From 1d3972772907cad5f6a1b1c9ae710029aafdffa6 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 26 Nov 2024 02:34:25 -0800 Subject: [PATCH 01/70] simplified 100 pct staking post, all else pulled out --- _posts/2024-12-01-issuance-fundamentals.md | 234 +++++++++++++++++++++ _posts/2024-12-05-gov-centralization.md | 18 ++ _posts/background.md | 68 ++++++ 3 files changed, 320 insertions(+) create mode 100644 _posts/2024-12-01-issuance-fundamentals.md create mode 100644 _posts/2024-12-05-gov-centralization.md create mode 100644 _posts/background.md diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md new file mode 100644 index 0000000..b7f56cc --- /dev/null +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -0,0 +1,234 @@ +# Will Reducing Issuance Avoid 100\% ETH Staked? + +## Acknowledgements + +This is the first blog post in a series summarizing our research +funded by EF GRANT INFO. E.M.D. acknowledges useful discussions with +Philipp Zahn, Eric Siu, Angsar Dietrichs, and Andrew Sudbury. + +## The Problem + +The share of Ether staked by Coinbase and LSTs such as Lido +("centralized" staking services) continues to grow quite considerably. +This has provoked fears, first raised by Ethereum researchers, and +which we share, that the future of Ethereum might involve (1) all of +its native asset being staked, such that (2) the de facto liquid Ether +is controlled by a confederation of centralized un-transparent +govrnance. In this blog post we address the first of these concerns. + +We use "stock and flow" differential equation models to study Ethereum +macroeconomics, specifically how changing issuance impacts these +questions. We will publish four blog posts on this topic: + +1. (This post) Will reducing issuance avoid 100% ETH Staked? +2. Will reducing issuance avoid Governance Centralization? +3. Other Levers besides Issuance, and a means of evaluating them. +4. Summary and community tools to help study and resolve issuance debates. + +We may also publish bonus posts, if there is sufficient community +engagement, so dear reader stay vigilant! + +## TLDR + +For the extraordinarily impatient our message is a bit of a downer: we +find that the proposed reduction of issuance, unfortunately, does not +necessarily solve either of problems (1) or (2) over parameter ranges +we judge to be plausible, and in fact, it could actually make things +worse. + +In some more detail, our conclusions on (1), the subject of this post, +for those who are only moderately impatient. + +The quantity of most importance to this debate is the staked ETH +fraction $$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is +"circulating" (unstaked, unburnt) ETH, and $$A=S+C$$ is the total +"accessible" (unburnt) ETH. Differential changes, such as changes in +time $$\frac{ds}{dt}:=\dot{s}$$ are given by the quotient rule +$$\dot{s}=\dot{S}/A-s\dot{A}{A}$$. The quantity $$\dot{A}/A$$ is the +inflation rate averaged on at-least-quarterly timescales. + +That is, an increase in staking fraction can be driven by more people +staking, and/or it can be driven by a reduction of the inflation rate. +The latter can be acheived by a reduction of issuance relative to the +base fee "burn rate". Because of this, there is a mathmatical regime +in which reducing issuance actually *increases* $$s^*$$, the market +equilibrium staking ratio. We argue this regime is likely to be +realistic, and thus worthy of attention. So, we urge caution before +rushing to reduce issuance. + +## 100% Ether Staked + +### Splitting Up A Growing Pie + +Consider the quantity $$E$$ of all Ether in exitence. Let the +issuance of new Ether as per The Merge be quantified by $$I$$. Then +the change in time of total Ether is $$\dot{E}:=dE/dt=I$$. + +-- FIGURE 1 (see notability) -- + +Now, let us separate $$E$$ into three smaller boxes; +$$S+C+\cancel{O}=E$$, representing (S)taked, (C)icrulating, and burnt +Ether $$\cancel{O}$$, respectively. By considering each quantity a +quarterly average we can plausibly ignore the detailed dynamics of the +staking, unstaking, and withdrawal queues. + +-- FIGURE 2 (see notability) -- + +### How Pie is Pushed Around The Plate + +This is our model: + +$$ +\begin{array} +\dot{E} &=& I\\ +\dot{S} &=& R+Q_+-Q_-\\ +\dot{A} &=& I-B\\ +\end{array} + +Where all quantities are assesed quarterly: +* $$I$$ -- Total Issuance +* $$R$$ -- Total Reinvestment of Validator Yields +* $$Q_\pm$$ -- Total (Un)Staking Queue Flows +* $$B$$ -- Total Base Fee, aka Burn Rate. + +Transaction fees $$F+B$$ are split into the base fee $$B$$ which is +burned, and priority fees plus MEV $$F$$ which go to validators via the +wihdrwal queue. Fees obey the inequality $$0\leq F+B\leq C$$. We ae +interested in the limit $$C\to0$$ and pull this dependence out by +defining $$f$$ such that $$fC = F+B$$. The fraction $$f$$ need not +be a constant, but whatever values it takes, $$f$$ must be a fraction: +$$00$$, the staking fraction receives a boost when +it is less than reinvestment rate $$sr$$. +Under deflation $$\dot{A}/A<0$$ the effects are reversed. The second +term combines new staking $$q_+$$ and the reinvestment of priority +fees and MEV $$rf$$, and is always non-negative. The third +term represents unstaking and is always non-positive. + +### Fixed Point + +If quarterly inflation $$\alpha:=\dot{A}/A$$, governed by the tension +between issuance and burn, changes quickly enough, we have a two +dimensional system. The varieties of qualitatuive behavior of this +system are however essentially similar to that of the one-dimensional +system, if all ones cares about is the stability of market equilibria, +and the response to issuance changes, which we do. + +If $$\alpha$$ varies lowly enough $$\alpha\approx\alpha^*$$, a fixed +point $$s^*$$ where $$\dot{s}=0$$ can obtain: + +$$s^*=\frac{r^*(\alpha^* + f^*) + q_+^*}{\alpha + r^*f^* - q^*_-}$$ + +Perhaps surprisingly we find that if $\dot{A}=0=\alpha$ (no inflation +nor deflation) then counter-intuitively, an interor market equilibrium +$s^*<1$ seems unlikely. We reason as follows. In the absence of +in/de-flation, an interior fixed point $$s^*<1$$ would require a +persistent unstaking/capitulation of existing validators $$q_->0$$. +This in turn either requires "churn", a persistent supply of new +validators to take their place $$q_+^*>0$$, or it is only a transient +and $$q_-^*\approx0$$; recall that reinvestment by existing validators is +not counted in $$q_+$$. + +Net unstaking $$q_->0$$ could only describe a market equilibrium if +one group of stakers was actively capitulating and withdrawing their +stake, while another group with a higher $r$ were aggressively +reinvesting in their business, and their reinvestment of fees and MEV +offset the unstaking, adjusting for inflation. This cannot maintain: +eventually there will be no new Capitulators left, and the fraction of +stake will grow again as Reinvestors take up more market share. +Similarly, at some point everyone who wants to stake should have +staked, and other than influxes due to humanity's issuance of new +humans etc., further validators count only toward $$r$$, so +$$q_+^*\approx0$$. + +Thus, the fixed point $$s^*$$ is likely to be + +$$s^*\approx r\frac{\alpha + f}{\alpha + rf}$$ + +A few limits are illustrative under the current regime of positive +inflation. If inflation dominates fees, $$\alpha\gg f$$ then +$$s^*\sim r^*<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and +$$s^*\to1$$. + +### Market Equilibrium + +The fixed-point $$x^*$$ represents a market equilibrium just when it +is unqiue and stable. + +Non-uniqueness would seem to require a shared and repeated root (the +other equilibrium point) among terms $$\alpha,rf+q_+,q_-$$. The root +must be repeated because these quantities are strictly non-negative. +It's unclear how this would occur. For now, lacking any supporting +data or a mechanism, we move on, assuming uniqueness. + +Stability requires that small perturbations shrink; +$$\frac{\partial\dot{x}}{\partial x}\big|_{x^*}<0$$. The full +condition is quite complicated, especially under deflation. Perhaps +onchain measurements and surveys/metrics of validator preferences will +allow us to estimate the full extent of the stability basin around +$$x^*$$. If however inflation is positive and $$r,\alpha,f$$ are +relatively nsensitive to small changes in $$x$$ near $$x^*$$, this +conditions is satisfied immediately; $$\alpha+rf>0$$. + +While we expect dependence on other macroeconomic factors, we find it +quite believable that reinvestment, inflation, and priority fees are +relatively insensitive to small changes in staking fraction near the +fixed point under small levels of inflation. + +So our takeaway is that this stable equilibrium is at least plausible, +and worthy of consideration. How does it behave? + +### Qualitative Behavior + +We have argued above that the fixed point +$$x^*=r(\alpha+f)/(\alpha+rf)$$ is then this behavior is quite distinct +from the arguments forwarded by Ethereum researchers, in which +inflation is held responsible for a 100% ETH Staked scenario. To be +clear, vey likely $$x_{now} Date: Tue, 26 Nov 2024 02:53:34 -0800 Subject: [PATCH 02/70] s/*/\star/g; fixing typos --- _posts/2024-12-01-issuance-fundamentals.md | 69 ++++++++++++---------- 1 file changed, 37 insertions(+), 32 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index b7f56cc..bed8658 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -51,7 +51,7 @@ That is, an increase in staking fraction can be driven by more people staking, and/or it can be driven by a reduction of the inflation rate. The latter can be acheived by a reduction of issuance relative to the base fee "burn rate". Because of this, there is a mathmatical regime -in which reducing issuance actually *increases* $$s^*$$, the market +in which reducing issuance actually *increases* $$s^\star$$, the market equilibrium staking ratio. We argue this regime is likely to be realistic, and thus worthy of attention. So, we urge caution before rushing to reduce issuance. @@ -77,13 +77,14 @@ staking, unstaking, and withdrawal queues. ### How Pie is Pushed Around The Plate This is our model: - $$ -\begin{array} -\dot{E} &=& I\\ -\dot{S} &=& R+Q_+-Q_-\\ -\dot{A} &=& I-B\\ -\end{array} +\displaystyle +\begin{align} +\dot{E} &= I\\ +\dot{S} &= R+Q_+-Q_-\\ +\dot{A} &= I-B\\ +\end{align} +$$ Where all quantities are assesed quarterly: * $$I$$ -- Total Issuance @@ -114,18 +115,19 @@ currently lies in the range $$.53\leq r\leq.75$$. The quarterly flows from the staking and unstaking queues must obey $$0\leq Q_+\leq C$$ and $$0\leq q_-\leq S$$, which we use below via -variable fractions $$q_+=Q_+/C,~q_-=Q_-/S$$. Finally, we rewrite -$$\dot{S}$$ the change in total staked ETH, anticipating that we wish -to understand the relationship to inflation rate $$\dot{A}/A$$: +variable fractions $$q_+=Q_+/C,~q_-=Q_-/S$$. -$$\dot{S} = r\dot{A} + r(B+F) + Q_+-Q_-$$. +Finally, we rewrite $$\dot{S}$$ the change in total staked ETH, +anticipating that we wish to understand the relationship to inflation +rate $$\dot{A}/A$$, obtaining $$\dot{S} = r\dot{A} + r(B+F) + Q_+-Q_-$$. The variable of primary interest is the staking fraction $$s=S/A$$; recalling the quotient rule $$\dot{S}=\dot{S}/A-s\dot{A}/A$$ and $$C=A-s$$, we obtain: -$$\displaystyle -\dot{s} = \frac{\dot{A}}{A}(r-s)~+~(rf+q_+)(1-s)~-~q_-s +$$ +\displaystyle +\dot{s} ~~=~~ \frac{\dot{A}}{A}(r-s)~~+~~(rf+q_+)(1-s)~~-~~q_-s $$ The first term captures the effect of inflation/deflation. Under @@ -145,45 +147,48 @@ system are however essentially similar to that of the one-dimensional system, if all ones cares about is the stability of market equilibria, and the response to issuance changes, which we do. -If $$\alpha$$ varies lowly enough $$\alpha\approx\alpha^*$$, a fixed -point $$s^*$$ where $$\dot{s}=0$$ can obtain: +If $$\alpha$$ varies lowly enough $$\alpha\approx\alpha^\star$$, a fixed +point $$s^\star$$ where $$\dot{s}=0$$ can obtain: -$$s^*=\frac{r^*(\alpha^* + f^*) + q_+^*}{\alpha + r^*f^* - q^*_-}$$ +$$ +\displaystyle +s^\star=\frac{r^\star(\alpha^\star + f^\star) + q_+^\star}{\alpha + r^\starf^\star - q^\star_-} +$$ Perhaps surprisingly we find that if $\dot{A}=0=\alpha$ (no inflation nor deflation) then counter-intuitively, an interor market equilibrium -$s^*<1$ seems unlikely. We reason as follows. In the absence of -in/de-flation, an interior fixed point $$s^*<1$$ would require a +$s^\star<1$ seems unlikely. We reason as follows. In the absence of +in/de-flation, an interior fixed point $$s^\star<1$$ would require a persistent unstaking/capitulation of existing validators $$q_->0$$. This in turn either requires "churn", a persistent supply of new -validators to take their place $$q_+^*>0$$, or it is only a transient -and $$q_-^*\approx0$$; recall that reinvestment by existing validators is +validators to take their place $$q_+^\star>0$$, or it is only a transient +and $$q_-^\star\approx0$$; recall that reinvestment by existing validators is not counted in $$q_+$$. Net unstaking $$q_->0$$ could only describe a market equilibrium if one group of stakers was actively capitulating and withdrawing their -stake, while another group with a higher $r$ were aggressively +stake, while another group with a higher $$r$$ were aggressively reinvesting in their business, and their reinvestment of fees and MEV -offset the unstaking, adjusting for inflation. This cannot maintain: +offset the unstaking, adjusted for inflation. This cannot maintain: eventually there will be no new Capitulators left, and the fraction of stake will grow again as Reinvestors take up more market share. Similarly, at some point everyone who wants to stake should have staked, and other than influxes due to humanity's issuance of new humans etc., further validators count only toward $$r$$, so -$$q_+^*\approx0$$. +$$q_+^\star\approx0$$. -Thus, the fixed point $$s^*$$ is likely to be +Thus, the fixed point $$s^\star$$ is likely to be -$$s^*\approx r\frac{\alpha + f}{\alpha + rf}$$ +$$s^\star\approx r\frac{\alpha + f}{\alpha + rf}$$ A few limits are illustrative under the current regime of positive inflation. If inflation dominates fees, $$\alpha\gg f$$ then -$$s^*\sim r^*<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and -$$s^*\to1$$. +$$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and +$$s^\star\to1$$. ### Market Equilibrium -The fixed-point $$x^*$$ represents a market equilibrium just when it +The fixed-point $$x^\star$$ represents a market equilibrium just when it is unqiue and stable. Non-uniqueness would seem to require a shared and repeated root (the @@ -193,12 +198,12 @@ It's unclear how this would occur. For now, lacking any supporting data or a mechanism, we move on, assuming uniqueness. Stability requires that small perturbations shrink; -$$\frac{\partial\dot{x}}{\partial x}\big|_{x^*}<0$$. The full +$$\frac{\partial\dot{x}}{\partial x}\big|_{x^\star}<0$$. The full condition is quite complicated, especially under deflation. Perhaps onchain measurements and surveys/metrics of validator preferences will allow us to estimate the full extent of the stability basin around -$$x^*$$. If however inflation is positive and $$r,\alpha,f$$ are -relatively nsensitive to small changes in $$x$$ near $$x^*$$, this +$$x^\star$$. If however inflation is positive and $$r,\alpha,f$$ are +relatively nsensitive to small changes in $$x$$ near $$x^\star$$, this conditions is satisfied immediately; $$\alpha+rf>0$$. While we expect dependence on other macroeconomic factors, we find it @@ -212,7 +217,7 @@ and worthy of consideration. How does it behave? ### Qualitative Behavior We have argued above that the fixed point -$$x^*=r(\alpha+f)/(\alpha+rf)$$ is then this behavior is quite distinct +$$x^\star=r(\alpha+f)/(\alpha+rf)$$ is then this behavior is quite distinct from the arguments forwarded by Ethereum researchers, in which inflation is held responsible for a 100% ETH Staked scenario. To be clear, vey likely $$x_{now} Date: Tue, 26 Nov 2024 17:57:53 -0800 Subject: [PATCH 03/70] much improved intro --- _posts/2024-12-01-issuance-fundamentals.md | 333 ++++++++++++++------- 1 file changed, 220 insertions(+), 113 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index bed8658..7203ec9 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -1,28 +1,31 @@ -# Will Reducing Issuance Avoid 100\% ETH Staked? +# Will Reducing Issuance Avoid Runaway Staking? ## Acknowledgements This is the first blog post in a series summarizing our research -funded by EF GRANT INFO. E.M.D. acknowledges useful discussions with -Philipp Zahn, Eric Siu, Angsar Dietrichs, and Andrew Sudbury. +funded by EF GRANT INFO. The author, Eric Downes, is grateful for +useful discussions with the 20 Squares team especially Danieli +Palombi and Philipp Zahn, as well as Eric Siu, Angsar Dietrichs, and +Andrew Sudbury. ## The Problem -The share of Ether staked by Coinbase and LSTs such as Lido -("centralized" staking services) continues to grow quite considerably. -This has provoked fears, first raised by Ethereum researchers, and -which we share, that the future of Ethereum might involve (1) all of -its native asset being staked, such that (2) the de facto liquid Ether -is controlled by a confederation of centralized un-transparent -govrnance. In this blog post we address the first of these concerns. +The share of Ether staked by exchanges such as Coinbase and Liquid +Staking Providers such as Lido ("centralized" staking services) +continues to grow. This has provoked concerns, first raised by +Ethereum researchers, and which we share, that the future of Ethereum +might involve (1) all of its native asset being staked, such that (2) +the de facto liquid Ether is controlled by a confederation of +centralized un-transparent govrnance. In this blog post we address the +first of these concerns. We use "stock and flow" differential equation models to study Ethereum macroeconomics, specifically how changing issuance impacts these questions. We will publish four blog posts on this topic: -1. (This post) Will reducing issuance avoid 100% ETH Staked? +1. (This post) Will reducing issuance avoid Runaway Staking? 2. Will reducing issuance avoid Governance Centralization? -3. Other Levers besides Issuance, and a means of evaluating them. +3. Other Levers besides Issuance, and a means of evaluating levers. 4. Summary and community tools to help study and resolve issuance debates. We may also publish bonus posts, if there is sufficient community @@ -30,15 +33,30 @@ engagement, so dear reader stay vigilant! ## TLDR -For the extraordinarily impatient our message is a bit of a downer: we -find that the proposed reduction of issuance, unfortunately, does not -necessarily solve either of problems (1) or (2) over parameter ranges -we judge to be plausible, and in fact, it could actually make things -worse. +### For the extraordinarily impatient. -In some more detail, our conclusions on (1), the subject of this post, -for those who are only moderately impatient. +* Under strong deflation, Ethereum faces an unenviable choice of futures: + - zero staking, or + - runaway staking +* Under no growth or weak deflation, runaway staking *must* occur. + +* Under inflation, runaway staking may still occur. + +* If runaway staking does *not* occur under inflation, it is because + -- (1) inflation is low enough, that concerns over inflation do not + dominate the microeconomic considerations governing the + reinvestment of profits by staking businesses, but simultaneously + -- (2) inflation is high enough to numerically dominate the quantity of + priority fees and MEV reinvested as profit. + +* That is, the staking reinvestment ratio is central to understanding + the future of Ethereum macroeconomics. Thankfully, it can be + well-estimated with onchain data. + +### For the merely moderately impatient. + +In some more detail, a sketch of the reasons behind our conclusions. The quantity of most importance to this debate is the staked ETH fraction $$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is "circulating" (unstaked, unburnt) ETH, and $$A=S+C$$ is the total @@ -49,18 +67,34 @@ inflation rate averaged on at-least-quarterly timescales. That is, an increase in staking fraction can be driven by more people staking, and/or it can be driven by a reduction of the inflation rate. -The latter can be acheived by a reduction of issuance relative to the -base fee "burn rate". Because of this, there is a mathmatical regime -in which reducing issuance actually *increases* $$s^\star$$, the market -equilibrium staking ratio. We argue this regime is likely to be -realistic, and thus worthy of attention. So, we urge caution before -rushing to reduce issuance. +The latter can be acheived in principle by a reduction of issuance +relative to the base fee "burn rate". Because of this quotient rule +tradeoff, low-but-positive inflation actually plays a positive +infrastructural role in moderating staking fraction. Because of this, +we find a "high fee + MEV" regime in which reducing issuance actually +*increases* $$s^\star$$, the market equilibrium staking ratio. +Clarity on the boundaries of this regime are worthy of attention if we +want to ensure policy interventions, such as issuance reduction, do +not backfire. + +In contrast, under current market conditions, very low fees and low +but positive inflation, the long-term equilibrium staking fraction +approaches the ratio at which validators reinvest their staking +rewards, which for LSTs is bounded below by the ratio of token yield +to total yield. Happily reinvestment ratio is intuitive for each +staking business to calculate, and measurable in detail with onchain +data. So long as the reduction in yield curve does not broadly alter +the market conditions (inflation, low fees) it will not change the +character of this fixed point, and reinvestment ratio thus becomes the +target which a policy intervention *must* target. ## 100% Ether Staked +Now the details! + ### Splitting Up A Growing Pie -Consider the quantity $$E$$ of all Ether in exitence. Let the +Consider the quantity $$E$$ of all Ether in existence. Let the issuance of new Ether as per The Merge be quantified by $$I$$. Then the change in time of total Ether is $$\dot{E}:=dE/dt=I$$. @@ -79,11 +113,11 @@ staking, unstaking, and withdrawal queues. This is our model: $$ \displaystyle -\begin{align} -\dot{E} &= I\\ -\dot{S} &= R+Q_+-Q_-\\ -\dot{A} &= I-B\\ -\end{align} +\begin{array}rcl} +\dot{E} &=& I\\ +\dot{S} &=& R+Q_+-Q_-\\ +\dot{A} &=& I-B\\ +\end{array} $$ Where all quantities are assesed quarterly: @@ -93,12 +127,16 @@ Where all quantities are assesed quarterly: * $$B$$ -- Total Base Fee, aka Burn Rate. Transaction fees $$F+B$$ are split into the base fee $$B$$ which is -burned, and priority fees plus MEV $$F$$ which go to validators via the -wihdrwal queue. Fees obey the inequality $$0\leq F+B\leq C$$. We ae -interested in the limit $$C\to0$$ and pull this dependence out by -defining $$f$$ such that $$fC = F+B$$. The fraction $$f$$ need not -be a constant, but whatever values it takes, $$f$$ must be a fraction: -$$00$$, the staking fraction receives a boost when it is less than