtp.gmail.com with ESMTPSA id a9-20020a17090a008900b001ef8397571asm3447226pja.35.2022.07.09.07.26.24 (version=TLS1_3 cipher=TLS_AES_128_GCM_SHA256 bits=128/128); Sat, 09 Jul 2022 07:26:24 -0700 (PDT) Content-Type: multipart/alternative; boundary=Apple-Mail-0A63F663-FFB5-4325-A4DD-CA9261D0E08C Content-Transfer-Encoding: 7bit From: Eric Voskuil Mime-Version: 1.0 (1.0) Date: Sat, 9 Jul 2022 07:26:22 -0700 Message-Id: References: In-Reply-To: To: Peter Todd , Bitcoin Protocol Discussion X-Mailer: iPhone Mail (19F77) Subject: Re: [bitcoin-dev] Surprisingly, Tail Emission Is Not Inflationary X-BeenThere: bitcoin-dev@lists.linuxfoundation.org X-Mailman-Version: 2.1.15 Precedence: list List-Id: Bitcoin Protocol Discussion List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , X-List-Received-Date: Sat, 09 Jul 2022 14:26:29 -0000 --Apple-Mail-0A63F663-FFB5-4325-A4DD-CA9261D0E08C Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable > Due to lost coins, a tail emission/fixed reward actually results in a stab= le money supply. Not an (monetarily) inflationary supply. This observation is not a proof of lost coins, that is an assumption. It is t= he provable consequence of market, as opposed to monopoly, production. https://github.com/libbitcoin/libbitcoin-system/wiki/Inflation-Principle Mises=E2=80=99 unfortunate error in the application of the Cantillon Effect t= o gold perpetuates this misperception. One could imagine applying this theor= y to all goods, not just money, and conclude perpetual loss of value in ever= ything produced, as a consequence of production. One might then be tempted t= o attribute the fact that this is not observable to loss/depreciation/consum= ption. While it is certainly possible that the amount of gold produced every= year is offset by the amount lost, this of course implies that all of it is= lost. =E2=80=9CCirculation=E2=80=9D does not determine demand, all money is always= held by someone. Changing hands only changes who owns the money, not its pu= rchasing power. See Rothbard=E2=80=99s critique of monetary =E2=80=9Cvelocit= y=E2=80=9D. e > On Jul 9, 2022, at 05:47, Peter Todd via bitcoin-dev wrote: >=20 > =EF=BB=BFNew blog post: >=20 > https://petertodd.org/2022/surprisingly-tail-emission-is-not-inflationary >=20 > tl;dr: Due to lost coins, a tail emission/fixed reward actually results in= a > stable money supply. Not an (monetarily) inflationary supply. >=20 > ...and for the purposes of reply/discussion, attached is the article itsel= f in > markdown format: >=20 > --- > layout: post > title: "Surprisingly, Tail Emission Is Not Inflationary" > date: 2022-07-09 > tags: > - bitcoin > - monero > --- >=20 > At present, all notable proof-of-work currencies reward miners with both a= block > reward, and transaction fees. With most currencies (including Bitcoin) pha= sing > 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 A= ug 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 b= lock > generation is unstable.[^instability-without-block-reward] To paraphrase A= ndrew > Poelstra, it's a scary phase change that no other coin has gone through.