{"type":"rich","version":"1.0","title":"Peter Todd [ARCHIVE] wrote","author_name":"Peter Todd [ARCHIVE] (npub1m2…a2np2)","author_url":"https://yabu.me/npub1m230cem2yh3mtdzkg32qhj73uytgkyg5ylxsu083n3tpjnajxx4qqa2np2","provider_name":"njump","provider_url":"https://yabu.me","html":"📅 Original date posted:2022-07-09\n📝 Original message:New blog post:\n\nhttps://petertodd.org/2022/surprisingly-tail-emission-is-not-inflationary\n\ntl;dr: Due to lost coins, a tail emission/fixed reward actually results in a\nstable money supply. Not an (monetarily) inflationary supply.\n\n...and for the purposes of reply/discussion, attached is the article itself in\nmarkdown format:\n\n---\nlayout: post\ntitle:  \"Surprisingly, Tail Emission Is Not Inflationary\"\ndate:   2022-07-09\ntags:\n- bitcoin\n- monero\n---\n\nAt present, all notable proof-of-work currencies reward miners with both a block\nreward, and transaction fees. With most currencies (including Bitcoin) phasing\nout block rewards over time. However in no currency have transaction fees\nconsistently been more than 5% to 10% of the total mining\nreward[^fee-in-reward], with the exception of Ethereum, from June 2020 to Aug 2021.\nTo date no proof-of-work currency has ever operated solely on transaction\nfees[^pow-tweet], and academic analysis has found that in this condition block\ngeneration is unstable.[^instability-without-block-reward] To paraphrase Andrew\nPoelstra, it's a scary phase change that no other coin has gone through.[^apoelstra-quote]\n\n[^pow-tweet]: [I asked on Twitter](https://twitter.com/peterktodd/status/1543231264597090304) and no-one replied with counter-examples.\n\n[^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)\n\n[^instability-without-block-reward]: [On the Instability of Bitcoin Without the Block Reward](https://www.cs.princeton.edu/~arvindn/publications/mining_CCS.pdf)\n\n[^apoelstra-quote]: [From a panel at TABConf 2021](https://twitter.com/peterktodd/status/1457066946898317316)\n\nMonero has chosen to implement what they call [tail\nemission](https://www.getmonero.org/resources/moneropedia/tail-emission.html):\na fixed reward per block that continues indefinitely. Dogecoin also has a fixed\nreward, which they widely - and incorrectly - refer to as an \"abundant\" supply[^dogecoin-abundant].\n\n[^dogecoin-abundant]: Googling \"dogecoin abundant\" returns dozens of hits.\n\nThis article will show that a fixed block reward does **not** lead to an\nabundant supply. In fact, due to the inevitability of lost coins, a fixed\nreward converges to a **stable** monetary supply that is neither inflationary\nnor deflationary, with the total supply proportional to rate of tail emission\nand probability of coin loss.\n\nCredit where credit is due: after writing the bulk of this article I found out\nthat Monero developer [smooth_xmr](https://www.reddit.com/user/smooth_xmr/)\nalso observed that tail emission results in a stable coin supply\n[a few years ago](https://www.reddit.com/r/Monero/comments/4z0azk/maam_28_monero_ask_anything_monday/d6sixyi/).\nThere's probably others too: it's a pretty obvious result.\n\n\n\u003cdiv markdown=\"1\" class=\"post-toc\"\u003e\n# Contents\n{:.no_toc}\n0. TOC\n{:toc}\n\u003c/div\u003e\n\n## Modeling the Fixed-Reward Monetary Supply\n\nSince the number of blocks is large, we can model the monetary supply as a\ncontinuous function $$N(t)$$, where $$t$$ is a given moment in time. If the\nblock reward is fixed we can model the reward as a slope $$k$$ added to an\ninitial supply $$N_0$$:\n\n$$\nN(t) = N_0 + kt\n$$\n\nOf course, this isn't realistic as coins are constantly being lost due to\ndeaths, forgotten passphrases, boating accidents, etc. These losses are\nindependent: I'm not any more or less likely to forget my passphrase because\nyou recently lost your coins in a boating accident — an accident I probably\ndon't even know happened. Since the number of individual coins (and their\nowners) is large — as with the number of blocks — we can model this loss as\nthough it happens continuously.