reinvestment rate $$sr$$. -Under deflation $$\dot{A}/A<0$$ the effects are reversed. The second -term combines new staking $$q_+$$ and the reinvestment of priority -fees and MEV $$rf$$, and is always non-negative. The third -term represents unstaking and is always non-positive. +Under deflation $$\dot{A}/A<0$$ the effects are reversed, though see +below. The second term combines new staking $$q_+$$ and the +reinvestment of priority fees and MEV $$rf$$, and is always +non-negative. The third term represents unstaking and is always +non-positive. ### Fixed Point If quarterly inflation $$\alpha:=\dot{A}/A$$, governed by the tension -between issuance and burn, changes quickly enough, we have a two -dimensional system. The varieties of qualitatuive behavior of this -system are however essentially similar to that of the one-dimensional -system, if all ones cares about is the stability of market equilibria, -and the response to issuance changes, which we do. - -If $$\alpha$$ varies lowly enough $$\alpha\approx\alpha^\star$$, a fixed -point $$s^\star$$ where $$\dot{s}=0$$ can obtain: - -$$ -\displaystyle -s^\star=\frac{r^\star(\alpha^\star + f^\star) + q_+^\star}{\alpha + r^\starf^\star - q^\star_-} +between issuance and burn, changes quickly enough, we have at least a +two dimensional system. Thankfully, if all ones cares about is the +(in)stability of market equilibria, and the potential response to +issuance changes, the varieties of qualitative behavior of this higher +dimensonal system are all present in the above one-dimensional system. +That is, we find the same supportive role is played by inflation in +moderating staking fraction, and that $$r$$ plays a prominent role in +the character of the fixed point. The primary difference, over +existing market conditions, is that oscillations are possible. + +If instead, $$\alpha$$ varies slowly enough +$$\alpha\approx\alpha^\star$$, a fixed point $$s^\star$$ where +$$\dot{s}=0$$ can obtain: +$$ \displaystyle +s^\star=\frac{r^\star(\alpha^\star + f^\star) + {q^\star}_+}{\alpha + +r^\star f^\star - {q^\star}_-} $$ -Perhaps surprisingly we find that if $\dot{A}=0=\alpha$ (no inflation -nor deflation) then counter-intuitively, an interor market equilibrium -$s^\star<1$ seems unlikely. We reason as follows. In the absence of -in/de-flation, an interior fixed point $$s^\star<1$$ would require a -persistent unstaking/capitulation of existing validators $$q_->0$$. +We find that if $\dot{A}=0=\alpha$ (no inflation +nor deflation) then an interor market equilibrium +$$s^\star<1$$ is impossible. We reason as follows. In the absence +of in/de-flation, an interior fixed point $$s^\star<1$$ would require +a persistent unstaking/capitulation of existing validators $$q_->0$$. This in turn either requires "churn", a persistent supply of new -validators to take their place $$q_+^\star>0$$, or it is only a transient -and $$q_-^\star\approx0$$; recall that reinvestment by existing validators is -not counted in $$q_+$$. +validators to take their place $$q_+^\star>0$$, or it is only a +transient and $$q_-^\star\approx0$$; recall that reinvestment by +existing validators is not counted in $$q_+$$. Net unstaking $$q_->0$$ could only describe a market equilibrium if one group of stakers was actively capitulating and withdrawing their stake, while another group with a higher $$r$$ were aggressively reinvesting in their business, and their reinvestment of fees and MEV -offset the unstaking, adjusted for inflation. This cannot maintain: -eventually there will be no new Capitulators left, and the fraction of -stake will grow again as Reinvestors take up more market share. -Similarly, at some point everyone who wants to stake should have -staked, and other than influxes due to humanity's issuance of new -humans etc., further validators count only toward $$r$$, so -$$q_+^\star\approx0$$. +offset the unstaking, adjusted for inflation. This cannot maintain +forever: eventually there will be no new Capitulators left, and +$$x^*$$ must once again grow as required by the Reinvestors' higher +$$r$$, so $$x^*$$ was not a fixed point at all. Similarly, at some +point everyone who wants to stake should have staked, and other than +influxes due to the issuance of new humans, further validators count +only toward $$r$$, so $$q_+^\star\approx0$$. Thus, the fixed point $$s^\star$$ is likely to be - $$s^\star\approx r\frac{\alpha + f}{\alpha + rf}$$ -A few limits are illustrative under the current regime of positive +A calculation is illustrative under the current regime of positive inflation. If inflation dominates fees, $$\alpha\gg f$$ then -$$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and -$$s^\star\to1$$. +$$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll +f$$ and $$s^\star\to1$$. For a numerical comparison, at present +$$.001\lesssim f\lesssim.02$$ so at inflation of 1-3\%, +$$f\ll\alpha$$. + +So if these conditions persist at long times, and to be clear there is +no reason to be certain they will, we should expect $$x^\star\approx +r\in(.52,.75)$$. This places the lower range of $$x^\star$$ near the +50% staking target proposed by Elowsson, which is hopeful, but little +weight can be attached to such back-of-envelope extrapolations. To +make believable projections we need systematic measurement of both the +statistics of individual validator reinvestment and metrics or at +least surveys reflecting validator sentiment and (in)capacity to absorb +reduced revenue. ### Market Equilibrium The fixed-point $$x^\star$$ represents a market equilibrium just when it is unqiue and stable. -Non-uniqueness would seem to require a shared and repeated root (the -other equilibrium point) among terms $$\alpha,rf+q_+,q_-$$. The root -must be repeated because these quantities are strictly non-negative. -It's unclear how this would occur. For now, lacking any supporting -data or a mechanism, we move on, assuming uniqueness. +Non-uniqueness requires a shared and repeated root (the other +equilibrium point) among all terms $$\alpha,rf+q_+,q_-$$. That is, +each economic flow would have to nearly disappear at a particular +value of staking fraction. Lacking any supporting data or a +mechanism, we cannot analyse such an alternate equilibria, and move +on, assuming uniqueness. Stability requires that small perturbations shrink; $$\frac{\partial\dot{x}}{\partial x}\big|_{x^\star}<0$$. The full -condition is quite complicated, especially under deflation. Perhaps -onchain measurements and surveys/metrics of validator preferences will -allow us to estimate the full extent of the stability basin around -$$x^\star$$. If however inflation is positive and $$r,\alpha,f$$ are -relatively nsensitive to small changes in $$x$$ near $$x^\star$$, this +condition is quite complicated, especially under deflation. +If however inflation is positive and $$r,\alpha,f$$ are +relatively insensitive to small changes in $$x$$ near $$x^\star$$, this conditions is satisfied immediately; $$\alpha+rf>0$$. -While we expect dependence on other macroeconomic factors, we find it -quite believable that reinvestment, inflation, and priority fees are -relatively insensitive to small changes in staking fraction near the -fixed point under small levels of inflation. - -So our takeaway is that this stable equilibrium is at least plausible, -and worthy of consideration. How does it behave? +This stable equilibrium seems plausible, and worthy of consideration. +How does it behave? ### Qualitative Behavior -We have argued above that the fixed point -$$x^\star=r(\alpha+f)/(\alpha+rf)$$ is then this behavior is quite distinct -from the arguments forwarded by Ethereum researchers, in which -inflation is held responsible for a 100% ETH Staked scenario. To be -clear, vey likely $$x_{now} Date: Wed, 27 Nov 2024 08:36:48 -0800 Subject: [PATCH 04/70] text mostly there just need to add figures --- _posts/2024-12-01-issuance-fundamentals.md | 179 +++++++++++++-------- 1 file changed, 113 insertions(+), 66 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 7203ec9..a172cc5 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -33,13 +33,14 @@ engagement, so dear reader stay vigilant! ## TLDR -### For the extraordinarily impatient. +### For the extraordinarily impatient reader. -* Under strong deflation, Ethereum faces an unenviable choice of futures: +* Roughly speaking, under strong deflation, Ethereum faces an + unenviable choice of futures: - zero staking, or - runaway staking -* Under no growth or weak deflation, runaway staking *must* occur. +* Under no growth or weak deflation, runaway staking is inevitable. * Under inflation, runaway staking may still occur. @@ -50,45 +51,87 @@ engagement, so dear reader stay vigilant! -- (2) inflation is high enough to numerically dominate the quantity of priority fees and MEV reinvested as profit. -* That is, the staking reinvestment ratio is central to understanding - the future of Ethereum macroeconomics. Thankfully, it can be - well-estimated with onchain data. +* Outside of the "low inflation, lower fees" regime, we expect + reductions of inflation to backfire, *raising* staking fraction. -### For the merely moderately impatient. +* Regardless, the staking reinvestment ratio is central to + understanding the quantitative future of Ethereum macroeconomics. + We believe it can be well-estimated with onchain data. + +### For the moderately impatient reader. In some more detail, a sketch of the reasons behind our conclusions. The quantity of most importance to this debate is the staked ETH -fraction $$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is -"circulating" (unstaked, unburnt) ETH, and $$A=S+C$$ is the total -"accessible" (unburnt) ETH. Differential changes, such as changes in -time $$\frac{ds}{dt}:=\dot{s}$$ are given by the quotient rule -$$\dot{s}=\dot{S}/A-s\dot{A}{A}$$. The quantity $$\dot{A}/A$$ is the -inflation rate averaged on at-least-quarterly timescales. +fraction $$s$$, currently roughly .33. Staking fraction is calculated +$$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is "circulating" +(unstaked, unburnt) ETH, and $$A=S+C$$ is the total "accessible" +(unburnt) ETH. Differential changes, such as changes in time +$$\frac{ds}{dt}:=\dot{s}$$ are given by the quotient rule +$$\dot{s}=\dot{S}/A-s\dot{A}{A}$$. The quantity $$\alpha = \dot{A}/A$$ +is the average on-paper inflation rate (supply expansion APY) averaged +on at-least-quarterly timescales. That is, an increase in staking fraction can be driven by more people staking, and/or it can be driven by a reduction of the inflation rate. The latter can be acheived in principle by a reduction of issuance relative to the base fee "burn rate". Because of this quotient rule tradeoff, low-but-positive inflation actually plays a positive -infrastructural role in moderating staking fraction. Because of this, -we find a "high fee + MEV" regime in which reducing issuance actually -*increases* $$s^\star$$, the market equilibrium staking ratio. -Clarity on the boundaries of this regime are worthy of attention if we -want to ensure policy interventions, such as issuance reduction, do -not backfire. - -In contrast, under current market conditions, very low fees and low -but positive inflation, the long-term equilibrium staking fraction -approaches the ratio at which validators reinvest their staking -rewards, which for LSTs is bounded below by the ratio of token yield -to total yield. Happily reinvestment ratio is intuitive for each -staking business to calculate, and measurable in detail with onchain -data. So long as the reduction in yield curve does not broadly alter -the market conditions (inflation, low fees) it will not change the -character of this fixed point, and reinvestment ratio thus becomes the -target which a policy intervention *must* target. - -## 100% Ether Staked +infrastructural role in moderating staking fraction. + +This can be seen in the market equilibrium staking fraction, which we +derive below: + +$$\displaystyle +s^\star = r\frac{\alpha + f}{\alpha + rf} +$$ + +Here $$0\leq r\leq 1$$ is the ratio of profits reinvested quarterly by +validators, $$0\leq f\leq 1$$ is the fraction of unstaked ETH spent on +transactions fees (base and priority) quarterly, and as above +$$\alpha$$ is inflation. The two extremes are $$\alpha\ll f$$ fees +dominate and $$\alpha\gg f$$ inflation dominates. In the former, +staking fraction is driven closer to 1, while in the latter, staking +is driven to match reinvestment $$x^*\approx r$$. + +Under current market conditions, low inflation and lower fees, the +long-term equilibrium staking fraction approaches the average ratio +$$r$$ at which validators reinvest their staking rewards. For LSTs +$$r$$ is bounded below by the ratio of token yield to total yield, and +we can use this to roughly estimate some real values. + +Approximate present values from YCharts are very roughly +$$f\approx.001$$/year, $$\alpha\approx.005$$/year, $$r\in(.5,.75)$$. +So if current market conditions (low inflation, lower fees) were to +persist at long times, we expect the staking ratio to convergence to +within +10\% of the the reinvestment ratio. If issuance is reduced by +half and fees maintain, $$x^\star$$ is still around $$r$$ plus 15\%. + +However, if issuance is reduced too much, such that fees dominate +inflation, we enter a regime in which reducing issuance *raises* +equilibrium staking fraction, and becomes insensitive to $$r$$. This +could occur by adopting too severe a yield curve, or by validaators +continuing to chase MEV yield, long after the yield from issuance has +become irrelevant. + +Under present conditions $$r$$ is the proximate mechanism by which +inflation must act, in affecting long term staking fraction. As per +all the arguments of researchers, a sufficient increase in inflation +$$\alpha^*\mapsto\alpha^*+\Delta\alpha$$ could well drive an runaway +increase in reinvestment $$\Delta{r}/\Delta\alpha\gesim 1-r$$. + +Thankfully, the reinvestment of profits $$r_i$$ is a microeconomic +quantity every staking business $$i$$ calculates, even if they do not +use those words. The global $$r$$ is then the market-share weighted +average of all these. If inflation pressures are indeed the dominant +consideraton for Ethereum users considering staking, this should +emerge from microeconomic surveys of validators' reinvestment +sentiments. + +We hope that this work can be built upon to focus inflationary +pressure arguments into empirically measurable assertions that can be +tracked as a metric for Ethereum health. + +## Modelling Staking Now the details! @@ -132,11 +175,10 @@ the wihdrwal queue. Fees obey the inequality $$0\leq F+B\leq C$$. We ae interested in the limit $$C\to0$$ and pull this dependence out by defining $$f$$ such that $$fC = F+B$$. The fraction $$f$$ need not be a constant, but whatever values it takes, $$f$$ must be a fraction: -$$00$$. -This in turn either requires "churn", a persistent supply of new -validators to take their place $$q_+^\star>0$$, or it is only a -transient and $$q_-^\star\approx0$$; recall that reinvestment by -existing validators is not counted in $$q_+$$. +We find that if $\dot{A}=0=\alpha$ (no inflation nor deflation) then +an interior market equilibrium $$s^\star<1$$ is impossible. We reason +as follows. In the absence of in/de-flation, an interior fixed point +$$s^\star<1$$ would require a persistent unstaking/capitulation of +existing validators $$q_->0$$. This in turn either requires "churn", +a persistent supply of new validators to take their place +$$q_+^\star>0$$, or it is only a transient and $$q_-^\star\approx0$$; +recall that reinvestment by existing validators is not counted in +$$q_+$$. Net unstaking $$q_->0$$ could only describe a market equilibrium if one group of stakers was actively capitulating and withdrawing their @@ -225,31 +270,32 @@ $$s^\star\approx r\frac{\alpha + f}{\alpha + rf}$$ A calculation is illustrative under the current regime of positive inflation. If inflation dominates fees, $$\alpha\gg f$$ then $$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll -f$$ and $$s^\star\to1$$. For a numerical comparison, at present -$$.001\lesssim f\lesssim.02$$ so at inflation of 1-3\%, -$$f\ll\alpha$$. +f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f +\approx .001\lessim .005\approx\alpha$$ so to within 10\% error +$$x^*\approx r$$ over the range of $$r$$ inferred from Lido yield +rate. So if these conditions persist at long times, and to be clear there is no reason to be certain they will, we should expect $$x^\star\approx -r\in(.52,.75)$$. This places the lower range of $$x^\star$$ near the -50% staking target proposed by Elowsson, which is hopeful, but little +r\in(.55,.77)$$. This places the lower range of $$x^\star$$ near the +50% staking target proposed by Elowsson. This is hopeful but little weight can be attached to such back-of-envelope extrapolations. To -make believable projections we need systematic measurement of both the +make reliable projections we need systematic measurement of both the statistics of individual validator reinvestment and metrics or at -least surveys reflecting validator sentiment and (in)capacity to absorb -reduced revenue. +least surveys reflecting validator sentiment and (in)capacity to +absorb reduced revenue. ### Market Equilibrium -The fixed-point $$x^\star$$ represents a market equilibrium just when it +The fixed-point $$x^\star$$ represents the market equilibrium when it is unqiue and stable. Non-uniqueness requires a shared and repeated root (the other equilibrium point) among all terms $$\alpha,rf+q_+,q_-$$. That is, each economic flow would have to nearly disappear at a particular value of staking fraction. Lacking any supporting data or a -mechanism, we cannot analyse such an alternate equilibria, and move -on, assuming uniqueness. +mechanism, we cannot really analyse such an alternate equilibria, and +intuitively it seems unlikely, so we move on, assuming uniqueness. Stability requires that small perturbations shrink; $$\frac{\partial\dot{x}}{\partial x}\big|_{x^\star}<0$$. The full @@ -288,7 +334,7 @@ We have argued, hopefully convincingly, that reducing issuance in a vacuum decreases inflation, and increases the relative strength of the $$rf(1-x)$$ term. If current conditions affecting $$f$$ etc. persist what will be the end effect on inflation $$\alpha$$ and $$x^\star$$? -Inflation in terms of the above variables: +Inflation in terms of the above variables is: $$\alpha = \dot{A}/A = (I - B)/A = ys~~-~~bf(1-s)$$ @@ -311,9 +357,9 @@ elevating priority and base fees moderately higher than present, while reinvestment is not strongly reduced (perhaps driven by breakthroughs in MEV, or an unexpectedly low elasticity in staking demand), such that $$\alpha\sim rf$$. In this hypothetical scenario, reinvestment -of fees will shrink $$C$$ consderably relative to $$S$$, pushing the -equilibrium staking fraction higher, requiring no need for more new -validators. +of fees will shrink $$C$$ consderably relative to $$S$$, potentially +leading to runaway staking. This mechanism requires no need for more +new validators. ## In Conclusion @@ -344,3 +390,4 @@ $$r_i$$. The global $$r$$ is then the market-share weighted averaged of all of these. We hope these calculations will bridge the macro- and micro-economics to help the community see a unified and actionable big picture on the future of Ethereum staking. + From d1d358bd02c910f21bcc3c036bab9b59eca42c62 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 01:42:33 -0500 Subject: [PATCH 05/70] penulatimate draft --- _posts/2024-12-01-issuance-fundamentals.md | 358 ++++++++++++++------- 1 file changed, 235 insertions(+), 123 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index a172cc5..5c92257 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -2,16 +2,15 @@ ## Acknowledgements -This is the first blog post in a series summarizing our research -funded by EF GRANT INFO. The author, Eric Downes, is grateful for -useful discussions with the 20 Squares team especially Danieli -Palombi and Philipp Zahn, as well as Eric Siu, Angsar Dietrichs, and -Andrew Sudbury. +This is a blog post on Runaway Staking in a series summarizing our +research funded by EF GRANT INFO. The author, Eric Downes, is +grateful for useful discussions with Eric Siu, Andrew Sudbury, and the +the 20 Squares team; especially Danieli Palombi and Philipp Zahn. ## The Problem The share of Ether staked by exchanges such as Coinbase and Liquid -Staking Providers such as Lido ("centralized" staking services) +Staking Providers (LSTs) such as Lido ("centralized" staking services) continues to grow. This has provoked concerns, first raised by Ethereum researchers, and which we share, that the future of Ethereum might involve (1) all of its native asset being staked, such that (2) @@ -21,45 +20,43 @@ first of these concerns. We use "stock and flow" differential equation models to study Ethereum macroeconomics, specifically how changing issuance impacts these -questions. We will publish four blog posts on this topic: +questions. We will publish several blog posts on this topic: +0. Basics of applying System Dynamics to Ethereum 1. (This post) Will reducing issuance avoid Runaway Staking? 2. Will reducing issuance avoid Governance Centralization? 