[^= apoelstra-quote] >=20 > [^pow-tweet]: [I asked on Twitter](https://twitter.com/peterktodd/status/1= 543231264597090304) and no-one replied with counter-examples. >=20 > [^fee-in-reward]: [Average Fee Percentage in Total Block Reward](https://b= itinfocharts.com/comparison/fee_to_reward-btc-eth-bch-ltc-doge-xmr-bsv-dash-= zec.html#alltime) >=20 > [^instability-without-block-reward]: [On the Instability of Bitcoin Withou= t the Block Reward](https://www.cs.princeton.edu/~arvindn/publications/minin= g_CCS.pdf) >=20 > [^apoelstra-quote]: [=46rom a panel at TABConf 2021](https://twitter.com/p= eterktodd/status/1457066946898317316) >=20 > Monero has chosen to implement what they call [tail > emission](https://www.getmonero.org/resources/moneropedia/tail-emission.ht= ml): > a fixed reward per block that continues indefinitely. Dogecoin also has a f= ixed > reward, which they widely - and incorrectly - refer to as an "abundant" su= pply[^dogecoin-abundant]. >=20 > [^dogecoin-abundant]: Googling "dogecoin abundant" returns dozens of hits.= >=20 > 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 inflation= ary > nor deflationary, with the total supply proportional to rate of tail emiss= ion > and probability of coin loss. >=20 > 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. >=20 >=20 >
> # Contents > {:.no_toc} > 0. TOC > {:toc} >
>=20 > ## Modeling the Fixed-Reward Monetary Supply >=20 > 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 th= e > block reward is fixed we can model the reward as a slope $$k$$ added to an= > initial supply $$N_0$$: >=20 > $$ > N(t) =3D N_0 + kt > $$ >=20 > 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 becau= se > you recently lost your coins in a boating accident =E2=80=94 an accident I= probably > don't even know happened. Since the number of individual coins (and their > owners) is large =E2=80=94 as with the number of blocks =E2=80=94 we can m= odel this loss as > though it happens continuously. >=20 > Since coins can only be lost once, the *rate* of coin loss at time $$t$$ i= s > proportional to the total supply *at that moment* in time. So let's look a= t the > *first derivative* of our fixed-reward coin supply: >=20 > $$ > \frac{dN(t)}{dt} =3D k > $$ >=20 > ...and subtract from it the lost coins, using $$\lambda$$ as our [coin los= s > constant](https://en.wikipedia.org/wiki/Exponential_decay): >=20 > $$ > \frac{dN(t)}{dt} =3D k - \lambda N(t) > $$ >=20 > That's a first-order differential equation, which can be easily solved wit= h > separation of variables to get: >=20 > $$ > N(t) =3D \frac{k}{\lambda} - Ce^{-\lambda t} > $$ >=20 > To remove the integration constant $$C$$, let's look at $$t =3D 0$$, where= the > coin supply is $$N_0$$: >=20 > $$ > \begin{align} > N_0 &=3D \frac{k}{\lambda} - Ce^{-\lambda 0} =3D \frac{k}{\lambda} - C \= \ > C &=3D \frac{k}{\lambda} - N_0 > \end{align} > $$ >=20 > Thus: >=20 > $$ > \begin{align} > N(t) &=3D \frac{k}{\lambda} - \left(\frac{k}{\lambda} - N_0 \right)e^{-= \lambda t} \\ > &=3D \frac{k}{\lambda} + \left(N_0 - \frac{k}{\lambda} \right)e^{-= \lambda t} > \end{align} > $$ >=20 >=20 > ## Long Term Coin Supply >=20 > It's easy to see that in the long run, the second half of the coin supply > equation goes to zero because $$\lim_{t \to \infty} e^{-\lambda t} =3D 0$$= : >=20 > $$ > \begin{align} > \lim_{t \to \infty} N(t) &=3D \lim_{t \to \infty} \left[ \frac{k}{\lamb= da} + \left(N_0 - \frac{k}{\lambda} \right)e^{-\lambda t} \right ] =3D \frac= {k}{\lambda} \\ > N(\infty) &=3D \frac{k}{\lambda} > \end{align} > $$ >=20 > An intuitive explanation for this result is that in the long