\n\nSince coins can only be lost once, the *rate* of coin loss at time $$t$$ is\nproportional to the total supply *at that moment* in time. So let's look at the\n*first derivative* of our fixed-reward coin supply:\n\n$$\n\\frac{dN(t)}{dt} = k\n$$\n\n...and subtract from it the lost coins, using $$\\lambda$$ as our [coin loss\nconstant](https://en.wikipedia.org/wiki/Exponential_decay):\n\n$$\n\\frac{dN(t)}{dt} = k - \\lambda N(t)\n$$\n\nThat's a first-order differential equation, which can be easily solved with\nseparation of variables to get:\n\n$$\nN(t) = \\frac{k}{\\lambda} - Ce^{-\\lambda t}\n$$\n\nTo remove the integration constant $$C$$, let's look at $$t = 0$$, where the\ncoin supply is $$N_0$$:\n\n$$\n\\begin{align}\n    N_0 \u0026= \\frac{k}{\\lambda} - Ce^{-\\lambda 0} = \\frac{k}{\\lambda} - C \\\\\n      C \u0026= \\frac{k}{\\lambda} - N_0\n\\end{align}\n$$\n\nThus:\n\n$$\n\\begin{align}\n    N(t) \u0026= \\frac{k}{\\lambda} - \\left(\\frac{k}{\\lambda} - N_0 \\right)e^{-\\lambda t} \\\\\n         \u0026= \\frac{k}{\\lambda} + \\left(N_0 - \\frac{k}{\\lambda} \\right)e^{-\\lambda t}\n\\end{align}\n$$\n\n\n## Long Term Coin Supply\n\nIt's easy to see that in the long run, the second half of the coin supply\nequation goes to zero because $$\\lim_{t \\to \\infty} e^{-\\lambda t} = 0$$:\n\n$$\n\\begin{align}\n    \\lim_{t \\to \\infty} N(t) \u0026= \\lim_{t \\to \\infty} \\left[ \\frac{k}{\\lambda} + \\left(N_0 - \\frac{k}{\\lambda} \\right)e^{-\\lambda t} \\right ] = \\frac{k}{\\lambda} \\\\\n                   N(\\infty) \u0026= \\frac{k}{\\lambda}\n\\end{align}\n$$\n\nAn intuitive explanation for this result is that in the long run, the initial\nsupply $$N_0$$ doesn't matter, because approximately all of those coins will\neventually be lost. Thus in the long run, the coin supply will converge towards\n$$\\frac{k}{\\lambda}$$, the point where coins are created just as fast as they\nare lost.\n\n\n## Short Term Dynamics and Economic Considerations\n\nOf course, the intuitive explanation for why supply converges to\n$$\\frac{k}{\\lambda}$$, also tells us that supply must converge fairly slowly:\nif 1% of something is lost per year, after 100 years 37% of the initial supply\nremains. It's not clear what the rate of lost coins actually is in a mature,\nvaluable, coin. But 1%/year is likely to be a good guess — quite possibly less.\n\nIn the case of Monero, they've introduced tail emission at a point where it\nrepresents a 0.9% apparent monetary inflation rate[^p2pool-tail]. Since the number of\npreviously lost coins, and the current rate of coin loss, is\nunknown[^unknowable] it's not possible to know exactly what the true monetary\ninflation rate is right now. But regardless, the rate will only converge\ntowards zero going forward.\n\n[^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.\n\n[^p2pool-tail]: P2Pool operates [a page with real-time date figures](https://p2pool.io/tail.html).\n\nIf an existing coin decides to implement tail emission as a means to fund\nsecurity, choosing an appropriate emission rate is simple: decide on the\nmaximum amount of inflation you are willing to have in the worst case, and set\nthe tail emission accordingly. In reality monetary inflation will be even lower\non day zero due to lost coins, and in the long run, it will converge towards\nzero.\n\nThe fact is, economic volatility dwarfs the effect of small amounts of\ninflation. Even a 0.5% inflation rate over 50 years only leads to a 22% drop.\nMeanwhile at the time of writing, Bitcoin has dropped 36% in the past year, and\ngained 993% over the past 5 years. While this discussion is a nice excuse to\nuse some mildly interesting math, in the end it's totally pedantic.\n\n## Could Bitcoin Add Tail Emission?\n\n...and why could Monero?\n\nAdding tail emission to Bitcoin would be a hard fork: a incompatible rule\nchange that existing Bitcoin nodes would reject as invalid. While Monero was\nable to get sufficiently broad consensus in the community to implement tail\nemission, it's unclear at best if it would ever be possible to achieve that for\nthe much larger[^btc-vs-xmr-market-cap] Bitcoin. Additionally, Monero has a\nculture of frequent hard forks that simply does not exist in Bitcoin.\n\n[^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.\n\nUltimately, as long as a substantial fraction of the Bitcoin community continue\nto run full nodes, the only way tail emission could ever be added to Bitcoin is\nby convincing that same community that it is a good idea.\n\n\n## Footnotes\n-------------- next part --------------\nA non-text attachment was scrubbed...\nName: signature.asc\nType: application/pgp-signature\nSize: 833 bytes\nDesc: not available\nURL: \u003chttp://lists.linuxfoundation.org/pipermail/bitcoin-dev/attachments/20220709/7f1a65d7/attachment.sig\u003e"}