3. Other Levers besides Issuance, and a means of evaluating levers. -4. Summary and community tools to help study and resolve issuance debates. - -We may also publish bonus posts, if there is sufficient community -engagement, so dear reader stay vigilant! +4. Some tools to help the community study and resolve policy debates. ## TLDR ### For the extraordinarily impatient reader. -* Roughly speaking, under strong deflation, Ethereum faces an - unenviable choice of futures: - - zero staking, or - - runaway staking - -* Under no growth or weak deflation, runaway staking is inevitable. - -* Under inflation, runaway staking may still occur. +* The staking reinvestment ratio is central to understanding the + quantitative future of Ethereum macroeconomics. We believe it can + be well-estimated with onchain data. -* If runaway staking does *not* occur under inflation, it is because - -- (1) inflation is low enough, that concerns over inflation do not - dominate the microeconomic considerations governing the - reinvestment of profits by staking businesses, but simultaneously - -- (2) inflation is high enough to numerically dominate the quantity of - priority fees and MEV reinvested as profit. +* Under weak inflation, runaway staking may occur. If it does *not*, + this is because + -- (1) inflation is low enough, that concerns over inflation + do not dominate the reinvestment of profits by staking businesses, + but simultaneously + -- (2) inflation is high enough to numerically dominate priority fees + and MEV, as a fraction of circulating Ether. -* Outside of the "low inflation, lower fees" regime, we expect - reductions of inflation to backfire, *raising* staking fraction. +* Outside of the "low inflation, lower fees" regime, convergence to a + reasonable staking future is not possible. + -- Under strong deflation, Ethereum faces a choice between zero staking + or runaway staking + -- Under no growth or weak deflation, runaway staking is inevitable. -* Regardless, the staking reinvestment ratio is central to - understanding the quantitative future of Ethereum macroeconomics. - We believe it can be well-estimated with onchain data. +* Givn the above, we emphasize that changing the yield curve can + backfire, *raising* equilibrium staking fraction. ### For the moderately impatient reader. +#### Dynamics Supporting Market Equilibrium + In some more detail, a sketch of the reasons behind our conclusions. The quantity of most importance to this debate is the staked ETH fraction $$s$$, currently roughly .33. Staking fraction is calculated @@ -76,10 +73,8 @@ staking, and/or it can be driven by a reduction of the inflation rate. The latter can be acheived in principle by a reduction of issuance relative to the base fee "burn rate". Because of this quotient rule tradeoff, low-but-positive inflation actually plays a positive -infrastructural role in moderating staking fraction. - -This can be seen in the market equilibrium staking fraction, which we -derive below: +infrastructural role in moderating staking fraction. This can be seen +in the market equilibrium staking fraction, derived below. $$\displaystyle s^\star = r\frac{\alpha + f}{\alpha + rf} @@ -87,11 +82,15 @@ $$ Here $$0\leq r\leq 1$$ is the ratio of profits reinvested quarterly by validators, $$0\leq f\leq 1$$ is the fraction of unstaked ETH spent on -transactions fees (base and priority) quarterly, and as above +transaction fees (base and priority) quarterly, and as above $$\alpha$$ is inflation. The two extremes are $$\alpha\ll f$$ fees dominate and $$\alpha\gg f$$ inflation dominates. In the former, staking fraction is driven closer to 1, while in the latter, staking -is driven to match reinvestment $$x^*\approx r$$. +is driven to match reinvestment $$x^*\approx r$$. A contour plot of +$$x^*$$ as a function of dimensionless fractions $$(\alpha/f,r)$$ is +below. + +-- FIGURE -- Under current market conditions, low inflation and lower fees, the long-term equilibrium staking fraction approaches the average ratio @@ -103,33 +102,67 @@ Approximate present values from YCharts are very roughly $$f\approx.001$$/year, $$\alpha\approx.005$$/year, $$r\in(.5,.75)$$. So if current market conditions (low inflation, lower fees) were to persist at long times, we expect the staking ratio to convergence to -within +10\% of the the reinvestment ratio. If issuance is reduced by -half and fees maintain, $$x^\star$$ is still around $$r$$ plus 15\%. - -However, if issuance is reduced too much, such that fees dominate -inflation, we enter a regime in which reducing issuance *raises* -equilibrium staking fraction, and becomes insensitive to $$r$$. This -could occur by adopting too severe a yield curve, or by validaators -continuing to chase MEV yield, long after the yield from issuance has -become irrelevant. - -Under present conditions $$r$$ is the proximate mechanism by which -inflation must act, in affecting long term staking fraction. As per -all the arguments of researchers, a sufficient increase in inflation -$$\alpha^*\mapsto\alpha^*+\Delta\alpha$$ could well drive an runaway -increase in reinvestment $$\Delta{r}/\Delta\alpha\gesim 1-r$$. - -Thankfully, the reinvestment of profits $$r_i$$ is a microeconomic -quantity every staking business $$i$$ calculates, even if they do not -use those words. The global $$r$$ is then the market-share weighted -average of all these. If inflation pressures are indeed the dominant -consideraton for Ethereum users considering staking, this should -emerge from microeconomic surveys of validators' reinvestment -sentiments. +no more than +10\% of the the reinvestment ratio. Good news: if +inflation is reduced by half, while fees etc. maintain, equilibrium is +still around $$r$$ + 15\%, so $$x^\star\in(.58,.8)$$. + +Wether this is acceptable, depends on $$r$$. That is, under current +market conditions, reinvestment ratio is the proxy by which inflation +must act, if it affects long term staking fraction. Thankfully, a +staking business $$i$$ should be able to calculate the reinvestment of +its own profits $$r_i$$. The global $$r$$ is then the market-share +weighted average of all these, a bridge between Ethereum micro- and +macro-economics. + +Consider this "backfire" scenario, though, in which reductions of +inflation decouple $$x^\star$$ from $$r$$. Issuance is reduced +further, as per the proposed yield curves. The burn rate and +transaction fees as a fraction of circulating Ether do not change +appreciably, such that inflation becomes negligible +$$\alpha/f\approx10^{-2}$$. Inflation hawks might even consider this +a desirable scenario. If $$r$$ does not change appreciably, we +predict that after (un)staking queue transients die out, staking +fraction would equilibrate to $$x^\star\in(.97,.99)$$; less than 3\% +of Ether would remain unstaked. + +How unrealistic is tht last assumption? Say that LSPs set their +yields at the current proportion of yield from issuance plus fees, and +LST-holders maintain their balances because they simply wish to +passively chase a small percentage yield. Solo validators reinvest at +about the same rate they have been, in order to chase priority fees +and MEV despite the reduction in issuance. This is all that is needed +for eventual 100\% ETH Staked; no net influx of stakers is required at +all. + +#### Macroeconomics at the Equilibrium Point + +Per the arguments of Ethereum researchers, high inflation could lead +to runaway staking. This has motivated the drive to reduce issuance, +thereby reducing inflation. Our model offers a perspective here as +well. + +In our model the net effect of inflation on staking fraction at +equilibrium is reflected by taking the derivative +$$0<\frac{dx^\star}{d\alpha}|_{\alpha^\star}$$ assuming $$r,f$$ are +implicit functions of $$\alpha$$. That is, the necessary condition +for inflation to push the market equilibrium $$x^\star$$ itself into +runaway staking is (see below for explanation): + +$$\displaystyle +1 < \frac{\partia\ln r}{\partial\ln\alpha}\big|^\star +\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} ++ \frac{\partial\ln f}{\partial\ln\alpha}\big|^\star +$$ We hope that this work can be built upon to focus inflationary pressure arguments into empirically measurable assertions that can be -tracked as a metric for Ethereum health. +tracked as a metric for Ethereum health. If inflation pressures are +indeed the dominant consideration for Ethereum users considering +staking, this should emerge from microeconomic surveys of validators' +reinvestment sentiments. A semi-empirical measurement of correlations +between transaction fees and inflation, using our model or similar to +control for staking queue flows, etc. should also shed light on wether +the above condition is satisfied or not. ## Modelling Staking @@ -221,23 +254,39 @@ reinvestment of priority fees and MEV $$rf$$, and is always non-negative. The third term represents unstaking and is always non-positive. +We understand the long-time behavior of staking by studying what +controls stable fixed points $$s^*$$ corresponding to $$\dot{s}=0$$. +Briefly, we check that at least the position of this fixed point will +not be altered by considering dynamics of inflation. + +### Aside on Inflation Dynamics + +Above we treat inflation as a parameter instead of as its own dynamic +variable. How dangerous is this? For a sanity-check, a quick look at +YCharts since The Merge show that $$s,\dot{s}$$ vary over a much +greater range than $$(\ln{A},\alpha)$$. + +-- FIGURES -- + +We expect the full dynamics to be more relevant for determining exact +stability criteria under a change of issuance. At present we judge +the lack of empiircal data on $$r$$ would obviate the added precision +of a more sophisticated treatment. + +We will proceed assuming $$\dot{\alpha}\approx0$$ and treating +$$\alpha$$ as a parameter. Curious or skeptical readers are +encouraged to study the full $$(\alpha,s,\beta)$$-system using the +modelling framework we have developed. + ### Fixed Point -If quarterly inflation $$\alpha:=\dot{A}/A$$, governed by the tension -between issuance and burn, changes quickly enough, we have at least a -two dimensional system. Thankfully, if all ones cares about is the -(in)stability of market equilibria, and the potential response to -issuance changes, the varieties of qualitative behavior of this higher -dimensonal system are all present in the above one-dimensional system. -That is, we find the same supportive role is played by inflation in -moderating staking fraction, and that $$r$$ plays a prominent role in -the character of the fixed point. The primary difference, over -existing market conditions, is that oscillations are possible. - -If instead, $$\alpha$$ varies slowly enough -$$\alpha\approx\alpha^\star$$, a fixed point $$s^\star$$ where -$$\dot{s}=0$$ can obtain: -$$ \displaystyle +Returning from our commercial break, recall we have found a supportive +role played by inflation in moderating staking fraction. We now wish +to study the fixed point $$s^*$$ given by solving +$$0=\dot{s}=\frac{\dot{A}}{A}(r-s)+(rf+q_+)(1-s)-q_-s$$ +yielding: + +$$ s^\star=\frac{r^\star(\alpha^\star + f^\star) + {q^\star}_+}{\alpha + r^\star f^\star - {q^\star}_-} $$ @@ -260,25 +309,29 @@ offset the unstaking, adjusted for inflation. This cannot maintain forever: eventually there will be no new Capitulators left, and $$x^*$$ must once again grow as required by the Reinvestors' higher $$r$$, so $$x^*$$ was not a fixed point at all. Similarly, at some -point everyone who wants to stake should have staked, and other than -influxes due to the issuance of new humans, further validators count -only toward $$r$$, so $$q_+^\star\approx0$$. +point everyone who wants to stake should have staked. If we judge the +quarterly fluxes due to the issuance of new humans and the burn rate +of legacy humans to be small and/or likely to take over existing +businesses, additional validators count overwhelmingly toward $$r$$, +so $$q_+^\star\approx0$$. -Thus, the fixed point $$s^\star$$ is likely to be -$$s^\star\approx r\frac{\alpha + f}{\alpha + rf}$$ +Thus, the fixed point $$s^\star$$ simplifies to +$$\displaystyle +s^\star = r\frac{\alpha + f}{\alpha + rf} +$$ A calculation is illustrative under the current regime of positive inflation. If inflation dominates fees, $$\alpha\gg f$$ then $$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f -\approx .001\lessim .005\approx\alpha$$ so to within 10\% error +\approx .001\lessim .005\approx\alpha$$ so to within 10\% error above, $$x^*\approx r$$ over the range of $$r$$ inferred from Lido yield rate. So if these conditions persist at long times, and to be clear there is no reason to be certain they will, we should expect $$x^\star\approx r\in(.55,.77)$$. This places the lower range of $$x^\star$$ near the -50% staking target proposed by Elowsson. This is hopeful but little +50% staking target proposed by Elowsson. This is hopeful, but little weight can be attached to such back-of-envelope extrapolations. To make reliable projections we need systematic measurement of both the statistics of individual validator reinvestment and metrics or at @@ -287,22 +340,29 @@ absorb reduced revenue. ### Market Equilibrium -The fixed-point $$x^\star$$ represents the market equilibrium when it -is unqiue and stable. +The fixed-point $$x^\star$$ represents "the" market equilibrium when it +is both unqiue and stable. -Non-uniqueness requires a shared and repeated root (the other -equilibrium point) among all terms $$\alpha,rf+q_+,q_-$$. That is, -each economic flow would have to nearly disappear at a particular -value of staking fraction. Lacking any supporting data or a -mechanism, we cannot really analyse such an alternate equilibria, and +Non-uniqueness requires another solution to $$\dot{s}=0$$, thus a +shared and repeated root among all terms $$\alpha,rf+q_+,q_-$$. That +is, each economic flow would have to nearly disappear at a particular +value of staking fraction. Lacking any supporting data or a mechanism +for this, we cannot really analyse such an alternate equilibria, but intuitively it seems unlikely, so we move on, assuming uniqueness. Stability requires that small perturbations shrink; -$$\frac{\partial\dot{x}}{\partial x}\big|_{x^\star}<0$$. The full +$$\frac{\partial\dot{x}}{\partial x}\big|^<0$$, where +$$\frac{d}{d\xi}|^\star$$ denotes a derivative with respect to $$\xi$$ +at the fixed point $$\xi^\star$$. + + + + +The full condition is quite complicated, especially under deflation. If however inflation is positive and $$r,\alpha,f$$ are relatively insensitive to small changes in $$x$$ near $$x^\star$$, this -conditions is satisfied immediately; $$\alpha+rf>0$$. +condition is satisfied immediately; $$\alpha+rf>0$$. This stable equilibrium seems plausible, and worthy of consideration. How does it behave? @@ -311,56 +371,81 @@ How does it behave? The fixed point $$x^\star=r(\alpha+f)/(\alpha+rf)$$ represents behavior distinct in flavor from the arguments forwarded by Ethereum -researchers, in which inflation is held to blame for a 100% ETH Staked -scenario. To be clear, very likely $$x_{now} Date: Thu, 5 Dec 2024 06:00:05 -0500 Subject: [PATCH 06/70] final draft, need add figures and fix typos --- _posts/2024-12-01-issuance-fundamentals.md | 375 ++++++++++----------- 1 file changed, 184 insertions(+), 191 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 5c92257..59214d8 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -64,7 +64,7 @@ $$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is "circulating" (unstaked, unburnt) ETH, and $$A=S+C$$ is the total "accessible" (unburnt) ETH. Differential changes, such as changes in time $$\frac{ds}{dt}:=\dot{s}$$ are given by the quotient rule -$$\dot{s}=\dot{S}/A-s\dot{A}{A}$$. The quantity $$\alpha = \dot{A}/A$$ +$$\dot{s}=\dot{S}/A-s\dot{A}/A$$. The quantity $$\alpha = \dot{A}/A$$ is the average on-paper inflation rate (supply expansion APY) averaged on at-least-quarterly timescales. @@ -74,37 +74,43 @@ The latter can be acheived in principle by a reduction of issuance relative to the base fee "burn rate". Because of this quotient rule tradeoff, low-but-positive inflation actually plays a positive infrastructural role in moderating staking fraction. This can be seen -in the market equilibrium staking fraction, derived below. +in the market equilibrium staking fraction $$s^\star$$, shown here and +derived below. $$\displaystyle -s^\star = r\frac{\alpha + f}{\alpha + rf} +s^\star = r^\star\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} $$ -Here $$0\leq r\leq 1$$ is the ratio of profits reinvested quarterly by +Here $$r^\star$$ etc. means the varable $$r(s,\ldots)$$ at the +equilibrium point $$s^\star$$. Speaking of which, the fraction +$$0\leq r\leq 1$$ is the ratio of profits reinvested quarterly by validators, $$0\leq f\leq 1$$ is the fraction of unstaked ETH spent on transaction fees (base and priority) quarterly, and as above -$$\alpha$$ is inflation. The two extremes are $$\alpha\ll f$$ fees -dominate and $$\alpha\gg f$$ inflation dominates. In the former, -staking fraction is driven closer to 1, while in the latter, staking -is driven to match reinvestment $$x^*\approx r$$. A contour plot of -$$x^*$$ as a function of dimensionless fractions $$(\alpha/f,r)$$ is -below. +$$\alpha$$ is inflation. The two extremes are $$\alpha^\star\ll +f^\star$$ fees dominate and $$\alpha^\star\gg f^\star$$ inflation +dominates. In the former, staking fraction is driven closer to 1, +while in the latter, staking is driven to match reinvestment +$$s^\star\approx r^\star$$. --- FIGURE -- +#### Good Scenario Under current market conditions, low inflation and lower fees, the long-term equilibrium staking fraction approaches the average ratio -$$r$$ at which validators reinvest their staking rewards. For LSTs -$$r$$ is bounded below by the ratio of token yield to total yield, and -we can use this to roughly estimate some real values. +$$r$$ at which validators reinvest their staking rewards. When there +are different types of validators, $$r$$ is the average of $$r_i$$ for +each validator type $$i$$, weighted by the amount of Ether each +stakes. For LSPs $$r_{LST}$$ is bounded below by the ratio of token +yield to total yield, and we can use this to roughly estimate some +full $$r$$ values. Approximate present values from YCharts are very roughly $$f\approx.001$$/year, $$\alpha\approx.005$$/year, $$r\in(.5,.75)$$. So if current market conditions (low inflation, lower fees) were to -persist at long times, we expect the staking ratio to convergence to -no more than +10\% of the the reinvestment ratio. Good news: if -inflation is reduced by half, while fees etc. maintain, equilibrium is -still around $$r$$ + 15\%, so $$x^\star\in(.58,.8)$$. +persist at long times, we expect the staking ratio converges to no +more than +10\% of the the reinvestment ratio. Good news: if issuance +is reduced such that inflation is reduced by half, while fees +etc. maintain, equilibrium is still around $$r$$ + 15\%, so +$$s^\star\in(.58,.8)$$. Wether this is acceptable, depends on $$r$$. That is, under current market conditions, reinvestment ratio is the proxy by which inflation @@ -114,44 +120,51 @@ its own profits $$r_i$$. The global $$r$$ is then the market-share weighted average of all these, a bridge between Ethereum micro- and macro-economics. +#### Bad Scenario + Consider this "backfire" scenario, though, in which reductions of -inflation decouple $$x^\star$$ from $$r$$. Issuance is reduced -further, as per the proposed yield curves. The burn rate and +inflation decouple $$s^\star$$ from $$r$$. + +Issuance is reduced, as +per the yield curve $$y(S)=kS^{-1/2}(1+k'S)$$. The burn rate and transaction fees as a fraction of circulating Ether do not change -appreciably, such that inflation becomes negligible -$$\alpha/f\approx10^{-2}$$. Inflation hawks might even consider this -a desirable scenario. If $$r$$ does not change appreciably, we -predict that after (un)staking queue transients die out, staking -fraction would equilibrate to $$x^\star\in(.97,.99)$$; less than 3\% -of Ether would remain unstaked. - -How unrealistic is tht last assumption? Say that LSPs set their -yields at the current proportion of yield from issuance plus fees, and -LST-holders maintain their balances because they simply wish to -passively chase a small percentage yield. Solo validators reinvest at -about the same rate they have been, in order to chase priority fees -and MEV despite the reduction in issuance. This is all that is needed -for eventual 100\% ETH Staked; no net influx of stakers is required at -all. +appreciably. + +LSPs adjust their yields to maintain the current +proportion $$r$$ of yield from issuance plus fees, and LST-holders +maintain their balances because they simply wish to passively chase a +small percentage yield. Solo validators reinvest at about the same +rate they have been, in order to chase priority fees and MEV despite +the reduction in issuance. So, we are pushed into the higher part of the +yield curve, keeping issuance negligible +$$\alpha^\star/f^\star\lesssim10^{-2}$$. Inflation hawks might even +consider this a desirable scenario. After (un)staking queue +transients die out, staking fraction would equilibrate to +$$s^\star\in(.97,.99)$$. + +Essentially, we really need to be sure of what drives validaor +behavior. If enough validators persist in reinvestment beyond where +we would expect them to, Ethereum could become deflationary, and this +is dangerous. We recommend surveys to gauge sentiment and +calculations of the statistics of $$r_i$$. #### Macroeconomics at the Equilibrium Point Per the arguments of Ethereum researchers, high inflation could lead -to runaway staking. This has motivated the drive to reduce issuance, -thereby reducing inflation. Our model offers a perspective here as -well. +to runaway staking. This has motivated the drive to reduce issuance +and thus inflation. Our model offers a perspective here as well. In our model the net effect of inflation on staking fraction at equilibrium is reflected by taking the derivative -$$0<\frac{dx^\star}{d\alpha}|_{\alpha^\star}$$ assuming $$r,f$$ are +$$0<\frac{ds^\star}{d\alpha}\big|_{\alpha^\star}$$ assuming $$r,f$$ are implicit functions of $$\alpha$$. That is, the necessary condition -for inflation to push the market equilibrium $$x^\star$$ itself into +for inflation to push the market equilibrium $$s^\star$$ itself into runaway staking is (see below for explanation): $$\displaystyle 1 < \frac{\partia\ln r}{\partial\ln\alpha}\big|^\star -\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} -+ \frac{\partial\ln f}{\partial\ln\alpha}\big|^\star +\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star}+ +\frac{\partial\ln f}{\partial\ln\alpha}\big|^\star $$ We hope that this work can be built upon to focus inflationary @@ -166,7 +179,7 @@ the above condition is satisfied or not. ## Modelling Staking -Now the details! +Now some more details! ### Splitting Up A Growing Pie @@ -174,16 +187,12 @@ Consider the quantity $$E$$ of all Ether in existence. Let the issuance of new Ether as per The Merge be quantified by $$I$$. Then the change in time of total Ether is $$\dot{E}:=dE/dt=I$$. --- FIGURE 1 (see notability) -- - Now, let us separate $$E$$ into three smaller boxes; $$S+C+\cancel{O}=E$$, representing (S)taked, (C)icrulating, and burnt Ether $$\cancel{O}$$, respectively. By considering each quantity a quarterly average we can plausibly ignore the detailed dynamics of the staking, unstaking, and withdrawal queues. --- FIGURE 2 (see notability) -- - ### How Pie is Pushed Around The Plate This is our model: @@ -263,20 +272,17 @@ not be altered by considering dynamics of inflation. Above we treat inflation as a parameter instead of as its own dynamic variable. How dangerous is this? For a sanity-check, a quick look at -YCharts since The Merge show that $$s,\dot{s}$$ vary over a much +YCharts since The Merge shows that $$s,\dot{s}$$ vary over a much greater range than $$(\ln{A},\alpha)$$. -- FIGURES -- -We expect the full dynamics to be more relevant for determining exact -stability criteria under a change of issuance. At present we judge -the lack of empiircal data on $$r$$ would obviate the added precision -of a more sophisticated treatment. - -We will proceed assuming $$\dot{\alpha}\approx0$$ and treating -$$\alpha$$ as a parameter. Curious or skeptical readers are -encouraged to study the full $$(\alpha,s,\beta)$$-system using the -modelling framework we have developed. +At present we judge the lack of empiircal data on $$r$$ would obviate +the added precision of a more sophisticated treatment. We will +proceed assuming $$\dot{\alpha}\approx0$$ and treating $$\alpha$$ as a +parameter. Curious or skeptical readers are encouraged to study the +full $$(\alpha,s,\beta)$$-system using the modelling framework we have +developed. ### Fixed Point @@ -307,8 +313,8 @@ stake, while another group with a higher $$r$$ were aggressively reinvesting in their business, and their reinvestment