run, the init= ial > supply $$N_0$$ doesn't matter, because approximately all of those coins wi= ll > eventually be lost. Thus in the long run, the coin supply will converge to= wards > $$\frac{k}{\lambda}$$, the point where coins are created just as fast as t= hey > are lost. >=20 >=20 > ## Short Term Dynamics and Economic Considerations >=20 > Of course, the intuitive explanation for why supply converges to > $$\frac{k}{\lambda}$$, also tells us that supply must converge fairly slow= ly: > if 1% of something is lost per year, after 100 years 37% of the initial su= pply > remains. It's not clear what the rate of lost coins actually is in a matur= e, > valuable, coin. But 1%/year is likely to be a good guess =E2=80=94 quite p= ossibly less. >=20 > In the case of Monero, they've introduced tail emission at a point where i= t > represents a 0.9% apparent monetary inflation rate[^p2pool-tail]. Since th= e 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 monet= ary > inflation rate is right now. But regardless, the rate will only converge > towards zero going forward. >=20 > [^unknowable]: Being a privacy coin with [shielded amounts](https://localm= onero.co/blocks/richlist), it's not even possible to get an estimate of the t= otal amount of XMR in active circulation. >=20 > [^p2pool-tail]: P2Pool operates [a page with real-time date figures](https= ://p2pool.io/tail.html). >=20 > 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 l= ower > on day zero due to lost coins, and in the long run, it will converge towar= ds > zero. >=20 > 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% dr= op. > 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 t= o > use some mildly interesting math, in the end it's totally pedantic. >=20 > ## Could Bitcoin Add Tail Emission? >=20 > ...and why could Monero? >=20 > Adding tail emission to Bitcoin would be a hard fork: a incompatible rule > change that existing Bitcoin nodes would reject as invalid. While Monero w= as > able to get sufficiently broad consensus in the community to implement tai= l > emission, it's unclear at best if it would ever be possible to achieve tha= t 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. >=20 > [^btc-vs-xmr-market-cap]: [As of writing](https://web.archive.org/web/2022= 0708143920/https://www.coingecko.com/), the apparent market cap of Bitcoin i= s $409 billion, almost 200x larger than Monero's $2.3 billion. >=20 > Ultimately, as long as a substantial fraction of the Bitcoin community con= tinue > to run full nodes, the only way tail emission could ever be added to Bitco= in is > by convincing that same community that it is a good idea. >=20 >=20 > ## Footnotes > _______________________________________________ > bitcoin-dev mailing list > bitcoin-dev@lists.linuxfoundation.org > https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev --Apple-Mail-0A63F663-FFB5-4325-A4DD-CA9261D0E08C Content-Type: text/html; charset=utf-8 Content-Transfer-Encoding: quoted-printable
> Due to lost coins, a tail e= mission/fixed reward actually results in a stable money supply. Not an (mone= tarily) inflationary supply.
<= span style=3D"caret-color: rgb(255, 255, 255); -webkit-text-size-adjust: aut= o;">
Th= is observation is not a proof of lost coins, that is an assumption. It is th= e provable consequence of market, as opposed to monopoly, production.