of fees and MEV offset the unstaking, adjusted for inflation. This cannot maintain forever: eventually there will be no new Capitulators left, and -$$x^*$$ must once again grow as required by the Reinvestors' higher -$$r$$, so $$x^*$$ was not a fixed point at all. Similarly, at some +$$s^\star$$ must once again grow as required by the Reinvestors' higher +$$r$$, so $$s^\star$$ was not a fixed point at all. Similarly, at some point everyone who wants to stake should have staked. If we judge the quarterly fluxes due to the issuance of new humans and the burn rate of legacy humans to be small and/or likely to take over existing @@ -325,12 +331,12 @@ inflation. If inflation dominates fees, $$\alpha\gg f$$ then $$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f \approx .001\lessim .005\approx\alpha$$ so to within 10\% error above, -$$x^*\approx r$$ over the range of $$r$$ inferred from Lido yield +$$s^*\approx r$$ over the range of $$r$$ inferred from Lido yield rate. So if these conditions persist at long times, and to be clear there is -no reason to be certain they will, we should expect $$x^\star\approx -r\in(.55,.77)$$. This places the lower range of $$x^\star$$ near the +no reason to be certain they will, we should expect $$s^\star\approx +r\in(.55,.77)$$. This places the lower range of $$s^\star$$ near the 50% staking target proposed by Elowsson. This is hopeful, but little weight can be attached to such back-of-envelope extrapolations. To make reliable projections we need systematic measurement of both the @@ -338,7 +344,7 @@ statistics of individual validator reinvestment and metrics or at least surveys reflecting validator sentiment and (in)capacity to absorb reduced revenue. -### Market Equilibrium +### Market Equilibrium: Stability The fixed-point $$x^\star$$ represents "the" market equilibrium when it is both unqiue and stable. @@ -348,93 +354,102 @@ shared and repeated root among all terms $$\alpha,rf+q_+,q_-$$. That is, each economic flow would have to nearly disappear at a particular value of staking fraction. Lacking any supporting data or a mechanism for this, we cannot really analyse such an alternate equilibria, but -intuitively it seems unlikely, so we move on, assuming uniqueness. +intuitively it seems unlikely, so for now we move on, assuming +uniqueness. Stability requires that small perturbations shrink; $$\frac{\partial\dot{x}}{\partial x}\big|^<0$$, where $$\frac{d}{d\xi}|^\star$$ denotes a derivative with respect to $$\xi$$ -at the fixed point $$\xi^\star$$. - - - - -The full -condition is quite complicated, especially under deflation. -If however inflation is positive and $$r,\alpha,f$$ are -relatively insensitive to small changes in $$x$$ near $$x^\star$$, this -condition is satisfied immediately; $$\alpha+rf>0$$. - -This stable equilibrium seems plausible, and worthy of consideration. -How does it behave? - -### Qualitative Behavior - -The fixed point $$x^\star=r(\alpha+f)/(\alpha+rf)$$ represents -behavior distinct in flavor from the arguments forwarded by Ethereum -researchers, in which concerns over inflation would drive the network -to a 100% ETH Staked scenario. To be clear, very likely -$$x_{now} +\frac{d\ln{r}}{d\ln s}\big|^\star + +(1 - \frac{r^\star}{s^\star}) +\frac{d\ln(\alpha/f)}{d\ln s}\big|^\star +$$ + +Where presently, we estimate the dimensionless quantity +$$\alpha/f\approx5$$ while its derivative depends on the slope of the +yield curve $$y$$ via $$\alpha/f=ys/f-b(1-s)=[I/(F+B)-b](1-s)$$. + +*If* the derivatives are all small in comparison (and there is no a +priori gaurantee of this), then we can say something useful. +Stability is automatic under inflation so long as +$$\alpha^\star+r^\star f^\star>0$$. If again derivatives are all +small but $$\alpha<0,~rf<|\alpha|0$$. The strong deflation +$$\alpha<-f<0$$ fixed point appears interior unstable, predicting +either runaway staking or a catastrophic loss of staking, depending on +initial considitions. Our interpretation of the latter behavior is +that if raw Ether is appreciating in value so quickly, and there has +been insufficien interest in staking thusfar, then why bother staking at +all? But we must emphasize that what instability *really* means is +that "market externalities take over", rather than supporting any +specific trajectory due to endogenous dynamics. + +Careful empirical data analysis will be necessary to evaluate the +above stability boundary with accuracy. We feel it is extremely +important that policy interventions, such as changes to issuance, +maintain a stable fixed point -- a fixed point where small changes +shrink instead of growing. Otherwise staking fraction is no longer +influenced by equilibrium arguments and becomes much harder to predict +or control. Potential for instability would manifest as large groups +of validators and other market participants taking very different bets +on the future of Ethereum. Which future manifests would probably +depend on details not observable from macroeconomics alone. In what +follows we will assume stability for the sake of discussion, but must +stress that while we judge it likely under current conditions, this +matter cannot be settled without further study. + +### Inflation Concerns + +As we have seen, reducing issuance in a vacuum decreases inflation, +and increases the relative strength of the $$rf(1-s)$$ term. This +leads to an increase in staking fraction if $$r$$ etc. do not reduce, +in response. Very likely $$x_{now}1$$; and there is no interior fixed point. + +Why does this happen, though? The reinvestment of issuance rewards +adds to both staked ETH $$S$$ and circulating ETH $$C$$, so the effect +on staking fraction depends on the relative size of these two. +However, the reinvestment of priority fees plus MEV is simply a net +value transfer from $$C$$ to $$S$$, which can only ever increase +staking fraction. + +Some good news is that under current conditions, we have some leeway. +A moderate reduction of inflation to half or even a third of its +present value would still result in a comfortably positive inflation +rate, $$\alpha>f$$ and maintain the dominance of reinvestment in the +long term $$s^\star\to r^\star$$. + +The effect of $$\alpha$$ on $$r$$ then becomes quite salient, and this +is where previous market equilibrium arguemnts come into play. We +agree with issuane hawks that there must be *some* $$(s,\alpha)$$ past +which fears of runaway inflation will drive $$r\to1$$ and therefore +$$s^\star\to1$$. How does our model account for such effects? + +The derivative of staking fraction by inflation near equilibrium +$$\frac{\partial x^\star}{\partial\alpha}\big|^\star$$ shows the response to +small endogenous changes, but not to a market crash, or protocol +hardfork. The condition that gently rising inflation causes an +*increase* in staking is that this derivative should be positive. Assuming $$r,f$$ are (unknown) implicit funcions of $$\alpha$$ the positive derivative condition becomes after some manipulation $$\displaystyle -1 < \frac{\partia\ln r}{\partial\ln\alpha}\big|^*\cdot\frac{1+\alpha^*/f^*}{1-r^*} -+ \frac{\partial\ln f}{\partial\ln\alpha}\big|^* +1 < \frac{\partia\ln r}{\partial\ln\alpha}\big|^\star\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} ++ \frac{\partial\ln f}{\partial\ln\alpha}\big|^\star $$ This gives us a condition expressing how the macroeconomic effects of @@ -448,58 +463,36 @@ this conditions. ## In Conclusion -We find a nuetral and even positive role for inflation in maintaining -moderate staking levels under various scenarios. We do not contest -the logic that runaway inflation could drive staking fraction -unreasonably high, but the picture emerging from our model appears far -more nuanced. +We find a potentially nuetral and even positive role for inflation in +maintaining moderate staking levels under various scenarios. We do +not contest the logic that runaway inflation *could* drive staking +fraction unreasonably high. If so, it must act through reinvestment, +and we emphasize that zero-growth or deflationary conditions would +only make matters worse. We find that the reinvestment of staking rewards plays a critical role in staking dynamics. This is especially true under present market -conditions post-EIP-1557, where inflation dominates reinvested fees, -and we expect the equilibrium staking fraction to approach the -reinvestment rate $$r$$. - -We propose that $$r$$ be added to the list of macroeconomic indicators -routinely tracked for Ethereum, and that the economic forces, and -internal variability affecting $$r$$ be modelled and measured. -Specifically, policy advisors need to know how and indeed if solo -validators and LSTs will alter their reinvestment strategies, +conditions post-EIP-1557, where inflation dominates fees, and we +expect the equilibrium staking fraction to approach the reinvestment +rate $$r$$. The formulas for the staking fixed point, stability +criteria, and criteria for inflation to contribute on net to runaway +staking, appear new and we hope they will likewise prove useful to the +community. In principle, all of these could be tracked. + +We propose at least that $$r$$ be added to the list of macroeconomic +indicators routinely tracked for Ethereum, and that the economic +forces, and internal variability affecting $$r$$ be modelled and +measured. Specifically, policy advisors need to know how and indeed +if solo validators and LSTs will alter their reinvestment strategies, especially under a regime of lower issuance yield. -It may well be that fears about runaway inflation will turn out to be -the dominant factor affecting reinvestment. If so this should emerge -from microeconomic surveys of large validators and LSTs. Each staking -business $$i$$ can, and probably already does, calculate its desired -$$r_i$$. The global $$r$$ is then the market-share weighted averaged -of all of these. We hope these calculations will bridge the macro- -and micro-economics to help the community see a unified and actionable -big picture on the future of Ethereum staking. - - - -# Unused Notes - -## $$\dot{\alpha}$$ - -Let's derive an expression for $$\dot{\alpha}=\frac{d}{dt}[(I-B)/A]$$. -$$S$$ is proportional to the number of validators, so we may write -$$I=yS$$ for a scaled yield curve $$y$$. Observe that $$y(S)=y(As) and $$ -and let $$y':=dy/dS$$ such that $$dy=. So in terms of the previous intensive variables -$$\alpha = ys-bf(1-s)$$, and we use $$\beta:=bf$$ to obtain -$$\dot{\alpha}(1-sy'A) = (y+\beta+y'/A)\dot{s}-(1-s)\dot{\beta}$$. - -Observe that for yield curves without derivative singularities, such -as all under serious discussion, $$sy'\ll A$$. Dividing by -$$1-sy'/A$$ obtains - -$$\dot{\alpha}=h(s,\beta,A)\dot{s}-\frac{1-s}{1-sy'/A}\dot{\beta}$$. - -We are interested in fixed points $$s^*$$ given by $$\dot{s}=0$$. -Thus $$s^*$$ is also a fixed point of $$\dot{\alpha}$$ when -$$\dot{\beta}=0$$ or $$s^*=1$$. That is, a market-equilibrium of -$$\dot{s}$$ is unaltered by considering inflation dyanmics due to -non-zero $$\dot{\alpha}=\frac{d^2\ln{A}}{dt^2}$$. Where one *may* wish -to return to studying a non-zero $$\dot{\alpha}$$, is to study the -full stability conditions of the $$(s,\alpha,\beta)$$-system. +It may well, as some have predicted, that fears about runaway +inflation will turn out to be the dominant factor affecting +reinvestment. If so we predict this must emerge from microeconomic +surveys of large validators and LSTs. Each staking business $$i$$ +can, and probably already does, calculate its desired $$r_i$$. The +global $$r$$ is then the market-share weighted averaged of all of +these. We hope these calculations will bridge the macro- and +micro-economics to help the community see a unified and actionable big +picture on the future of Ethereum staking. From 57123e0754329f1385d126e4e325eed165f90486 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 06:18:09 -0500 Subject: [PATCH 07/70] final draft, need add figures and fix typos --- _posts/2024-12-01-issuance-fundamentals.md | 119 ++++++++++----------- 1 file changed, 58 insertions(+), 61 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 59214d8..0c93dd3 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -1,4 +1,4 @@ -# Will Reducing Issuance Avoid Runaway Staking? +# Some danger in reducing Issuance to avoid Runaway Staking. ## Acknowledgements @@ -22,7 +22,7 @@ We use "stock and flow" differential equation models to study Ethereum macroeconomics, specifically how changing issuance impacts these questions. We will publish several blog posts on this topic: -0. Basics of applying System Dynamics to Ethereum +0. Basics of applying Dynamics to Ethereum 1. (This post) Will reducing issuance avoid Runaway Staking? 2. Will reducing issuance avoid Governance Centralization? 3. Other Levers besides Issuance, and a means of evaluating levers. @@ -50,8 +50,8 @@ questions. We will publish several blog posts on this topic: or runaway staking -- Under no growth or weak deflation, runaway staking is inevitable. -* Givn the above, we emphasize that changing the yield curve can - backfire, *raising* equilibrium staking fraction. +* Given the above, we emphasize that changing the yield curve could + backfire, *raising* long term staking fraction. ### For the moderately impatient reader. @@ -81,16 +81,16 @@ $$\displaystyle s^\star = r^\star\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} $$ -Here $$r^\star$$ etc. means the varable $$r(s,\ldots)$$ at the -equilibrium point $$s^\star$$. Speaking of which, the fraction -$$0\leq r\leq 1$$ is the ratio of profits reinvested quarterly by -validators, $$0\leq f\leq 1$$ is the fraction of unstaked ETH spent on -transaction fees (base and priority) quarterly, and as above -$$\alpha$$ is inflation. The two extremes are $$\alpha^\star\ll -f^\star$$ fees dominate and $$\alpha^\star\gg f^\star$$ inflation -dominates. In the former, staking fraction is driven closer to 1, -while in the latter, staking is driven to match reinvestment -$$s^\star\approx r^\star$$. +Here $$r^\star$$ etc. means the function $$r(s,\alpha,\ldots)$$ at the +equilibrium coordinates $$(s^\star,\alpha^\star,\ldots)$$. Speaking +of which, the fraction $$0\leq r\leq 1$$ is the ratio of profits +reinvested quarterly by validators, $$0\leq f\leq 1$$ is the fraction +of unstaked ETH spent on transaction fees (base and priority) +quarterly, and as above $$\alpha$$ is inflation. The two extremes are +$$\alpha^\star\ll f^\star$$ fees dominate and $$\alpha^\star\gg +f^\star$$ inflation dominates. In the former, staking fraction is +driven closer to 1, while in the latter, staking is driven to match +reinvestment $$s^\star\approx r^\star$$. #### Good Scenario @@ -125,28 +125,25 @@ macro-economics. Consider this "backfire" scenario, though, in which reductions of inflation decouple $$s^\star$$ from $$r$$. -Issuance is reduced, as -per the yield curve $$y(S)=kS^{-1/2}(1+k'S)$$. The burn rate and -transaction fees as a fraction of circulating Ether do not change -appreciably. - -LSPs adjust their yields to maintain the current -proportion $$r$$ of yield from issuance plus fees, and LST-holders -maintain their balances because they simply wish to passively chase a -small percentage yield. Solo validators reinvest at about the same -rate they have been, in order to chase priority fees and MEV despite -the reduction in issuance. So, we are pushed into the higher part of the -yield curve, keeping issuance negligible +Issuance is reduced, as per the yield curve $$y(S)=kS^{-1/2}(1+k'S)$$. +The burn rate and transaction fees as a fraction of circulating Ether +do not change appreciably. LSPs adjust their yields to maintain the +current proportion $$r$$ of the yield from issuance plus fees, and +LST-holders maintain their balances because they simply wish to +passively chase a small percentage yield. Solo validators reinvest at +about the same rate they have been, in order to chase priority fees +and MEV despite the reduction in issuance. So, we are pushed into the +higher part of the yield curve, forcing inflation below fees $$\alpha^\star/f^\star\lesssim10^{-2}$$. Inflation hawks might even -consider this a desirable scenario. After (un)staking queue -transients die out, staking fraction would equilibrate to -$$s^\star\in(.97,.99)$$. +have considered this a desirable scenario before reading this post! +After (un)staking queue transients die out, staking fraction would +equilibrate to $$s^\star\in(.97,.99)$$. Essentially, we really need to be sure of what drives validaor behavior. If enough validators persist in reinvestment beyond where we would expect them to, Ethereum could become deflationary, and this is dangerous. We recommend surveys to gauge sentiment and -calculations of the statistics of $$r_i$$. +calculations of the distribution of $$r_i$$. #### Macroeconomics at the Equilibrium Point @@ -331,18 +328,18 @@ inflation. If inflation dominates fees, $$\alpha\gg f$$ then $$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f \approx .001\lessim .005\approx\alpha$$ so to within 10\% error above, -$$s^*\approx r$$ over the range of $$r$$ inferred from Lido yield +$$s^\star\approx r$$ over the range of $$r$$ inferred from Lido yield rate. So if these conditions persist at long times, and to be clear there is no reason to be certain they will, we should expect $$s^\star\approx r\in(.55,.77)$$. This places the lower range of $$s^\star$$ near the -50% staking target proposed by Elowsson. This is hopeful, but little -weight can be attached to such back-of-envelope extrapolations. To -make reliable projections we need systematic measurement of both the -statistics of individual validator reinvestment and metrics or at -least surveys reflecting validator sentiment and (in)capacity to -absorb reduced revenue. +50% staking target proposed by Elowsson. This is hopeful! However +ittle weight can be attached to such back-of-envelope extrapolations +as these. To make reliable projections we need systematic measurement +of both the statistics of individual validator reinvestment and +metrics or at least surveys reflecting validator sentiment and +(in)capacity to absorb reduced revenue. ### Market Equilibrium: Stability @@ -391,34 +388,34 @@ all? But we must emphasize that what instability *really* means is that "market externalities take over", rather than supporting any specific trajectory due to endogenous dynamics. -Careful empirical data analysis will be necessary to evaluate the -above stability boundary with accuracy. We feel it is extremely -important that policy interventions, such as changes to issuance, -maintain a stable fixed point -- a fixed point where small changes -shrink instead of growing. Otherwise staking fraction is no longer -influenced by equilibrium arguments and becomes much harder to predict -or control. Potential for instability would manifest as large groups -of validators and other market participants taking very different bets -on the future of Ethereum. Which future manifests would probably -depend on details not observable from macroeconomics alone. In what -follows we will assume stability for the sake of discussion, but must -stress that while we judge it likely under current conditions, this -matter cannot be settled without further study. +We feel it is extremely important that policy interventions, such as +changes to issuance, maintain a stable fixed point -- a fixed point +where small changes shrink instead of growing. Otherwise staking +fraction is no longer influenced by equilibrium arguments and becomes +much harder to predict or control. Potential for instability would +manifest as large groups of validators and other market participants +taking very different bets on the future of Ethereum. Which future +manifests would probably depend on details not observable from +macroeconomics alone. In what follows we will assume stability for +the sake of discussion, but must stress that while we judge it likely +under current conditions, this matter is not settled without further +and more careful study. ### Inflation Concerns As we have seen, reducing issuance in a vacuum decreases inflation, and increases the relative strength of the $$rf(1-s)$$ term. This -leads to an increase in staking fraction if $$r$$ etc. do not reduce, -in response. Very likely $$x_{now}1$$; and there is no interior fixed point. +leads to an increase in staking fraction if $$r$$ etc. do not reduce +sufficiently, in response. Very likely $$x_{now}1$$; and there is no +interior fixed point. Why does this happen, though? The reinvestment of issuance rewards adds to both staked ETH $$S$$ and circulating ETH $$C$$, so the effect @@ -488,7 +485,7 @@ especially under a regime of lower issuance yield. It may well, as some have predicted, that fears about runaway inflation will turn out to be the dominant factor affecting -reinvestment. If so we predict this must emerge from microeconomic +reinvestment. If so we expect this will emerge from microeconomic surveys of large validators and LSTs. Each staking business $$i$$ can, and probably already does, calculate its desired $$r_i$$. The global $$r$$ is then the market-share weighted averaged of all of From f4264828d142b65b5f9d7ecececc8b03404a3d34 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 06:25:34 -0500 Subject: [PATCH 08/70] fiixing typos... --- _posts/2024-12-01-issuance-fundamentals.md | 20 +++++++++----------- 1 file changed, 9 insertions(+), 11 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 0c93dd3..dc0af3d 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -159,9 +159,9 @@ for inflation to push the market equilibrium $$s^\star$$ itself into runaway staking is (see below for explanation): $$\displaystyle -1 < \frac{\partia\ln r}{\partial\ln\alpha}\big|^\star +1 < \frac{\partial\ln r}{\partial\ln\alpha}\Big|^\star \cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star}+ -\frac{\partial\ln f}{\partial\ln\alpha}\big|^\star +\frac{\partial\ln f}{\partial\ln\alpha}\Big|^\star $$ We hope that this work can be built upon to focus inflationary @@ -193,9 +193,10 @@ staking, unstaking, and withdrawal queues. ### How Pie is Pushed Around The Plate This is our model: + $$ \displaystyle -\begin{array}rcl} +\begin{array}{rcl} \dot{E} &=& I\\ \dot{S} &=& R+Q_+-Q_-\\ \dot{A} &=& I-B\\ @@ -215,7 +216,7 @@ ae interested in the limit $$C\to0$$ and pull this dependence out by defining $$f$$ such that $$fC = F+B$$. The fraction $$f$$ need not be a constant, but whatever values it takes, $$f$$ must be a fraction: $$0 Date: Thu, 5 Dec 2024 06:37:09 -0500 Subject: [PATCH 09/70] fiixing moar typos... --- _posts/2024-12-01-issuance-fundamentals.md | 53 ++++++++++++---------- 1 file changed, 28 insertions(+), 25 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index dc0af3d..5bd1ae0 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -159,9 +159,9 @@ for inflation to push the market equilibrium $$s^\star$$ itself into runaway staking is (see below for explanation): $$\displaystyle -1 < \frac{\partial\ln r}{\partial\ln\alpha}\Big|^\star -\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star}+ -\frac{\partial\ln f}{\partial\ln\alpha}\Big|^\star +1 < \left.\frac{\partial\ln r}{\partial\ln\alpha}\right|^\star +\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} + +\left.\frac{\partial\ln f}{\partial\ln\alpha}\right|^\star $$ We hope that this work can be built upon to focus inflationary @@ -229,8 +229,8 @@ a certain fraction of yield is the value proposition for the token-holder; so any model with LSTs must include reinvestment. Again we define a variable fraction $$r:=R/(I+F)$$ which obeys $$0\leq{r}\leq1$$. As of Nov 23, 2024 the stETH token yield is -$$\approx$$3\%. Over the same period, without MEV-Boost validator -yield is $$\approx$$4\%, and with MEV-Boost it is $$\approx$$5.7\%. +$$\approx3$$\%. Over the same period, without MEV-Boost validator +yield is $$\approx4$$\%, and with MEV-Boost it is $$\approx5.7$$\%. So probably, $$r$$ currently lies in the range $$.5\leq r\leq.75$$. The quarterly flows from the staking and unstaking queues must obey @@ -240,8 +240,8 @@ $$\dot{S}$$ the change in total staked ETH, anticipating that we wish to understand the relationship to inflation rate $$\dot{A}/A$$, obtaining $$\dot{S} = r\dot{A} + r(B+F) + Q_+-Q_-$$. For about the past 6 months Ethereum supply has been growing pretty linearly, -corresponding to an inflation rate of about .5\%, or -$$\alpha=.005$$/year. +corresponding to an inflation rate of about .5\%, that is +$$\alpha\approx.005$$/year. The variable of primary interest is the staking fraction $$s=S/A$$; recalling the quotient rule $$\dot{S}=\dot{S}/A-s\dot{A}/A$$ and @@ -308,15 +308,15 @@ stake, while another group with a higher $$r$$ were aggressively reinvesting in their business, and their reinvestment of fees and MEV offset the unstaking, adjusted for inflation. This cannot maintain forever: eventually there will be no new Capitulators left, and -$$s^\star$$ must once again grow as required by the Reinvestors' higher -$$r$$, so $$s^\star$$ was not a fixed point at all. Similarly, at some -point everyone who wants to stake should have staked. If we judge the -quarterly fluxes due to the issuance of new humans and the burn rate -of legacy humans to be small and/or likely to take over existing -businesses, additional validators count overwhelmingly toward $$r$$, -so $$q_+^\star\approx0$$. - -Thus, the fixed point $$s^\star$$ simplifies to +$$s^\star$$ must once again grow as required by the Reinvestors' +higher $$r$$, so $$s^\star$$ was not a fixed point at all. Similarly, +at some point everyone who wants to stake should have staked. If we +judge the quarterly fluxes due to the issuance of new humans and the +burn rate of legacy humans to be small and/or likely to take over +existing businesses, additional validators count overwhelmingly toward +$$r$$, so $$q_+^\star\approx0$$. Thus, the fixed point $$s^\star$$ +simplifies to + $$\displaystyle s^\star = r\frac{\alpha + f}{\alpha + rf} $$ @@ -353,15 +353,17 @@ intuitively it seems unlikely, so for now we move on, assuming uniqueness. Stability requires that small perturbations shrink; -$$\frac{\partial\dot{x}}{\partial x}\big|^<0$$, where -$$\frac{d}{d\xi}|^\star$$ denotes a derivative with respect to $$\xi$$ -at the fixed point $$\xi^\star$$. The full stability condition is +$$\left.\frac{\partial\dot{x}}{\partial x}\right|^\star<0$$, where +$$\left.\frac{d}{d\xi}\right|^\star$$ denotes a derivative with +respect to $$\xi$$ at the fixed point $$\xi^\star$$. The full +stability condition is $$\displaystyle -1 + \frac{r^\star}{\alpha^\star/f^\star} > -\frac{d\ln{r}}{d\ln s}\big|^\star + -(1 - \frac{r^\star}{s^\star}) -\frac{d\ln(\alpha/f)}{d\ln s}\big|^\star +1 + \left.\frac{r^\star}{\alpha^\star/f^\star} +~~ > ~~ +\frac{\partial\,\log r}{\partial,\log s}\right|^\star + +\left.(1 - \frac{r^\star}{s^\star}) +\frac{\partial\,\log(\alpha/f)}{\partial\,\log s}\right|^\star $$ Where presently, we estimate the dimensionless quantity @@ -443,8 +445,9 @@ Assuming $$r,f$$ are (unknown) implicit funcions of $$\alpha$$ the positive derivative condition becomes after some manipulation $$\displaystyle -1 < \frac{\partia\ln r}{\partial\ln\alpha}\big|^\star\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} -+ \frac{\partial\ln f}{\partial\ln\alpha}\big|^\star +1 ~~ < ~~ \left.\frac{\partial\,\log r}{\partial\,\log\alpha}\right|^\star +\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} ++ \left.\frac{\partial\,\log f}{\partial\,\log\alpha}\right|^\star $$ This gives us a condition expressing how the macroeconomic effects of From 30a20ad2a50635cfb54698dcf68dd7fff4ee3be7 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 06:38:18 -0500 Subject: [PATCH 10/70] fiixing moar typos... --- _posts/2024-12-01-issuance-fundamentals.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 5bd1ae0..4fa831a 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -368,7 +368,7 @@ $$ Where presently, we estimate the dimensionless quantity $$\alpha/f\approx5$$ while its derivative depends on the slope of the -yield curve $$y$$ via $$\alpha/f=ys/f-b(1-s)=[I/(F+B)-b](1-s)$$. +yield curve $$y$$ via $$\alpha/f=ys/f-b(1-s)=\left(I/(F+B)-b\right)(1-s)$$. *If* the derivatives are all small in comparison (and there is no a priori gaurantee of this), then we can say something useful. From bdda21a76d131b69325f18a7a5cb5f3a4c2aacf8 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 06:39:43 -0500 Subject: [PATCH 11/70] fiixing even moar typos... --- _posts/2024-12-01-issuance-fundamentals.md | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 4fa831a..89a54e3 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -437,17 +437,18 @@ which fears of runaway inflation will drive $$r\to1$$ and therefore $$s^\star\to1$$. How does our model account for such effects? The derivative of staking fraction by inflation near equilibrium -$$\frac{\partial x^\star}{\partial\alpha}\big|^\star$$ shows the response to +$$\left.\frac{\partial x^\star}{\partial\alpha}\right|^\star$$ shows the response to small endogenous changes, but not to a market crash, or protocol hardfork. The condition that gently rising inflation causes an *increase* in staking is that this derivative should be positive. Assuming $$r,f$$ are (unknown) implicit funcions of $$\alpha$$ the positive derivative condition becomes after some manipulation -$$\displaystyle +$$ +\displaystyle 1 ~~ < ~~ \left.\frac{\partial\,\log r}{\partial\,\log\alpha}\right|^\star -\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} -+ \left.\frac{\partial\,\log f}{\partial\,\log\alpha}\right|^\star +\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} + +\left.\frac{\partial\,\log f}{\partial\,\log\alpha}\right|^\star $$ This gives us a condition expressing how the macroeconomic effects of From a9dc814fc38eb74f62ec42fbb2cb3a31f2dff96e Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 06:47:21 -0500 Subject: [PATCH 12/70] fixing rendering errs --- _posts/2024-12-01-issuance-fundamentals.md | 23 +++++++++++----------- 1 file changed, 12 insertions(+), 11 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 89a54e3..6bffbcf 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -153,15 +153,15 @@ and thus inflation. Our model offers a perspective here as well. In our model the net effect of inflation on staking fraction at equilibrium is reflected by taking the derivative -$$0<\frac{ds^\star}{d\alpha}\big|_{\alpha^\star}$$ assuming $$r,f$$ are +$$0<\left.\frac{ds^\star}{d\alpha}\right|^\star$$ assuming $$r,f$$ are implicit functions of $$\alpha$$. That is, the necessary condition for inflation to push the market equilibrium $$s^\star$$ itself into runaway staking is (see below for explanation): $$\displaystyle -1 < \left.\frac{\partial\ln r}{\partial\ln\alpha}\right|^\star +1 < \left.\frac{\partial\log r}{\partial\log\alpha}\right|^\star \cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} + -\left.\frac{\partial\ln f}{\partial\ln\alpha}\right|^\star +\left.\frac{\partial\log f}{\partial\log\alpha}\right|^\star $$ We hope that this work can be built upon to focus inflationary @@ -275,7 +275,7 @@ greater range than $$(\ln{A},\alpha)$$. -- FIGURES -- -At present we judge the lack of empiircal data on $$r$$ would obviate +At present we judge the lack of empirical data on $$r$$ would obviate the added precision of a more sophisticated treatment. We will proceed assuming $$\dot{\alpha}\approx0$$ and treating $$\alpha$$ as a parameter. Curious or skeptical readers are encouraged to study the @@ -288,9 +288,9 @@ Returning from our commercial break, recall we have found a supportive role played by inflation in moderating staking fraction. We now wish to study the fixed point $$s^*$$ given by solving $$0=\dot{s}=\frac{\dot{A}}{A}(r-s)+(rf+q_+)(1-s)-q_-s$$ -yielding $$ -s^\star = \frac{r^\star(\alpha^\star + f^\star) + q_+^\star}{ -\alpha^\star + r^\star f^\star - q_-^\star}$$ +yielding +$$s^\star = \frac{r^\star (\alpha^\star + f^\star) + {q_+}^\star}{ +\alpha^\star + r^\star f^\star - {q_-}^\star}$$ We find that if $\dot{A}=0=\alpha$ (no inflation nor deflation) then an interior market equilibrium $$s^\star<1$$ is impossible. We reason @@ -318,7 +318,8 @@ $$r$$, so $$q_+^\star\approx0$$. Thus, the fixed point $$s^\star$$ simplifies to $$\displaystyle -s^\star = r\frac{\alpha + f}{\alpha + rf} +s^\star ~~ = ~~ r^\star ~ +\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} $$ A calculation is illustrative under the current regime of positive @@ -361,9 +362,9 @@ stability condition is $$\displaystyle 1 + \left.\frac{r^\star}{\alpha^\star/f^\star} ~~ > ~~ -\frac{\partial\,\log r}{\partial,\log s}\right|^\star + -\left.(1 - \frac{r^\star}{s^\star}) -\frac{\partial\,\log(\alpha/f)}{\partial\,\log s}\right|^\star +\frac{\partial\log r}{\partial\log s}\right|^\star + +\left(1 - \frac{r^\star}{s^\star}\right) +\left.\frac{\partial\log(\alpha/f)}{\partial\log s}\right|^\star $$ Where presently, we estimate the dimensionless quantity From 791fde8cbb0b2ca8389fc9c249787de83c5a3820 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 06:51:21 -0500 Subject: [PATCH 13/70] typooooooos... --- _posts/2024-12-01-issuance-fundamentals.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-01-issuance-fundamentals.md index 6bffbcf..cca2186 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-01-issuance-fundamentals.md @@ -326,7 +326,7 @@ A calculation is illustrative under the current regime of positive inflation. If inflation dominates fees, $$\alpha\gg f$$ then $$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f -\approx .001\lessim .005\approx\alpha$$ so to within 10\% error above, +\approx .001<.005\approx\alpha$$ so to within 10\% error above, $$s^\star\approx r$$ over the range of $$r$$ inferred from Lido yield rate. @@ -362,9 +362,9 @@ stability condition is $$\displaystyle 1 + \left.\frac{r^\star}{\alpha^\star/f^\star} ~~ > ~~ -\frac{\partial\log r}{\partial\log s}\right|^\star + +\frac{\partial\log\ r}{\partial\log\ s}\right|^\star + \left(1 - \frac{r^\star}{s^\star}\right) -\left.\frac{\partial\log(\alpha/f)}{\partial\log s}\right|^\star +\left.\frac{\partial\log(\alpha/f)}{\partial\log\ s}\right|^\star $$ Where presently, we estimate the dimensionless quantity @@ -415,7 +415,7 @@ looked at the fixed point $$s^\star=r(\alpha+f)/(\alpha+rf)$$, and note it reflects a role for inflation more nuanced than has usually been communicated. In fact, we want to emphasize that deflation or zero growth could actually be quite dangerous; for instance when -$$\alpha^*<0,~|\alpha^*|1$$; and there is no +$$\alpha^\star<0,~|\alpha^\star|1$$; and there is no interior fixed point. Why does this happen, though? The reinvestment of issuance rewards From 6f670e428aabb72d6ff011e2f0e05356539e3849 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 07:16:22 -0500 Subject: [PATCH 14/70] added images and correct header --- ...md => 2024-12-05-issuance-fundamentals.md} | 53 ++++++++++-------- .../YCharts-alpha.jpg | Bin 0 -> 101899 bytes .../YCharts-x.jpg | Bin 0 -> 429537 bytes 3 files changed, 29 insertions(+), 24 deletions(-) rename _posts/{2024-12-01-issuance-fundamentals.md => 2024-12-05-issuance-fundamentals.md} (93%) create mode 100644 assetsPosts/2024-12-05-issuance-fundamentals/YCharts-alpha.jpg create mode 100644 assetsPosts/2024-12-05-issuance-fundamentals/YCharts-x.jpg diff --git a/_posts/2024-12-01-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md similarity index 93% rename from _posts/2024-12-01-issuance-fundamentals.md rename to _posts/2024-12-05-issuance-fundamentals.md index cca2186..984c905 100644 --- a/_posts/2024-12-01-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -1,22 +1,25 @@ -# Some danger in reducing Issuance to avoid Runaway Staking. - -## Acknowledgements - -This is a blog post on Runaway Staking in a series summarizing our -research funded by EF GRANT INFO. The author, Eric Downes, is -grateful for useful discussions with Eric Siu, Andrew Sudbury, and the -the 20 Squares team; especially Danieli Palombi and Philipp Zahn. +--- +layout: post +title: Some danger in reducing Issuance to avoid Runaway Staking. +author: Eric Downes +categories: [ethereum, issuance] +excerpt: This is a blog post on Runaway Staking in a series summarizing our +research funded by EF GRANT INFO. +usemathjax: true +thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Danieli Palombi and Philipp Zahn. +--- ## The Problem The share of Ether staked by exchanges such as Coinbase and Liquid Staking Providers (LSTs) such as Lido ("centralized" staking services) -continues to grow. This has provoked concerns, first raised by -Ethereum researchers, and which we share, that the future of Ethereum -might involve (1) all of its native asset being staked, such that (2) -the de facto liquid Ether is controlled by a confederation of -centralized un-transparent govrnance. In this blog post we address the -first of these concerns. +continues to grow. This has provoked +[concerns](https://issuance.wtf/), first raised by Ethereum +researchers, and which we share, that the future of Ethereum might +involve (1) all of its native asset being staked, such that (2) the de +facto liquid Ether is controlled by a confederation of centralized +un-transparent govrnance. In this blog post we address the first of +these concerns. We use "stock and flow" differential equation models to study Ethereum macroeconomics, specifically how changing issuance impacts these @@ -125,15 +128,16 @@ macro-economics. Consider this "backfire" scenario, though, in which reductions of inflation decouple $$s^\star$$ from $$r$$. -Issuance is reduced, as per the yield curve $$y(S)=kS^{-1/2}(1+k'S)$$. -The burn rate and transaction fees as a fraction of circulating Ether -do not change appreciably. LSPs adjust their yields to maintain the -current proportion $$r$$ of the yield from issuance plus fees, and -LST-holders maintain their balances because they simply wish to -passively chase a small percentage yield. Solo validators reinvest at -about the same rate they have been, in order to chase priority fees -and MEV despite the reduction in issuance. So, we are pushed into the -higher part of the yield curve, forcing inflation below fees +Issuance is reduced, as per a new yield curve such as the proposed +$$y(S)=k\cdot S^{-1/2}\cdot(1+k'S)^{-1}$$. The burn rate and +transaction fees as a fraction of circulating Ether do not change +appreciably. LSPs adjust their yields to maintain the current +proportion $$r$$ of the yield from issuance plus fees, and LST-holders +maintain their balances because they simply wish to passively chase a +small percentage yield. Solo validators reinvest at about the same +rate they have been, in order to chase priority fees and MEV despite +the reduction in issuance. So, we are pushed into the higher part of +the yield curve, forcing inflation below fees $$\alpha^\star/f^\star\lesssim10^{-2}$$. Inflation hawks might even have considered this a desirable scenario before reading this post! After (un)staking queue transients die out, staking fraction would @@ -273,7 +277,8 @@ variable. How dangerous is this? For a sanity-check, a quick look at YCharts since The Merge shows that $$s,\dot{s}$$ vary over a much greater range than $$(\ln{A},\alpha)$$. --- FIGURES -- +![The staking fraction from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-x.jpg) +![The inflation rate from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-alpha.jpg) At present we judge the lack of empirical data on $$r$$ would obviate the added precision of a more sophisticated treatment. 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z?|p`tw{q6Q(p}<;+@IF(iC0N6|B1~Nc);1j@}p%nH`6mrwSs{F-3I!=O&QgW?W&f_ zg*vIBQpCM#8d%eF7qE|Y*mjLMzNR+(cG{ET(NKrfW3MQVtogrn?nVPED#z5fsb7ow zB6~Tqk~^NLFB#f}f*CsQ^q1Z(+OwPeX^QTKq5h=d+B)=W&Ss#F<_qQf-f;((f5YBY zu#w;RUgid7M@RmHZ+_MvtdDr#Xp4Ut>j@al`-61_>gdZYbM%jjA*Cd%cx1*iyJR5R zd+*K2=;G+mhcxJH$dWf9VQbHW?C|^(kZUoRE4~kFT5)#w)r^OCh$Qx2 z1CO#CZo+czN#N4Zq5gMBw9j3@^A1jr<2n4nY7b~Y+U}~I zeGhiW)^GdP=)~WR`lZK~z0cE%w+Ej5V%60@O#kd0lo=N~0L5_zn?|w_lfj+Iw8izwaX*W zGsNB5d`H#Qq2*;R@4hGOo=tR9-k#|ke}+;tYG^p{B}AizkDj)Znkr^4B0!xcj%w%4 z1$gAxyu6}!x6Vt}*8q3TXoV3D(3Xrue#HoMJs2CSEiaq$3EH?hD)Fa3lTF`AGq}jE ze@&7p>(Gfd&@65+Jb=!XG;mneu3fI?+OWEzcXt3DXRyNsMQ=MTtkrbe_Hrx<1-$1kZZGiBy#DPAr zv3?E%dBt8S^3Kk=>5x_?Jl3mqP5)XXNoldc3Wzp?GU#WpuhM2~;iHmmzUq;w3-!DH zjM5r}YHs1G!D7q&~Q@M+G9x|Bo6 zTVW2HRKy}S&%AFbho42cw=bw}6WWH>_PLjPI#b-Xyz4*MRa{^7)Wq=OzYBfn{qM8} qCjaaIF&9jkz<~UJ{`mivfckIW|19GF>9(N$+y6g{_ Date: Thu, 5 Dec 2024 07:17:44 -0500 Subject: [PATCH 15/70] fixing header line err --- _posts/2024-12-05-issuance-fundamentals.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md index 984c905..eed6bba 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -3,8 +3,7 @@ layout: post title: Some danger in reducing Issuance to avoid Runaway Staking. author: Eric Downes categories: [ethereum, issuance] -excerpt: This is a blog post on Runaway Staking in a series summarizing our -research funded by EF GRANT INFO. +excerpt: This is a blog post on Runaway Staking in a series summarizing our research funded by EF GRANT INFO. usemathjax: true thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Danieli Palombi and Philipp Zahn. --- From 7af70dd4a1b297eeca7168cbc8564aff53b40a44 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 07:22:03 -0500 Subject: [PATCH 16/70] fixing lack of emdash using colon --- _posts/2024-12-05-issuance-fundamentals.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md index eed6bba..9b557d7 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -207,10 +207,10 @@ $$ $$ Where all quantities are assesed quarterly: -* $$I$$ -- Total Issuance -* $$R$$ -- Total Reinvestment of Validator Yields -* $$Q_\pm$$ -- Total (Un)Staking Queue Flows -* $$B$$ -- Total Base Fee, aka Burn Rate. +* $$I$$: Total Issuance in a certain quarter +* $$R$$: Total Reinvestment of Validator Yields, in that quarter +* $$Q_\pm$$: Total (Un)Staking Queue Flows, in that quarter +* $$B$$: Total Base Fee (burn) assed, in that quarter. Transaction fees $$F+B$$ are split into the base fee $$B$$ which is burned, and priority fees plus MEV $$F$$ which go to validators via From 706d73b04a17f9c655c99d66bdd11ceb99b4d9d8 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 07:23:16 -0500 Subject: [PATCH 17/70] rm any use of $ to mean dollars --- _posts/2024-12-05-issuance-fundamentals.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md index 9b557d7..0d346ee 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -219,7 +219,7 @@ ae interested in the limit $$C\to0$$ and pull this dependence out by defining $$f$$ such that $$fC = F+B$$. The fraction $$f$$ need not be a constant, but whatever values it takes, $$f$$ must be a fraction: $$0 Date: Thu, 5 Dec 2024 07:35:06 -0500 Subject: [PATCH 18/70] fixing minor typo and adding note on stability and inflation-led runaway staking --- _posts/2024-12-05-issuance-fundamentals.md | 15 ++++++++++----- 1 file changed, 10 insertions(+), 5 deletions(-) diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md index 0d346ee..b570ae4 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -219,9 +219,9 @@ ae interested in the limit $$C\to0$$ and pull this dependence out by defining $$f$$ such that $$fC = F+B$$. The fraction $$f$$ need not be a constant, but whatever values it takes, $$f$$ must be a fraction: $$0 Date: Thu, 5 Dec 2024 07:39:17 -0500 Subject: [PATCH 19/70] good enough --- _posts/2024-12-05-issuance-fundamentals.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md index b570ae4..7717156 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -452,7 +452,8 @@ positive derivative condition becomes after some manipulation $$ \displaystyle 1 ~~ < ~~ \left.\frac{\partial\log\ r}{\partial\log\ \alpha}\right|^\star -\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} + +\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} +~~ + ~~ \left.\frac{\partial\log\ f}{\partial\log\ \alpha}\right|^\star $$ From 5b7f848f26cee40149209abe7c2c0a490233ce5b Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 07:45:53 -0500 Subject: [PATCH 20/70] attempt to fix bad kerning --- _posts/2024-12-05-issuance-fundamentals.md | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md index 7717156..a2af29f 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -162,9 +162,10 @@ for inflation to push the market equilibrium $$s^\star$$ itself into runaway staking is (see below for explanation): $$\displaystyle -1 < \left.\frac{\partial\log r}{\partial\log\alpha}\right|^\star -\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} + -\left.\frac{\partial\log f}{\partial\log\alpha}\right|^\star +1 ~~ < ~~ +\left.\frac{\partial\log\ r}{\partial\log\ \alpha}\right|^\star +\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} ~~ + ~~ +\left.