https://github.com/libbitcoin= /libbitcoin-system/wiki/Inflation-Principle

Mises=E2=80=99 unfortunate error in the application of the Ca= ntillon Effect to gold perpetuates this misperception. One could imagine app= lying this theory to all goods, not just money, and conclude perpetual loss o= f value in everything produced, as a consequence of production. One might th= en be tempted to attribute the fact that this is not observable to loss/depr= eciation/consumption. While it is certainly possible that the amount of gold= produced every year is offset by the amount lost, this of course implies th= at all of it is lost.

=E2=80= =9CCirculation=E2=80=9D does not determine demand, all money is always held b= y someone. Changing hands only changes who owns the money, not its purchasin= g power. See Rothbard=E2=80=99s critique of monetary =E2=80=9Cvelocity=E2=80= =9D.

e

On Jul 9, 2022, at 05:47, Peter T= odd via bitcoin-dev <bitcoin-dev@lists.linuxfoundation.org> wrote:
=
=EF=BB=BF<= span>New blog post:

https://petertodd.org/2= 022/surprisingly-tail-emission-is-not-inflationary
tl;dr: Due to lost coins, a tail emission/fixed reward actually resu= lts in a
stable money supply. Not an (monetarily) inflationa= ry supply.

...and for the purposes of reply= /discussion, attached is the article itself in
markdown form= at:

---
layout: post=
title:  "Surprisingly, Tail Emission Is Not Inflationary"
date:   2022-07-09
tags:
- bitcoin
- monero
---
<= /span>
At present, all notable proof-of-work currencies reward mine= rs with both a block
reward, and transaction fees. With most= currencies (including Bitcoin) phasing
out block rewards ov= er time. However in no currency have transaction fees
consis= tently 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.<= /span>
To date no proof-of-work currency has ever operated solely o= n transaction
fees[^pow-tweet], and academic analysis has fo= und that in this condition block
generation is unstable.[^in= stability-without-block-reward] To paraphrase Andrew
Poelstr= a, 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 replie= d with counter-examples.

[^fee-in-reward]: [= Average Fee Percentage in Total Block Reward](https://bitinfocharts.com/comp= arison/fee_to_reward-btc-eth-bch-ltc-doge-xmr-bsv-dash-zec.html#alltime)

[^instability-without-block-reward]: [On the I= nstability of Bitcoin Without the Block Reward](https://www.cs.princeton.edu= /~arvindn/publications/mining_CCS.pdf)

[^ap= oelstra-quote]: [=46rom a panel at TABConf 2021](https://twitter.com/peterkt= odd/status/1457066946898317316)

Monero has c= hosen to implement what they call [tail
emission](https://ww= w.getmonero.org/resources/moneropedia/tail-emission.html):
a= fixed reward per block that continues indefinitely. Dogecoin also has a fix= ed
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 re= ward does **not** lead to an
abundant supply. In fact, due t= o the inevitability of lost coins, a fixed
reward converges t= o a **stable** monetary supply that is neither inflationary
= nor deflationary, with the total supply proportional to rate of tail emissio= n
and probability of coin loss.

<= span>Credit where credit is due: after writing the bulk of this article I fo= und out
that Monero developer [smooth_xmr](https://www.reddi= t.com/user/smooth_xmr/)
also observed that tail emission res= ults in a stable coin supply
[a few years ago](https://www.r= eddit.com/r/Monero/comments/4z0azk/maam_28_monero_ask_anything_monday/d6sixy= i/).
There's probably others too: it's a pretty obvious resu= lt.


<div markdown=3D"1"= class=3D"post-toc">
# Contents
{:.no_toc= }
0. TOC
{:toc}
</div>=

## Modeling the Fixed-Reward Monetary Supp= ly

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

$$
<= span>N(t) =3D N_0 + kt
$$

O= f course, this isn't realistic as coins are constantly being lost due to
deaths, forgotten passphrases, boating accidents, etc. These lo= sses are
independent: I'm not any more or less likely to for= get my passphrase because
you recently lost your coins in a b= oating accident =E2=80=94 an accident I probably
don't even k= now happened. Since the number of individual coins (and their
owners) is large =E2=80=94 as with the number of blocks =E2=80=94 we can m= odel 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 t= hat moment* in time. So let's look at the
*first derivative*= of our fixed-reward coin supply:

$$=
\frac{dN(t)}{dt} =3D k
$$
<= br>...and subtract from it the lost coins, using $$\lambda$$ as our [c= oin loss
constant](https://en.wikipedia.org/wiki/Exponential= _decay):

$$
\frac{dN(t)}{dt= } =3D k - \lambda N(t)
$$

T= hat's a first-order differential equation, which can be easily solved with
separation of variables to get:

$$
N(t) =3D \frac{k}{\lambda} - Ce^{-\lambda t}$$

To remove the integration consta= nt $$C$$, let's look at $$t =3D 0$$, where the
coin supply i= s $$N_0$$:

$$
\begin{align}=
   N_0 &=3D \frac{k}{\lambda} - Ce^{-\l= ambda 0} =3D \frac{k}{\lambda} - C \\
   &nb= sp; C &=3D \frac{k}{\lambda} - N_0
\end{align}
$$