\frac{\partial\log\ f}{\partial\log\ \alpha}\right|^\star $$ We hope that this work can be built upon to focus inflationary @@ -176,7 +177,7 @@ reinvestment sentiments. A semi-empirical measurement of correlations between transaction fees and inflation, using our model or similar to control for staking queue flows, etc. should also shed light on wether the above condition is satisfied or not. - + ## Modelling Staking Now some more details! @@ -388,7 +389,7 @@ $$\alpha<-f<0$$ fixed point appears interior unstable, predicting either runaway staking or a catastrophic loss of staking, depending on initial considitions. Our interpretation of the latter behavior is that if raw Ether is appreciating in value so quickly, and there has -been insufficien interest in staking thusfar, then why bother staking at +been insufficient interest in staking thusfar, then why bother staking at all? But we must emphasize that what instability *really* means is that "market externalities take over", rather than supporting any specific trajectory due to endogenous dynamics. From 23e6ac63f81fa25eefb2d09ae9cc4a93b75ce980 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 07:52:07 -0500 Subject: [PATCH 21/70] should publish later --- _posts/2024-12-05-gov-centralization.md | 18 ------------------ 1 file changed, 18 deletions(-) delete mode 100644 _posts/2024-12-05-gov-centralization.md diff --git a/_posts/2024-12-05-gov-centralization.md b/_posts/2024-12-05-gov-centralization.md deleted file mode 100644 index 3943319..0000000 --- a/_posts/2024-12-05-gov-centralization.md +++ /dev/null @@ -1,18 +0,0 @@ -Governance Centralization. - - - -The share of Ether staked by Coinbase and -LSTs such as Lido ("centralized" staking services) continues to grow -quite considerably -. For the share of Ether staked by -individually-operated validators to outgrow centralized staking -services in the long term, one of two things must happen. Either there -is a greater continuous influx of new solo stakers relative to -centralized services, or solo stakers reinvest relatively more of -their profits, en masse. At present, due to economies of scale, solo -staking has at least as high a cost floor per node as centralized -staking, and less access to new capital flows. So reducing issuance -requires solo stakers to essentially subsidize the protocol relative -to the yield available by validating for an LST. - From a72787d42eda7c0bbfe72074cac4170f9e41d6d1 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 5 Dec 2024 07:57:09 -0500 Subject: [PATCH 22/70] rm notes --- _posts/background.md | 68 -------------------------------------------- 1 file changed, 68 deletions(-) delete mode 100644 _posts/background.md diff --git a/_posts/background.md b/_posts/background.md deleted file mode 100644 index 29728a1..0000000 --- a/_posts/background.md +++ /dev/null @@ -1,68 +0,0 @@ - -# Notes - -- There is a problem: Centralized stakers and major LSTs could replace - Ethereum's decentralized governance with a oligopoly of staking - interests. - -- These concerns have been felt for a few years as people watched the - relatve [Lido stake grow](LINK) - -- As the percentage of stake increases, so does issuance, and if this - [on-paper inflation]LINK investopedia) translates into real price - inflation, the real rate of validator yields will decrease. - -- Changing the yield curve [was proposed](LINK) by some ethereum - researchers as a potential solution to both of these interlocking - problems. - -- The arguments are laid out [in an introductory talk here](LINK), in - more technical form [here](LINK), and in general resources on this - issue have been collected [here](LINK to issuance.wtf) - -- In this post we derive, using [dynamical systems](link to strogaz) - and very mild economic assumptions, conditions under which changing - the the yield curve would solve the "100% Staking Issue" and the - "Governance Centralization" issue, vs. exaccerbate it further. - -- Unfortunately, we cannot support changing the yield curve at this - time. We find the dynamics to be mre nuanced than the proposals - have suggested, and express concern. - -- We cannot say for certain what the parameters will be when measured, - but we see reason for concern. Under certain believable parameter - ranges, changing the yield curve could have no meaninful impact on - centralizaton, and would actally exacerbate the 100% staking - situation. - -- There are other parameter ranges that are more dangerous, but - thankfully these can be easily designed against. - -- We close with some thoughts on exploring the design space of - possible dynamical behaviors, and propose conditions for evaluating - potential long and short-term solutions. - -- governance centralization tldr. If a reduction in issuance reduces - the relative staking ratio of centralized to decentralized staking, - this occurs because solo stakers are less elastic than centralized - stakers like Coinbase and LSTs. Due to economy of scale, it is hard - to believe that solo stakers have a lower cost floor per validator, - than centralized operations. So, it seems likely that if the growth - of solo stakers exceeds centralized operations, it is because the - solo stakers are deferring profit to temporarily subsidize the - protocol. Unless we can devise some hashcash-lke mechanism to - disproportionately increase costs for centralized stakers this is - probably unsustainable long term. Even as a short term means of - buying time, discussions with validators are strongly recommended to - ensure they have the capital and appetite to do this. - -## Background - -### Ethereum Issuance Curve, Staking, and Liquid Staking - -### Quarterly Averaging and Difference vs. Differential Equations - -### Fixed Points, Stability, Market Equilibria - -## Governance Centralization - From 93de0e8b33f31d9a9a9abeb16c364312e5997b87 Mon Sep 17 00:00:00 2001 From: Eric M Downes Date: Thu, 5 Dec 2024 12:15:08 -0800 Subject: [PATCH 23/70] Update 2024-12-05-issuance-fundamentals.md Co-authored-by: Daniele <45427760+dpl0a@users.noreply.github.com> --- _posts/2024-12-05-issuance-fundamentals.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-05-issuance-fundamentals.md index a2af29f..39b420b 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-05-issuance-fundamentals.md @@ -5,7 +5,7 @@ author: Eric Downes categories: [ethereum, issuance] excerpt: This is a blog post on Runaway Staking in a series summarizing our research funded by EF GRANT INFO. usemathjax: true -thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Danieli Palombi and Philipp Zahn. +thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Daniele Palombi and Philipp Zahn. --- ## The Problem From d3deff1f733d15671ee74fe04cbce7b893ea216b Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 16 Dec 2024 07:32:30 -0500 Subject: [PATCH 24/70] updating dates for likely release --- _posts/2024-12-19-issuance-basics.md | 110 ++++++++++ ...md => 2024-12-20-issuance-fundamentals.md} | 205 +++++++++++++----- .../staking-fixpoint.png | Bin 0 -> 59503 bytes 3 files changed, 261 insertions(+), 54 deletions(-) create mode 100644 _posts/2024-12-19-issuance-basics.md rename _posts/{2024-12-05-issuance-fundamentals.md => 2024-12-20-issuance-fundamentals.md} (76%) create mode 100644 assetsPosts/2024-12-05-issuance-fundamentals/staking-fixpoint.png diff --git a/_posts/2024-12-19-issuance-basics.md b/_posts/2024-12-19-issuance-basics.md new file mode 100644 index 0000000..b173287 --- /dev/null +++ b/_posts/2024-12-19-issuance-basics.md @@ -0,0 +1,110 @@ +±s±--- +layout: post +title: Basics of applying dynamics to Ethereum Issuance +author: Eric Downes +categories: [ethereum, issuance] +excerpt: This is a pedagogical blog post on differential equtions for ethereum in a series summarizing our research funded by EF GRANT INFO. +usemathjax: true +thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Danieli Palombi and Philipp Zahn. +--- + +## The Problem + +The share of Ether staked by exchanges such as Coinbase and Liquid +Staking Providers (LSTs) such as Lido ("centralized" staking services) +continues to grow. This has provoked +[concerns](https://issuance.wtf/), first raised by Ethereum +researchers, and which we share, that the future of Ethereum might +involve (1) all of its native asset being staked, such that (2) the de +facto liquid Ether is controlled by a confederation of centralized +un-transparent govrnance. In this blog post we setup some basics for +readers who can read python but are unfamiliar with differential +equations. + +We use "stock and flow" differential equation models to study Ethereum +macroeconomics, specifically how changing issuance impacts these +questions. We will publish several blog posts on this topic: + +0. (This post) Basics of applying Dynamics to Ethereum +1. [Danger in reducing issuance to avoid Runaway Staking](./2024-12-05-issuance-fundamentals.md) +2. Will reducing issuance avoid Governance Centralization? +3. Other Levers besides Issuance, and a means of evaluating levers. +4. Some tools to help the community study and resolve policy debates. + +We wrapped the `scipy.odesim` tool and provided some examples +[here](https://github.com/20squares/issuance-fundamentals) for +researchers wishing to simulate their own models. We will return to +this tool in greater depth later, and demonstrate some basic usage +below. + +# TLDR for the impatient + +* Under the present yield curve, assuming weak Ethereum supply should eventually + stabilize: nearly all Ether is staked with zero inflation; issuance + matches the burn. + +-- include styuff from notion.so + + +# Why Diff Eq + +On the face of it, differential equations, such as the reader may have +suffered through in highschool or college does not seem up to the task +of saying anything useful about Ethereum. +-- Everything in Ethereum is discrete, not continuous +-- Diferential Equations are cookbook recipes solving very limited solutions. + +As often, everything is a matter of perspective. Let's illustrate by +modeling the burn rate from [EIP +1559](https://ethereum.github.io/abm1559/notebooks/eip1559.html). +This EIP split Ethereum transaction fees into base and priority fees, +where the base fee is destroyed forever, and determined dynamically by +the amount of block congestion, using gas price as proxy. The idea is +that traders can still compete for priority in blocks but they must +pay a deflationary penalty when "excessive" competition for +blockspace occurs. + +$$\displaystyle +\beta_{t+\tau} = \left(1+\frac{1}{8} +\left(\frac{g_{t}}{g^\star}-1\right)\right)\beta_t +$$ + +Where: +- $\tau$ is the blocktime between successive blocks $$\approx12$$s/block +- $\beta_t$$ is the burn rate in ETH/block at time $t$ +- $g_t$$ is the gas price in Ether at time $$t$$ +- $g^\star$ is the target gas price $$15\times10^6$$ETH/block. + +The above equation is a "difference equation", solved by + + + +But, if we are +studying policy questions like issuance, we are looking at much longer +times, such as quarters. + + +# A Toy Model + +Let’s start with two assumptions about Ethereum + +1. All Ether ever issued can be split into “Staked Ether” $S$ “Unstaked Ether” $U$ + + $(1)~~~~~E = S + U$ + + $E$ — Total Ether in existence + + $S$ — Ether staked as per the Shanghai hard-fork + + $U$ — Everything else; burned Ether, liquid Ether, Ether locked in smart contracts, etc. + + $U=C+B$ circulating and burned + + +1. Ethereum is Proof-of-Stake with a protocol-level yield curve. + + The existing yield curve, post-Shanghai hardfork, $y_0(S)$ determines issuance of new Ether, and EIP-1559 determines Ether burned “lost” $\ell$ from the system. We count burned Ether as part of total Ether $E$ so: + + $(2)~~~~~\dot{E}:=\frac{dE}{dt}=y(S)S$ + + These staking rewards occur directly to $U$ and may be reinvested into $S$ at a reinvestment rate $r$. diff --git a/_posts/2024-12-05-issuance-fundamentals.md b/_posts/2024-12-20-issuance-fundamentals.md similarity index 76% rename from _posts/2024-12-05-issuance-fundamentals.md rename to _posts/2024-12-20-issuance-fundamentals.md index 39b420b..ff2a12f 100644 --- a/_posts/2024-12-05-issuance-fundamentals.md +++ b/_posts/2024-12-20-issuance-fundamentals.md @@ -3,9 +3,9 @@ layout: post title: Some danger in reducing Issuance to avoid Runaway Staking. author: Eric Downes categories: [ethereum, issuance] -excerpt: This is a blog post on Runaway Staking in a series summarizing our research funded by EF GRANT INFO. +excerpt: This is a blog post on Runaway Staking in a series summarizing our issuance research for the Ethereum Foundation. usemathjax: true -thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Daniele Palombi and Philipp Zahn. +thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Daniele Palombi and Philipp Zahn. We are grateful to the EF for funding this research. --- ## The Problem @@ -16,8 +16,8 @@ continues to grow. This has provoked [concerns](https://issuance.wtf/), first raised by Ethereum researchers, and which we share, that the future of Ethereum might involve (1) all of its native asset being staked, such that (2) the de -facto liquid Ether is controlled by a confederation of centralized -un-transparent govrnance. In this blog post we address the first of +facto liquid Ether is controlled by a confederation of centralized entities wiith +less transparent govrnance. In this blog post we address the first of these concerns. We use "stock and flow" differential equation models to study Ethereum @@ -28,32 +28,55 @@ questions. We will publish several blog posts on this topic: 1. (This post) Will reducing issuance avoid Runaway Staking? 2. Will reducing issuance avoid Governance Centralization? 3. Other Levers besides Issuance, and a means of evaluating levers. -4. Some tools to help the community study and resolve policy debates. +4. Tools to study and resolve policy debates. + +Even if runaway staking does not directly concern you, this post is +essential to understand our view of the interaction between inflation +and staking, which is in tension with other research + ## TLDR -### For the extraordinarily impatient reader. +### For the extraordinarily impatient reader, our conclusions. -* The staking reinvestment ratio is central to understanding the - quantitative future of Ethereum macroeconomics. We believe it can - be well-estimated with onchain data. +* We identify a "low inflation; even lower fees" regime (LI;ELF) in + which there the amount of Ether staked is stable and (potentially) + less than 100%; $$s^\star<1$$. -* Under weak inflation, runaway staking may occur. If it does *not*, - this is because - -- (1) inflation is low enough, that concerns over inflation - do not dominate the reinvestment of profits by staking businesses, - but simultaneously - -- (2) inflation is high enough to numerically dominate priority fees - and MEV, as a fraction of circulating Ether. +* Staking rewards do *not* simply enter circulation. So long as LSTs + exist some fraction of these are necessarily "reinvested" in + staking at a non-zero ratio $$r$$. + -- When LI;ELF persists, the fraction of Ether staked approaches the + reinvestment ratio from above $$s\to s^\star= r^\star +\epsilon$$. + -- Reinvestment can be estimated with onchain data, and its + sensitivity to inflation measured with + validator surveys. -* Outside of the "low inflation, lower fees" regime, convergence to a - reasonable staking future is not possible. - -- Under strong deflation, Ethereum faces a choice between zero staking - or runaway staking - -- Under no growth or weak deflation, runaway staking is inevitable. +* Outside of LI;ELF, convergence to a reasonable staking future is not + possible. + -- Under high deflation, Ethereum faces a choice between zero staking + $$s^\star\to0$$ or runaway staking $$s^\star\to1$$. + -- Under no growth or low deflation, runaway staking *is inevitable*. -* Given the above, we emphasize that changing the yield curve could - backfire, *raising* long term staking fraction. +* Within LI;ELF, runaway staking may still occur. If it does *not*, + this is because + -- (1) inflation is low enough, that concerns over inflation + do not dominate the reinvestment of profits by staking businesses + at equilibrium, $$d_\alpha{r}|^\star\leq0$$, but simultaneously + -- (2) inflation is high enough at equilibrium to numerically + dominate priority fees and MEV, as a fraction of circulating + Ether; $$\alpha^\star\gg f^\star$$. + +* The medium term future of ethereum we see is one in which inflation + $$\alpha$$, while higher than that prefered by inflation hawks, + plays a *positive* role to mediate staking fraction and + centralization. In long $$t\to\infty$$ term, $$\alpha\to0$$ forcing + $$s\to1$$. We will revisit the consequences of this for Ethereum in + a future blog post. + +* Given all the above, we advise caution. Intervening to change the + yield curve seems dangerously capable of exaccerbating the very + problems we seek to avoid. ### For the moderately impatient reader. @@ -74,45 +97,119 @@ That is, an increase in staking fraction can be driven by more people staking, and/or it can be driven by a reduction of the inflation rate. The latter can be acheived in principle by a reduction of issuance relative to the base fee "burn rate". Because of this quotient rule -tradeoff, low-but-positive inflation actually plays a positive -infrastructural role in moderating staking fraction. This can be seen -in the market equilibrium staking fraction $$s^\star$$, shown here and -derived below. +tradeoff, low-but-positive inflation actually plays a positive almost +"infrastructure" role in moderating staking fraction. + +This positive role for inflation can be seen in the contours of the +market equilibrium staking fraction $$s^\star$$ corresponding to +$$\dot{s}=0$$, shown in [figure +1](../assetsPosts/2024-12-05-issuance-fundamentals/staking-fixpoint.png). +The equation graphed is as follows: $$\displaystyle s^\star = r^\star\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} $$ -Here $$r^\star$$ etc. means the function $$r(s,\alpha,\ldots)$$ at the -equilibrium coordinates $$(s^\star,\alpha^\star,\ldots)$$. Speaking -of which, the fraction $$0\leq r\leq 1$$ is the ratio of profits -reinvested quarterly by validators, $$0\leq f\leq 1$$ is the fraction -of unstaked ETH spent on transaction fees (base and priority) -quarterly, and as above $$\alpha$$ is inflation. The two extremes are -$$\alpha^\star\ll f^\star$$ fees dominate and $$\alpha^\star\gg -f^\star$$ inflation dominates. In the former, staking fraction is -driven closer to 1, while in the latter, staking is driven to match -reinvestment $$s^\star\approx r^\star$$. - -#### Good Scenario - -Under current market conditions, low inflation and lower fees, the -long-term equilibrium staking fraction approaches the average ratio -$$r$$ at which validators reinvest their staking rewards. When there -are different types of validators, $$r$$ is the average of $$r_i$$ for -each validator type $$i$$, weighted by the amount of Ether each -stakes. For LSPs $$r_{LST}$$ is bounded below by the ratio of token -yield to total yield, and we can use this to roughly estimate some -full $$r$$ values. +The fraction $$0\leq r\leq 1$$ is the ratio of profits reinvested +quarterly by validators, $$0\leq f\leq 1$$ is the fraction of unstaked +ETH spent on transaction fees (base and priority) quarterly, and as +above $$\alpha$$ is inflation. Here $$r^\star$$ etc. means the +function $$r(s,\alpha,\ldots)$$ at the equilibrium coordinates +$$(s^\star,\alpha^\star,\ldots)$$. To find the equilibrium values +$$(\alpha^\star/f^\star,\,r^\star)$$ necessary to acheive a desired +staking fraction $$x^\star$$, simply pick a colored contour in the +figure: these are the values of constant $$x^\star$$. For every point +on this curve, the equilibrium inflation:fee ratio +$$\alpha^\star/f^\star$$ is the x-coordinate, and the equilibrium +reinvestment ratio $$r^\star$$ is the y-value. + +The two extremes are $$\alpha^\star\ll f^\star$$ fees dominate and +$$\alpha^\star\gg f^\star$$ inflation dominates. In the former, +staking fraction becomes insensitive to reinvestment, raising +$$x^\star$$. Equivalntly, for any value of positive inflation, +$$r^\star$$ is a lower bound for the equilibrium staking fraction we +should expect. Approximate present values from YCharts are very roughly $$f\approx.001$$/year, $$\alpha\approx.005$$/year, $$r\in(.5,.75)$$. -So if current market conditions (low inflation, lower fees) were to -persist at long times, we expect the staking ratio converges to no -more than +10\% of the the reinvestment ratio. Good news: if issuance -is reduced such that inflation is reduced by half, while fees -etc. maintain, equilibrium is still around $$r$$ + 15\%, so -$$s^\star\in(.58,.8)$$. +How these transient values $$(\alpha/f,\,r)$$ relate to their +equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ depends on +some considerations. + +#### Long term staking ratio dominated by LSTs. + +We will return to reinvestment in a future post; here is a sketch. +When there are different types of validators, $$r$$ is the average of +$$r_i$$ for each validator type $$i$$, weighted by the amount of Ether +each stakes. For LSPs $$r_{LST}$$ is bounded below by the ratio of +token yield to total yield, and we can use this to roughly estimate +some full $$r$$ values. Should conditions persist $$r_{now}\leq +r_{stETH}\leq r^\star$$; at long times, the stakers who reinvest more +of their revenue in their staking business will acheieve a larger +share. Essentially at long times, $$r^\star$$ should approach the +highest sustainable reinvestment ratio that the market can bear, which +likely corresponds to the highest sustainable LST yield. + +#### Long term inflation looks plausibly stable under this yield curve + +Quarterly Inflation is related to issuance by $$\alpha = ys-B/A$$, +where $$B$$ is the amount of Ether burned in a quarter, and $$y$$ is +the total issuance in the same quarter. More work will be required to +determine how we should expect qaurterly burn to evolve. However so +long as $$\dot{B}/A>-\alpha^2$$, inflation at long times decays slowly +according to the current yield curve $$0\leq \alpha^\star\leq +y(S)/2$$. + +So long as the quarterly burn +does not respond to supply changes more intensely than the inflation +rate $$|\dot{B}/A|\lessim\alpha^2$$ and + + +$$|dB/dA|\lessim\alpha +$$\left|\frac{\dot{B}}{A}\right|\lessim \alpha^2$$, a slowly decaying +$$\alpha^\star$$ obtains and $$\alpha^\star=. + +leads to an slowly decreasing +inflation rate depending on $$\gamma:=y(1+\frac{d\log\ y}{d\log\ S}$$, +which is $$y^\star/2$$ under the curent curve. + +$$\displaystyle +0=-\alpha^\star+\gamma^\star\alpha^\star-\frac{\dot{B}}{A} +$$ + + + + +If for instance $$dB/dA$$ is constant, + +How does this quantity behave in +the long term? Suppose that the ratio $$B/A$$ were a constant, then + +after some calculus (see below), this leads to a stable +value $$\alpha^\star$$ + + +Maintaining LI;ELF while changing Issuance + +Under current market conditions, low inflation and even lower fees +(LI;ELF), the long-term equilibrium staking fraction approaches the +average ratio $$r$$ at which validators reinvest their staking +rewards. This makes staking fraction responsive to economic +interventions that affect the reinvestment decisions of participants. + +So if LI;ELF conditions persist at long times, we expect the staking +ratio converges to no more than +10\% of the the equilibrium +reinvestment ratio. Good news: if issuance is reduced such that +inflation is reduced by half, while fees etc. maintain, equilibrium is +still around $$r$$ + 15\%, so $$s^\star\in(.58,.8)$$, at present values. + +If the yield curve is reduced from $$y_0=\frac{k}{\sqrt{S}}$$ to +$$y'=\frac{y_0}{1+k'S}$$ as proposed in ----------, what is likely to +happen? + + + + Wether this is acceptable, depends on $$r$$. 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zPtNi3^2&9f_U8w6!Sc#JGs*>j36b_8W}{2&W?pWDEFse2iw~p-kph^qRVBS=zQ8u) z6P}=hIK>ZNtcgf**4p{Y Date: Fri, 27 Dec 2024 06:40:09 -0500 Subject: [PATCH 25/70] saving work before switchng to single-post + guide --- _posts/2024-12-20-issuance-fundamentals.md | 245 +++++---------------- 1 file changed, 55 insertions(+), 190 deletions(-) diff --git a/_posts/2024-12-20-issuance-fundamentals.md b/_posts/2024-12-20-issuance-fundamentals.md index ff2a12f..da2234c 100644 --- a/_posts/2024-12-20-issuance-fundamentals.md +++ b/_posts/2024-12-20-issuance-fundamentals.md @@ -12,13 +12,13 @@ thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and The share of Ether staked by exchanges such as Coinbase and Liquid Staking Providers (LSTs) such as Lido ("centralized" staking services) -continues to grow. This has provoked -[concerns](https://issuance.wtf/), first raised by Ethereum -researchers, and which we share, that the future of Ethereum might -involve (1) all of its native asset being staked, such that (2) the de -facto liquid Ether is controlled by a confederation of centralized entities wiith -less transparent govrnance. In this blog post we address the first of -these concerns. +[is considerable](https://dune.com/queries/2394100/3928083) continues +to grow. This has provoked [concerns](https://issuance.wtf/), first +raised by Ethereum researchers, and which we share, that the future of +Ethereum might involve (1) all of its native asset being staked, such +that (2) the de facto currency is controlled by a confederation of +centralized entities wiith less transparent govrnance. In this blog +post we address the first of these concerns. We use "stock and flow" differential equation models to study Ethereum macroeconomics, specifically how changing issuance impacts these @@ -28,11 +28,11 @@ questions. We will publish several blog posts on this topic: 1. (This post) Will reducing issuance avoid Runaway Staking? 2. Will reducing issuance avoid Governance Centralization? 3. Other Levers besides Issuance, and a means of evaluating levers. -4. Tools to study and resolve policy debates. +4. Conclusions, tools to study and resolve policy debates. Even if runaway staking does not directly concern you, this post is essential to understand our view of the interaction between inflation -and staking, which is in tension with other research +and staking, a picture in tension with other research. ## TLDR @@ -47,7 +47,7 @@ and staking, which is in tension with other research exist some fraction of these are necessarily "reinvested" in staking at a non-zero ratio $$r$$. -- When LI;ELF persists, the fraction of Ether staked approaches the - reinvestment ratio from above $$s\to s^\star= r^\star +\epsilon$$. + reinvestment ratio from above. -- Reinvestment can be estimated with onchain data, and its sensitivity to inflation measured with validator surveys. @@ -60,23 +60,23 @@ and staking, which is in tension with other research * Within LI;ELF, runaway staking may still occur. If it does *not*, this is because - -- (1) inflation is low enough, that concerns over inflation + -- (1) inflation is held low enough, that concerns over inflation do not dominate the reinvestment of profits by staking businesses at equilibrium, $$d_\alpha{r}|^\star\leq0$$, but simultaneously - -- (2) inflation is high enough at equilibrium to numerically + -- (2) inflation is held high enough at equilibrium to numerically dominate priority fees and MEV, as a fraction of circulating Ether; $$\alpha^\star\gg f^\star$$. * The medium term future of ethereum we see is one in which inflation $$\alpha$$, while higher than that prefered by inflation hawks, plays a *positive* role to mediate staking fraction and - centralization. In long $$t\to\infty$$ term, $$\alpha\to0$$ forcing + centralization. In long $$t\to\infty$$ term, though $$\alpha\to0$$ forcing $$s\to1$$. We will revisit the consequences of this for Ethereum in a future blog post. -* Given all the above, we advise caution. Intervening to change the - yield curve seems dangerously capable of exaccerbating the very - problems we seek to avoid. +* Given all the above, we advise caution. Intervening to reeduce the + issuance yield curve seems dangerously capable of exaccerbating the + very problems we seek to avoid. ### For the moderately impatient reader. @@ -84,11 +84,12 @@ and staking, which is in tension with other research In some more detail, a sketch of the reasons behind our conclusions. The quantity of most importance to this debate is the staked ETH -fraction $$s$$, currently roughly .33. Staking fraction is calculated -$$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is "circulating" -(unstaked, unburnt) ETH, and $$A=S+C$$ is the total "accessible" -(unburnt) ETH. Differential changes, such as changes in time -$$\frac{ds}{dt}:=\dot{s}$$ are given by the quotient rule +fraction $$s$$, as of Dec 2024 is [roughly +.28](https://dune.com/queries/1933048/3188490). Staking fraction is +calculated $$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is +"circulating" (unstaked, unburnt) ETH, and $$A=S+C$$ is the total +"accessible" (unburnt) ETH. Differential changes, such as changes in +time $$\frac{ds}{dt}:=\dot{s}$$ are given by the quotient rule $$\dot{s}=\dot{S}/A-s\dot{A}/A$$. The quantity $$\alpha = \dot{A}/A$$ is the average on-paper inflation rate (supply expansion APY) averaged on at-least-quarterly timescales. @@ -107,7 +108,7 @@ $$\dot{s}=0$$, shown in [figure The equation graphed is as follows: $$\displaystyle -s^\star = r^\star\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} +s^\star = r^\star\frac{\alpha^\star/f^\star + 1}{\alpha^\star/f^\star + r^\star} $$ The fraction $$0\leq r\leq 1$$ is the ratio of profits reinvested @@ -136,60 +137,7 @@ How these transient values $$(\alpha/f,\,r)$$ relate to their equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ depends on some considerations. -#### Long term staking ratio dominated by LSTs. - -We will return to reinvestment in a future post; here is a sketch. -When there are different types of validators, $$r$$ is the average of -$$r_i$$ for each validator type $$i$$, weighted by the amount of Ether -each stakes. For LSPs $$r_{LST}$$ is bounded below by the ratio of -token yield to total yield, and we can use this to roughly estimate -some full $$r$$ values. Should conditions persist $$r_{now}\leq -r_{stETH}\leq r^\star$$; at long times, the stakers who reinvest more -of their revenue in their staking business will acheieve a larger -share. Essentially at long times, $$r^\star$$ should approach the -highest sustainable reinvestment ratio that the market can bear, which -likely corresponds to the highest sustainable LST yield. - -#### Long term inflation looks plausibly stable under this yield curve - -Quarterly Inflation is related to issuance by $$\alpha = ys-B/A$$, -where $$B$$ is the amount of Ether burned in a quarter, and $$y$$ is -the total issuance in the same quarter. More work will be required to -determine how we should expect qaurterly burn to evolve. However so -long as $$\dot{B}/A>-\alpha^2$$, inflation at long times decays slowly -according to the current yield curve $$0\leq \alpha^\star\leq -y(S)/2$$. - -So long as the quarterly burn -does not respond to supply changes more intensely than the inflation -rate $$|\dot{B}/A|\lessim\alpha^2$$ and - - -$$|dB/dA|\lessim\alpha -$$\left|\frac{\dot{B}}{A}\right|\lessim \alpha^2$$, a slowly decaying -$$\alpha^\star$$ obtains and $$\alpha^\star=. - -leads to an slowly decreasing -inflation rate depending on $$\gamma:=y(1+\frac{d\log\ y}{d\log\ S}$$, -which is $$y^\star/2$$ under the curent curve. - -$$\displaystyle -0=-\alpha^\star+\gamma^\star\alpha^\star-\frac{\dot{B}}{A} -$$ - - - - -If for instance $$dB/dA$$ is constant, - -How does this quantity behave in -the long term? Suppose that the ratio $$B/A$$ were a constant, then - -after some calculus (see below), this leads to a stable -value $$\alpha^\star$$ - - -Maintaining LI;ELF while changing Issuance +### Maintaining LI;ELF while changing Issuance Under current market conditions, low inflation and even lower fees (LI;ELF), the long-term equilibrium staking fraction approaches the @@ -197,9 +145,9 @@ average ratio $$r$$ at which validators reinvest their staking rewards. This makes staking fraction responsive to economic interventions that affect the reinvestment decisions of participants. -So if LI;ELF conditions persist at long times, we expect the staking +So if LI;ELF conditions persist, we expect the staking ratio converges to no more than +10\% of the the equilibrium -reinvestment ratio. Good news: if issuance is reduced such that +reinvestment ratio. Good news: if issuance can be reduced such that inflation is reduced by half, while fees etc. maintain, equilibrium is still around $$r$$ + 15\%, so $$s^\star\in(.58,.8)$$, at present values. @@ -209,42 +157,6 @@ happen? - - -Wether this is acceptable, depends on $$r$$. That is, under current -market conditions, reinvestment ratio is the proxy by which inflation -must act, if it affects long term staking fraction. Thankfully, a -staking business $$i$$ should be able to calculate the reinvestment of -its own profits $$r_i$$. The global $$r$$ is then the market-share -weighted average of all these, a bridge between Ethereum micro- and -macro-economics. - -#### Bad Scenario - -Consider this "backfire" scenario, though, in which reductions of -inflation decouple $$s^\star$$ from $$r$$. - -Issuance is reduced, as per a new yield curve such as the proposed -$$y(S)=k\cdot S^{-1/2}\cdot(1+k'S)^{-1}$$. The burn rate and -transaction fees as a fraction of circulating Ether do not change -appreciably. LSPs adjust their yields to maintain the current -proportion $$r$$ of the yield from issuance plus fees, and LST-holders -maintain their balances because they simply wish to passively chase a -small percentage yield. Solo validators reinvest at about the same -rate they have been, in order to chase priority fees and MEV despite -the reduction in issuance. So, we are pushed into the higher part of -the yield curve, forcing inflation below fees -$$\alpha^\star/f^\star\lesssim10^{-2}$$. Inflation hawks might even -have considered this a desirable scenario before reading this post! -After (un)staking queue transients die out, staking fraction would -equilibrate to $$s^\star\in(.97,.99)$$. - -Essentially, we really need to be sure of what drives validaor -behavior. If enough validators persist in reinvestment beyond where -we would expect them to, Ethereum could become deflationary, and this -is dangerous. We recommend surveys to gauge sentiment and -calculations of the distribution of $$r_i$$. - #### Macroeconomics at the Equilibrium Point Per the arguments of Ethereum researchers, high inflation could lead @@ -281,9 +193,10 @@ Now some more details! ### Splitting Up A Growing Pie -Consider the quantity $$E$$ of all Ether in existence. Let the -issuance of new Ether as per The Merge be quantified by $$I$$. Then -the change in time of total Ether is $$\dot{E}:=dE/dt=I$$. +Consider the quantity $$E$$ of all Ether in existence, averaged +qaurterly. Let the issuance of new Ether as per The Merge be +quantified by $$I$$. Then the change in time of total Ether is +$$\dot{E}:=dE/dt=I$$. Now, let us separate $$E$$ into three smaller boxes; $$S+C+\cancel{O}=E$$, representing (S)taked, (C)icrulating, and burnt @@ -329,7 +242,7 @@ Reinvestment is also a part of every LST smart contract; via rebaisng, a certain fraction of yield is the value proposition for the token-holder; so any model with LSTs must include reinvestment. Again we define a variable fraction $$r:=R/(I+F)$$ which obeys -$$0\leq{r}\leq1$$. As of Nov 23, 2024 the stETH token yield is +$$0\leq{r}\leq1$$. As of Nov 23, 2024 the stETH token yield is $$\approx3$$\%. Over the same period, without MEV-Boost validator yield is $$\approx4$$\%, and with MEV-Boost it is $$\approx5.7$$\%. So probably, $$r$$ currently lies in the range $$.5\leq r\leq.75$$. @@ -362,35 +275,22 @@ reinvestment of priority fees and MEV $$rf$$, and is always non-negative. The third term represents unstaking and is always non-positive. -We understand the long-time behavior of staking by studying what -controls stable fixed points $$s^*$$ corresponding to $$\dot{s}=0$$. -Briefly, we check that at least the position of this fixed point will -not be altered by considering dynamics of inflation. +### Equilibrium -### Aside on Inflation Dynamics +#### Fixed Point -Above we treat inflation as a parameter instead of as its own dynamic -variable. How dangerous is this? For a sanity-check, a quick look at -YCharts since The Merge shows that $$s,\dot{s}$$ vary over a much -greater range than $$(\ln{A},\alpha)$$. +We now wish to study the fixed point $$s^\star$$ given by solving +$$0=\dot{s}=\frac{\dot{A}}{A}(r-s)+(rf+q_+)(1-s)-q_-s$$. Recall from +our last post that since the Merge it seems +$$|\dot{\alpha}|\ll|\dot{s}|$$ and at medium timescales there is +support for a stable slowly changing value of $$\alpha^\star$$, +supporting a study of the fixed point $$x^\star$$ *given* a value for +$$\alpha^\star$$. -![The staking fraction from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-x.jpg) -![The inflation rate from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-alpha.jpg) -At present we judge the lack of empirical data on $$r$$ would obviate -the added precision of a more sophisticated treatment. We will -proceed assuming $$\dot{\alpha}\approx0$$ and treating $$\alpha$$ as a -parameter. Curious or skeptical readers are encouraged to study the -full $$(\alpha,s,\beta)$$-system using the modelling framework we have -developed. -### Fixed Point +This yields the solution -Returning from our commercial break, recall we have found a supportive -role played by inflation in moderating staking fraction. We now wish -to study the fixed point $$s^*$$ given by solving -$$0=\dot{s}=\frac{\dot{A}}{A}(r-s)+(rf+q_+)(1-s)-q_-s$$ -yielding $$s^\star = \frac{r^\star (\alpha^\star + f^\star) + {q_+}^\star}{ \alpha^\star + r^\star f^\star - {q_-}^\star}$$ @@ -470,8 +370,8 @@ $$\displaystyle $$ Where presently, we estimate the dimensionless quantity -$$\alpha/f\approx5$$ while its derivative depends on the slope of the -yield curve $$y$$ via $$\alpha/f=ys/f-b(1-s)=\left(I/(F+B)-b\right)(1-s)$$. +$$\alpha/f\in(2,5)$$ while its derivative depends on the slope of the +yield curve $$y$$. *If* the derivatives are all small in comparison (and there is no a priori gaurantee of this), then we can say something useful. @@ -492,17 +392,16 @@ that "market externalities take over", rather than supporting any specific trajectory due to endogenous dynamics. We feel it is extremely important that policy interventions, such as -changes to issuance, maintain a stable fixed point -- a fixed point -where small changes shrink instead of growing. Otherwise staking -fraction is no longer influenced by equilibrium arguments and becomes -much harder to predict or control. Potential for instability would -manifest as large groups of validators and other market participants -taking very different bets on the future of Ethereum. Which future -manifests would probably depend on details not observable from -macroeconomics alone. In what follows we will assume stability for -the sake of discussion, but must stress that while we judge it likely -under current conditions, this matter is not settled without further -and more careful study. +changes to issuance, not disrupt desirable stable fixed points. +Otherwise staking fraction is in danger of no longer influenced by +equilibrium arguments and becomes much harder to predict or control. +Potential for instability would manifest as large groups of validators +and other market participants taking very different bets on the future +of Ethereum. Which future manifests would probably depend on details +not observable from macroeconomics alone. In what follows we will +assume stability for the sake of discussion, but must stress that +while we judge it likely under current conditions, this matter is not +settled without further and more careful study. ### Inflation Concerns @@ -511,8 +410,8 @@ and increases the relative strength of the $$rf(1-s)$$ term. This leads to an increase in staking fraction if $$r$$ etc. do not reduce sufficiently, in response. Very likely $$x_{now}f$$ and maintain the dominance of reinvestment in the @@ -571,36 +469,3 @@ for inflation-led runaway-staking. ## In Conclusion -We find a potentially nuetral and even positive role for inflation in -maintaining moderate staking levels under various scenarios. We do -not contest the logic that runaway inflation *could* drive staking -fraction unreasonably high. If so, it must act through reinvestment, -and we emphasize that zero-growth or deflationary conditions would -only make matters worse. - -We find that the reinvestment of staking rewards plays a critical role -in staking dynamics. This is especially true under present market -conditions post-EIP-1557, where inflation dominates fees, and we -expect the equilibrium staking fraction to approach the reinvestment -rate $$r$$. The formulas for the staking fixed point, stability -criteria, and criteria for inflation to contribute on net to runaway -staking, appear new and we hope they will likewise prove useful to the -community. In principle, all of these could be tracked. - -We propose at least that $$r$$ be added to the list of macroeconomic -indicators routinely tracked for Ethereum, and that the economic -forces, and internal variability affecting $$r$$ be modelled and -measured. Specifically, policy advisors need to know how and indeed -if solo validators and LSTs will alter their reinvestment strategies, -especially under a regime of lower issuance yield. - -It may well, as some have predicted, that fears about runaway -inflation will turn out to be the dominant factor affecting -reinvestment. If so we expect this will emerge from microeconomic -surveys of large validators and LSTs. Each staking business $$i$$ -can, and probably already does, calculate its desired $$r_i$$. The -global $$r$$ is then the market-share weighted averaged of all of -these. We hope these calculations will bridge the macro- and -micro-economics to help the community see a unified and actionable big -picture on the future of Ethereum staking. - From aa6012b44afb0b218bfac6cbba2c5c8dcf58d9a0 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Fri, 3 Jan 2025 09:30:14 -0500 Subject: [PATCH 26/70] saving work before flight --- _posts/2025-01-10-issuance-dynamics.md | 580 ++++++++++++++++++ .../YCharts-alpha.jpg | Bin .../YCharts-x.jpg | Bin .../eth-balloon.jpg | Bin 0 -> 808212 bytes .../staking-fixpoint.png | Bin 5 files changed, 580 insertions(+) create mode 100644 _posts/2025-01-10-issuance-dynamics.md rename assetsPosts/{2024-12-05-issuance-fundamentals => 2024-12-30-issuance-dynamics}/YCharts-alpha.jpg (100%) rename assetsPosts/{2024-12-05-issuance-fundamentals => 2024-12-30-issuance-dynamics}/YCharts-x.jpg (100%) create mode 100644 assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg rename assetsPosts/{2024-12-05-issuance-fundamentals => 2024-12-30-issuance-dynamics}/staking-fixpoint.png (100%) diff --git a/_posts/2025-01-10-issuance-dynamics.md b/_posts/2025-01-10-issuance-dynamics.md new file mode 100644 index 0000000..1b2448c --- /dev/null +++ b/_posts/2025-01-10-issuance-dynamics.md @@ -0,0 +1,580 @@ +# Ethereum Macroeconomics + +This is the first of two posts on Ethereum macroeconomics. + +The share of Ether staked by exchanges such as Coinbase and Liquid +Staking Providers (LSTs) such as Lido ("centralized" staking services) +[is considerable](https://dune.com/queries/2394100/3928083) and continues +to grow. This has provoked [concerns](https://issuance.wtf/), among +Ethereum researchers, and which we share, that the future of Ethereum +might involve (1) all of its native asset being staked, such that (2) +the de facto currency will be controlled by a confederation of +centralized entities wiith less transparent govrnance. + +## Lookahead + +In this blog post we address runaway staking $s\to1$, the first of +these concerns using a "stock and flow" macroecnomics model built with +guidance from dynamical system theory. We find inflation plays a +positive but temporary role in moderating runaway staking. In the +second post, we look more closely at governance centralization and +discuss a means for evaluating macroeconomic interventions inspred by +bifurcation theory. Briefly, we are not optimstic that reducing +issuance will prevent governance centralization. + +In both posts, we provide a few code examples using `ethode` a thin +units-aware wrapper we built around `scipy.odeint` to streamline model +evaluation. Readers desiring to follow our derivations, dive into +technical mathematical points not covered here, run their own +simulations, or learn some dynamical systems are recommended to look +at our ethode [guide](), and references therein. It is certainly a +work in progress, but should have enough to get you going. + +Finally, a warning! In an act of hubris, but not without +reasons[^reasons], we have chosen $S$ to refer to Staked ETH, while +others have at times used $S$ for "circulating (S)upply", which we +call instead accessible ETH supply $A$, so $s=S/A$. [Here]() is a +python script for converting between our variables and those most +commonly used on [issuance.wtf](https://issuance.wtf). + +## For The Impatient! + +We use our macroeconmics model to identify a "low inflation; even +lower fees" regime (LI;ELF). Outside of LI;ELF convergence to a +desirable staking future without runaway staking is not possible; high +deflation pushes staking toward 0 or 100\%, while under zero or low +deflation, the tendency toward runaway staking can only be moderated +by very high churn. + +In contrast within LI;ELF staked ETH fraction approaches near the +reinvestment ratio. Thus, runaway staking can be avoided only while +inflation is held + +1. low enough, that concerns over inflation do not dominate the +reinvestment of profits by staking businesses at equilibrium, +$$\frac{dr}{d\alpha}|^\star\leq0$$ (no news here) but simultaneously + +1. high enough to numerically dominate priority fees and MEV, as a +fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. + +The bad news for opponents of runaway staking is that long +$$t\to\infty$$ term, $$\alpha^\star$$ approaches zero, suggesting +$$s^\star\to1$$ anyway. Ethereum would undergo cycles of +inflation/deflation or obtain the "L2 future" that is sometimes +described where most Ether is staked, with the majority used for +settlement of L2 rollups. + +Given all the above, we advise caution. Intervening to reduce the +issuance yield curve seems quite capable of exaccerbating the +very problems we seek to avoid. + +## Modeling an Open Zeppelin[^humor] + +![Ethereum as a balloon with compartments.](../assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg) + +### Stocks + +Consider a balloon with variable internal compartments. The averaged +size of each is measured by *stocks* + +- ($S$)taked Ether (participating in [consensus]()) is a compartment, as is +- ($U$)nstaked unburnt Ether, + - containing the ($V$)alidator reward queue, while +- ($\cancel{O}$) is all irrecoverable (burned, lost, etc.) Ether, and call +- ($A$)ccessible Ether, $A=S+U\approx120.4\times10^6$ in Dec 2024. + +The net change in time of a stock is written using a dot, such as +$\frac{dA}{dt}:=\dot{A}$[^partial], the net change in accessible +Ether. The staking $\mathcal{Q}_+$, unstaking $\nathcal{Q}_-$, and +reward $V$ queues could also have been treated as stocks. By +quarterly averaging[^aves] we *approximate* these queues as +equilibrated $\dot{V}\approx0$ etc. This gives rise to $Q_\pm,R$ below. + +Stocks grow or shrink based on flows which add to or subtract from +their derivatives. Here all flows are positive real numbers with +units ETH/yr. Flows have a domain (where its coming from) and a +codomain (where is going to).