Thus:

$$
\begin{align}
  &nb= sp;N(t) &=3D \frac{k}{\lambda} - \left(\frac{k}{\lambda} - N_0 \right)e^= {-\lambda t} \\
       &= nbsp;&=3D \frac{k}{\lambda} + \left(N_0 - \frac{k}{\lambda} \right)e^{-\= lambda t}
\end{align}
$$


## Long Term Coin Supply

It's easy to see that in the long run, the second half of the c= oin supply
equation goes to zero because $$\lim_{t \to \inft= y} e^{-\lambda t} =3D 0$$:

$$
\begin{align}
   \lim_{t \to \infty} N(t)= &=3D \lim_{t \to \infty} \left[ \frac{k}{\lambda} + \left(N_0 - \frac{k= }{\lambda} \right)e^{-\lambda t} \right ] =3D \frac{k}{\lambda} \\            &n= bsp;      N(\infty) &=3D \frac{k}{\lambda}=
\end{align}
$$

<= span>An intuitive explanation for this result is that in the long run, the i= nitial
supply $$N_0$$ doesn't matter, because approximately a= ll of those coins will
eventually be lost. Thus in the long r= un, the coin supply will converge towards
$$\frac{k}{\lambda= }$$, the point where coins are created just as fast as they
= are lost.


## Short Term Dy= namics and Economic Considerations

Of cours= e, 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 t= he initial supply
remains. It's not clear what the rate of l= ost coins actually is in a mature,
valuable, coin. But 1%/ye= ar is likely to be a good guess =E2=80=94 quite possibly less.

In the case of Monero, they've introduced tail emission a= t a point where it
represents a 0.9% apparent monetary infla= tion rate[^p2pool-tail]. Since the number of
previously lost= coins, and the current rate of coin loss, is
unknown[^unkno= wable] 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://localm= onero.co/blocks/richlist), it's not even possible to get an estimate of the t= otal 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 d= ecides to implement tail emission as a means to fund
securit= y, 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 i= nflation will be even lower
on day zero due to lost coins, a= nd in the long run, it will converge towards
zero.
The fact is, economic volatility dwarfs the effect o= f small amounts of
inflation. Even a 0.5% inflation rate ove= r 50 years only leads to a 22% drop.
Meanwhile at the time o= f writing, Bitcoin has dropped 36% in the past year, and
gai= ned 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 peda= ntic.

## Could Bitcoin Add Tail Emission?

...and why could Monero?

Adding tail emission to Bitcoin would be a hard fork: a incomp= atible rule
change that existing Bitcoin nodes would reject a= s invalid. While Monero was
able to get sufficiently broad c= onsensus in the community to implement tail
emission, it's u= nclear at best if it would ever be possible to achieve that for
the much larger[^btc-vs-xmr-market-cap] Bitcoin. Additionally, Monero ha= s a
culture of frequent hard forks that simply does not exis= t in Bitcoin.

[^btc-vs-xmr-market-cap]: [As= of writing](https://web.archive.org/web/20220708143920/https://www.coingeck= o.com/), the apparent market cap of Bitcoin is $409 billion, almost 200x lar= ger than Monero's $2.3 billion.

Ultimately,= as long as a substantial fraction of the Bitcoin community continue<= br>to run full nodes, the only way tail emission could ever be added t= o Bitcoin is
by convincing that same community that it is a g= ood idea.


## Footnotes
_______________________________________________
bitcoin-dev mailing list
bitcoin-dev@lists.linuxfoundation.= org
https://lists.linuxfoundation.org/mailman/listinfo/bitco= in-dev
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