[^cats] They also obey (in)equalities, usually as +a fraction of the source, but sometimes as a fraction of other +flows.[^flowfrac] + +We convert these inequalities; for each uppercase *extensive* flow +$J,F,R\ldots$ we define a lowercase *intensive variable* $j,f,r$: the +fractions \[1\] and fractional rates \[1/yr\]. In forming these, +the ideal is to apply the tightest available bounds that still capture +the [asymptotic behavior]() in the limit of interest $S\to A$; see below. + +### Flows and Equations + +These intensives are not generally constant, we are supressing their +dependence for readability. Instead the intensives are assumed to be a +function of the dynamical variables and time, so the burn: +$b(A,S,t)=B/F$. We can often use the dependence on $t$ to smuggle in +any forces, like market panics, etc. that we neglected to include as +dynamical variables. If not, we must add a dynamical variable. + +| Name | Symbol | Source$\to$Target(s) | Constraint | Intensive | Range \[Units\] | +| :-- | :-- | :-: | :-- | :-- | :-- | +| Tx Fees | $F$ | $U\to\cancel{O},SS$ | $B+P=F$ | $f:=F/U$ | $f\in(\varepsilon_b,1)$ [1/yr] | +| Base Fees[^aves] | $B$ | $U\to\cancel{O}$ | .. | $b:=B/F$ | $b\in[\varepsilon_b,1) \[1\] | +| Priority Fees | $P$ | $U\to S$ | .. | $1-b=P/F | $1-b\in(0,1)$ \[1\] | +| Issuance[^aves] | $I$ | $\cdot\to V$ | $yS\leq I$ | $\imath:=I/S= y-\kappa$ | $y\in(0,1)$\[1/yr\] | +| Slashing | $J$ | $S\to\cancel{O}$ | $J\leq S$ | $j:=J/S$ | $j\in(0,1)$ \[1/yr\] | +| Reinvestment[^whencer] | $R$ | $V\to S$ | $R+K\approx I+P$ | $r:=R/(I+P)$ \[1\] | $r\in(0,1)$ \[1\] | +| Costs & Profit | $K$ | $V\to U$ | .. | $1-r=K/(I+P)$ | $1-r\in(0,1)$ \[1\] | +| New Staking | $Q_+$ | $U\to S$ | $Q_+\leq U-V$ | $q_+:=Q_+/U | $q_+\in(0,1)$ \[1/yr\] | +| Unstaking | $Q_-$ | $S\to U$ | $Q_-\leq S$ | $q_-:=Q_-/S | $q_-\in(0,1)$ \[1/yr\] | + +The result of all these decisions, about which you can read more in +our [modeling guide](), is to create and transform this conceptual +model + +$$\dislaystyle +\begn{array}{rcl} +\dot{A} &=& I - B - J\\ +\dot{S} &=& R + \Delta{Q}_\pm - J\\ +\dot{V} &=& I + P - R - K +\dot{U} - \dot{V} &=& K - F - \Delta{Q}_\pm +$$ + +Into one that is defined in its own dynamical and intensve variables, +a *dynamical system*. + +$$\dislaystyle +\begn{array}{rcrlcrl} +\dot{S} &=& (ry-\jmath-q_-) & S & + & \left(q_++r(1-b)f\right) & U\\ +\dot{U} &=& \left((1-r)y+q_-\right) & S & - & \left(rf+(1-r)bf+q_+\right) & U\\ +\end{array} +$$ + +#### Simulating the $(S,U)$ Model + +You can explore some behaviors of this model using a simulation; see +[guide](). Here is an unrealistically simplified example where all +coefficients are held constant that still extracts some broad +features. + +```python +from ethode import * +class Toy(ODESim): + cSS: 1/Yr + cSU: 1/Yr & Pos + cUS: 1/Yr & Pos + cSS: 1/Yr & Pos + @staticmethod + def func(v:UnitTuple(ETH, 2), t:Yr, p:Params) -> UnitTuple(ETH/Yr, 2): + S, U = v + dS = p.cSS * S + p.cSU * U + dU = p.cUS * S - p.cUU * U + return dS, dU +Toy(1, 1, 1, 1).sim() +Toy(-1, 1, 1, 1).sim() +``` + +Specifically, so long as $ry>\jmath+q_-$ staked ETH $S$ just continues +growing and growing, while $U$ cannot get too big, before its own loss +term $-\left(rf+(1-r)bf+q_+\right)\cdot U$ dominates. At some point +$S$ becomes big enough that $ry<\jmath+q_-$ and the system is capable +of oscillation, depending on a zoo of partial derivatves. Here is a +less trivial code example where all intensves other than issuance yield are +held constant. + +```python +from math import sqrt +from ethode import * +class SUconst(ODESim): + y1: 1/Yr & Pos + f: 1/Yr & PosFrac + b: NoDim & PosFrac + r: NoDim & PosFrac + qu: 1/Yr & PosFrac + qs: 1/Yr & PosFrac + j: 1/Yr & PosFrac + S1: ETH = 1 * ETH + def y(self, S:ETH, *args) -> ETH/Yr: + return self.y1 * sqrt(self.S1 / S) + @staticmethod + def func(v:UnitTuple(ETH,2), t:Yr, p:Params) -> UnitTuple(EPY, 2): + x = {'S': (S = v[0]), + 'U': (U = v[1])} + y = p.y(**x) + beta = p.b * p.f + rf = p.r * p.f + dS = (p.r * y - p.j - p.qu) * S + (rf * (1 - p.b) + p.qs) * U + dU = ((1 - p.r) * y + p.qu) * S - (rf + (1 - p.r) * beta + p.qs) * U + return dS, dU +su = SUConst.sim(y1 = 166.3 / Yr, f = .002 / Yr, \ + b = .8, r = .5, \ + qu = 1e-4 / Yr, qs = 1e-2 / Yr, j = 1e-6 / Yr) +su.verify() +su.sim() +``` + +If you have an opinion about how, for instance $r$ should depend on +$(S,U)$, you can define, instead of `r: NoDim` your own function +within the class declaration. + +```python + def rnvst(self, S:ETH, U:ETH, *args) -> NoDim: + return S * U / (S + U) ** 2 / 2 +``` + +Of course the variables we really care about are inflaion and staking +fraction. You can designate these as outputs of the model withn +the class declaration. + +```python + @output + def alpha(self, S:ETH, U:ETH, *args) -> 1/Yr: + return p.y(S) * S - p.b * p.f * U - p.j * S + @output + def sfrac(self, S:ETH, U:ETH, *args) -> 1/Yr: + return S / (S + U) + ... +su.sim(graph_outputs = ('alpha', 'sfrac')) +``` + +Below we transform from staked/unstaked variables $(S,U)$ into +modeling directly accessible supply, inflation, and staking $(A,\alpha,s)$. +But first we need to emphasize what, in our view, makes our model +different from others. + +#### Reinvestment; Only F@%$ Up the limits when it doesn't matter! + +When forming intensve variables it is *critical* that the modeler's +choices reflect the correct asymptotic behavior in the limits of +concern, in his case $U\to 0$. + +If for instance, instead of assuming $R\leq I+P$, we had assumed $R$ +is bounded by unstaked ETH $U$ (which is necessary but not +sufficient), and so defined $r_{bad}=R/U$ we get a *very* different +model, but this is wrong in a way that matters. It is wrong because +validator rewards $I+P$ can go to zero *independently* of the size of +$U$. It matters because reinvestment +$S\overset{+}{\rightsquigarrow}S$ is quite evidently related to the +positive-feedback loop between staking and issuance that people are so +worried about. Other examples abound.[^elowex] + +Concerns over staking-issuance feedback are in fact exactly why we +have split staking into new investment $Q_+$ and reinvestment $R$. +The quantity $r=R/(I+P)$ is one of the distinguishing feature of our model, and +important for staking fraction under "low inflation; even lower fee" +conditions, see below. + +1. Modelling $r$ is necessary to model LSTs,[^rlst] and +2. we want to separate the transient external forcing $Q_+$ from the +long-term feedback $R$,[^rdyn] and. +3. $r$ should be measurable with onchain data. + +## Infation and Staking Fraction + +Inflation is used to refer to many things, but here we mean +specifically the quarterly fractional change in accessible +Ether. Let's express inflation $\alpha:=\frac{d\log~A}{dt}$ in +terms of $s$ and the intensives + +$$\displaystyle +\alpha = (I-B-J)/A \approx y(sA)s - bf(1-s) - js +$$ + +### Positive Inflation cannot maintain + +A key feature of $\dot{A} concerns discouragement [Buterin](), which +requires sublinear issuance $I\lesssim S$ to avoid instability to +certain attacks. Because of this, the positive term continues to get +smaller as Accessible ETH supply grows; consistent with the +$(S,U)$-model above, eventually average inflation tends to zero. + +In contrast, while slashing can go to zero, so long as blocks are +produced, these is some minimal burn. $A$ could get quite large; for +$s=1-\epsilon$ and the current yield curve we have $A^\star\sim +(y(1)/b^\star f^\star)^2\epsilon^{-2}$; as $\epsilon$ is very small +$\epsilon^{-2}$ is very large. Nonetheless, we should expect +$\alpha^\star=0$ to be the inflation fixed point as $t\to\infty$. + +### Changing Variables + +So, our new dynamical variables are $(A,\alpha,s)$. Two important +points now require mention. + +* The quotient rule $\dot{s}=\frac{\dot{S}{A}-s\alpha$ emphasizes that + changes in a fraction occur from *both* changes in the numerator $S$ + and its denominator $A$. + +* Making conclusions about market equilibria requires us to look at *all* + of the dynamical variables, and perform a [local stability analysis](). + - Solve $\dot{x}:=(\dot{A},\dot{\alpha},\dot{s})=(0,0,0)$, obtaining a fixed point + $x^\star=(A^\star,\alpha^\star,s^\star)$. + - This $x^\star$ is a market equilibrium when it is (possibly meta-)stable, and + *the* market equiibrium just when it is unique.[^metastable] + +That is, an increase in staking fraction can be driven by more people +staking, and/or it can be driven by a reduction of the inflation rate. +The latter can be acheived in principle by a reduction of issuance +relative to the base fee "burn rate". Because of this quotient rule +tradeoff, low-but-positive inflation actually plays a positive almost +"infrastructure" role in moderating staking fraction. + +After some calculus, and a lot of algebra, we obtain the following + +$$\displaystyle +\begin{array}{rcl} +\dot{A} &=& \alpha A\\ +\dot{\alpha} &=& \zeta\dot{s}-\gamma\alpha s + X\\ +\dot{s} &=& \alpha\cdot(r-s)~+~(rf+q_+)\cdot(1-s)~-~\eft(q_-+(1-r)j\right)\cdot s\\ +\end{array} +$$ + +* $X= ...$ represents externallities to the $(A,\alpha,s)$ system +* $\zeta = ... $ + +At intremediate times we believe inflation can be treated as a parameter instead of as its own dynamic +variable + +$\dot{s}\ll\dot{\alpha}$ + +For a sanity-check, a quick look at +YCharts since The Merge shows that $$s,\dot{s}$$ vary over a much +greater range than $$(\ln{A},\alpha)$$. + +![The staking fraction from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-x.jpg) +![The inflation rate from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-alpha.jpg) + +At present we judge the lack of empirical data on $$r$$ would obviate +the added precision of a more sophisticated treatment. We will +proceed assuming $$\dot{\alpha}\approx0$$ and treating $$\alpha$$ as a +parameter. + +So, let us examine the fixed point $x^\star$ + +We find that if $\dot{A}=0=\alpha$ (no inflation nor deflation) then +an interior market equilibrium $$s^\star<1$$ requires ver high slashing. We reason +as follows. In the absence of in/de-flation, an interior fixed point +$$s^\star<1$$ would require a persistent unstaking/capitulation of +existing validators $$q_->0$$. This in turn either requires "churn", +a persistent supply of new validators to take their place +$$q_+^\star>0$$, or it is only a transient and $$q_-^\star\approx0$$; +recall that reinvestment by existing validators is not counted in +$$q_+$$. + +Net unstaking $$q_->0$$ could only describe a market equilibrium if +one group of stakers was actively capitulating and withdrawing their +stake, while another group with a higher $$r$$ were aggressively +reinvesting in their business, and their reinvestment of fees and MEV +offset the unstaking, adjusted for inflation. This cannot maintain +forever: eventually there will be no new Capitulators left, and +$$s^\star$$ must once again grow as required by the Reinvestors' +higher $$r$$, so $$s^\star$$ was not a fixed point at all. Similarly, +at some point everyone who wants to stake should have staked. If we +judge the quarterly fluxes due to the issuance of new humans and the +burn rate of legacy humans to be small and/or likely to take over +existing businesses, additional validators count overwhelmingly toward +$$r$$, so $$q_+^\star\approx0$$. Thus, the fixed point $$s^\star$$ +simplifies to + +$$\displaystyle +s^\star ~~ = ~~ r^\star ~ +\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} +$$ + +A calculation is illustrative under the current regime of positive +inflation. If inflation dominates fees, $$\alpha\gg f$$ then +$$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll +f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f +\approx .001<.005\approx\alpha$$ so to within 10\% error above, +$$s^\star\approx r$$ over the range of $$r$$ inferred from Lido yield +rate. + +This positive role for inflation can be seen in the contours of the +market equilibrium staking fraction $$s^\star$$ corresponding to +$$\dot{s}=0$$, shown in [figure +1](../assetsPosts/2024-12-05-issuance-fundamentals/staking-fixpoint.png). +The equation graphed is as follows: + +$$\displaystyle +s^\star = r^\star\frac{\alpha^\star/f^\star + 1}{\alpha^\star/f^\star + r^\star} +$$ + +The fraction $$0\leq r\leq 1$$ is the ratio of profits reinvested +quarterly by validators, $$0\leq f\leq 1$$ is the fraction of unstaked +ETH spent on transaction fees (base and priority) quarterly, and as +above $$\alpha$$ is inflation. Here $$r^\star$$ etc. means the +function $$r(s,\alpha,\ldots)$$ at the equilibrium coordinates +$$(s^\star,\alpha^\star,\ldots)$$. To find the equilibrium values +$$(\alpha^\star/f^\star,\,r^\star)$$ necessary to acheive a desired +staking fraction $$x^\star$$, simply pick a colored contour in the +figure: these are the values of constant $$x^\star$$. For every point +on this curve, the equilibrium inflation:fee ratio +$$\alpha^\star/f^\star$$ is the x-coordinate, and the equilibrium +reinvestment ratio $$r^\star$$ is the y-value. + +The two extremes are $$\alpha^\star\ll f^\star$$ fees dominate and +$$\alpha^\star\gg f^\star$$ inflation dominates. In the former, +staking fraction becomes insensitive to reinvestment, raising +$$x^\star$$. Equivalntly, for any value of positive inflation, +$$r^\star$$ is a lower bound for the equilibrium staking fraction we +should expect. + +Approximate present values from YCharts are very roughly +$$f\approx.001$$/year, $$\alpha\approx.005$$/year, $$r\in(.5,.75)$$. +How these transient values $$(\alpha/f,\,r)$$ relate to their +equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ depends on +some considerations. + + + + +# Conclusions + +Why does this happen, though? The reinvestment of issuance rewards +adds to both staked ETH $$S$$ and circulating ETH $$C$$, so the effect +on staking fraction depends on the relative size of these two. +However, the reinvestment of priority fees plus MEV is simply a net +value transfer from $$C$$ to $$S$$, which can only ever increase +staking fraction. + +In a future post we turn to the challenges facing governance +centralization, sketch a potential solution, and propose a framework +by which macroeconomics interventions can be assesed. +Briefly, for the impatient: +- LST market is indeed winner take all, as many have described +- canging the yield curve doesn't change this at all +- if decreasng issuance does prevent gov capture by CEXes and LSTs then it is because + solo staers are subsidizing the protocol +- bifurcaton theory quantifies our intuitions + + + +[^humor]: [Open Zeppelin]() is an early icon of smart contract best +practices, and continues to provide templates and auditing services in +high demand. They have absolutely no connection to this post, our +models, etc. and hopefully they will not sue us for using their name +in a bad dynamical systems joke. + +[^aves]: We use moving quarterly averages, though any timescale +$\tau$ sufficiently long that the [fast dynammics]() of [the base]() +fee are [integrated out](), and the lags from (un)staking queues are +not appreciable; see [guide]() for details. As we are averaging +quarterly, we set the staking $\dot{\mathcal{Q}}_+$, unstaking +$\dot{\mathcal{Q}}_-$, and reward $\dot{W}$ queues to zero, including +their respective flows ($R+Q_+,Q_-,I+P$) in their target stocks +($S,U,V$); even if ethereum produces empty blocks, so long as the +reward queue is not empty $U>0$. + +[^IyS]: The approximation is due to the use of quarterly averages. A +more precise approximation is $$\displaystyle +I=\frac{1}{\tau}\int_{t-\tau}^ty^\bullet S^\bullet dt'=yS-|\kappa| $$ +where $$S^\bullet,y^\bullet$ are spot values and the quarterly +covariance between staking and yield is +$\kappa=\int_{t-\tau}^t(y^\bullet-y)(S^\bullet-S)dt'<0$. See the +[ethode guide]() for more discussion + +[^regex]: This regex script is provided to translate the $\LaTeX$ +within the markdown source to (our esimatimate of) terminology more +common at [issuance.wtf](https://issuance.wtf) + +[^partial]: Sometimes $\dot{x}$ is used e.g. for the partial +derivative $\frac{\partial x}{\partial t}$, *but not here*. For a model +in which you assume dynamical variables $(x,y,z)$, these derivatives +are thus related + +$$\displaystyle \frac{dx}{dt} := \dot{x} = +\frac{\partial x}{\partial t} + +\frac{\partial x}{\partial y}\dot{y} + +\frac{\partial x}{\partial z}\dot{z} +$$ + +in which each partial is taken holding the other variables constant, +and $\frac{\partial x}{\partial t}$ is used in practice to capture any +variability from non-dynamical variables. + +[^yield]: For anyone from finance, this is *not* the same as a [bond +yield curve](https://www.investopedia.com/terms/y/yieldcurve.asp); +there is essentially no necessary lag for vaidator rewards, accounting +quarterly. + + +[^reasons]: The entire human project of using symbols to refer to +things is deeply cursed and probably hopeless. Nonetheless, we all +presist in providing excuses for + +[^cats]: We use domain/codomain in imprecise analogy with category +theory mainly because we want to reserve "source" for an attractor, as +per dynamical systems. The analogy, whie imprecise is not +inappropriate. It is routine to implicitly use associativity to +account for fibers of flows through multiple steps; "electricity from +wind/nuclear/gas" even though the electrons are indistinguishable. +Flows such as tx fees $U\overset{F}{\longrightarrow}V,\cancel{O}$ +involve a categorical product $V\times\cancel{O}$ in that the +fractional flows $U\overset{B}{\longrightarrow}\cancel{O}$ and must +factor through $V\times\cancel{O}$. For any flow with contributions +to stocks $V,\cancel{O}$ it is not hard imagining a uu2i universal +object $V\times\cancel{O}$ mediating. Similarly the staking queue +$V+(U-V)\overset{R+Q_+}{\longrightarrow}S$ should involve a +categorical coproduct. Wether there is content here beyond "flows are +functions in $\sf Set$" is unclear. None of this matters in the least +for Ethereum dynamics, of course. If you're reading it consider this +an easter egg / attempt to detect a living and alert audience. + +[^elowex]: Why not simply choose \"$B=bA$\" as was done more simply (modulo +syntax) in [Elowsson 2020]()?[^syntax] Obviously if there is no +unstaked Ether no one can afford tx fees, but this might not matter. +More importantly $S$ is a *dynamical variable*, so $b=B(A-S)$ is more +appropriate here. The function $B$ might do all kinds of complicated +nonsense, but it can never go negative and it can never exceed $U$. + +[^whencer]: Intensives expressed as fractions of flows such as $R/(I+P)$), instead +of fractional rates of sources (like $J/S$ or $Q_-/S$) occur when the +source dynamical variable, here $V$, is assumed to equilibrate +$\dot{V}\approx0$. Then the outging flows $R+K$ must equal the +incoming flows $I+P$, so we choose $R=r(I+P)$. If onchain data +indicates %\approx$70\% reinvestment of staking rewards into $S$ takes +a lot longer than a quarter we would revisit this assumption, though +we do not expect our qualitative results to change re inflation and +staking fraction. + +[^rlst]: A non-zero $r=R/(I+P)$ is built into the smart contract of every +Liquid Staking Provider (LSP). Here, token-holders provide Ether +and receive a redeemable token (LST) that shares some staking rewards +with them. This fraction of rewards $r_{LST}$ is a lower bound on +the long term behavior $r_{LST}\leq r_{LSP}\leq +R^\star(I^\star+P^\star)$; more on this next time. + +[^rdyn]: Splitting the staking queue into $R+Q_+$ allows us to +somewhat separate short-term *transient* behavior from long-term +dynamics. Speculative investment in staking by venture capitalists +and novice stakers is expected to die down eventually +$\lim_{t\to\infty}Q_+\approx0$; either they give up or they run +staking like a business where making a profit matters. Every business +that wants to stay in business reinvests some portion of its profits, +so $r,R>0$ and is what matters in the long run, once most everyone who +wants to stake is indeed staking. + +[^metastable]: Metastability involves a region of space where +trajectories are bounded surrounding a fixed point with at least one +eigenvalue with positive real part. We won't rule it out, but look +first for stable fixed points. + + +# Open Questions + +Reinvestment within each quarter may not be realistic, and could +introduce interesting oscillations. Some averaging timescale $\tau$, +at least monthly, is necessary to integrate over the short-term burn +dynamics, and (un)staking queue lags. The shorter $\tau$, the better +the approximations $I\approx yS$ and +$\frac{d\log~y}{d\log~S}\approx\frac{d\log~y^\bullet}{d\log~S^\bullet}$. +Yet, $\tau$ must be long enough so that $\dot{V}\approx0$ holds, +allowing us to define reinvestment without a time lag $r:=R/(I+P)$. +Once we have better data on how long the tail of reinvestment is we +could model selling rewards for revenue $K$ and reinvestment $R$ in a +more nuanced fashion, allowing $V$ to be its own dynamical variable. + +We would like to pay more closer attention to how the Deneb hardfork's +limit on staking queue interacts with inflation and reinvestment. We +have ignored this affect in our dynamics, because it does not +presently limit staking flows, and we see inflation decreasing long +term. However for finer questions, it depends on the resources and +patience of the validators. How does a potential race condition +change the composition of the staking queue, in terms of old $R$ +vs. new $Q_+$ and LSTs $L$ vs. everybody else $S-L$. + diff --git a/assetsPosts/2024-12-05-issuance-fundamentals/YCharts-alpha.jpg b/assetsPosts/2024-12-30-issuance-dynamics/YCharts-alpha.jpg similarity index 100% rename from assetsPosts/2024-12-05-issuance-fundamentals/YCharts-alpha.jpg rename to assetsPosts/2024-12-30-issuance-dynamics/YCharts-alpha.jpg diff --git a/assetsPosts/2024-12-05-issuance-fundamentals/YCharts-x.jpg b/assetsPosts/2024-12-30-issuance-dynamics/YCharts-x.jpg similarity index 100% rename from assetsPosts/2024-12-05-issuance-fundamentals/YCharts-x.jpg rename to assetsPosts/2024-12-30-issuance-dynamics/YCharts-x.jpg diff --git a/assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg b/assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg new file mode 100644 index 0000000000000000000000000000000000000000..130617e11db1ed56a29dbe106bca9443eb06067b GIT binary 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++++++++++++++++---------- 1 file changed, 50 insertions(+), 32 deletions(-) diff --git a/_posts/2025-01-10-issuance-dynamics.md b/_posts/2025-01-10-issuance-dynamics.md index 1b2448c..547933a 100644 --- a/_posts/2025-01-10-issuance-dynamics.md +++ b/_posts/2025-01-10-issuance-dynamics.md @@ -2,25 +2,45 @@ This is the first of two posts on Ethereum macroeconomics. -The share of Ether staked by exchanges such as Coinbase and Liquid -Staking Providers (LSTs) such as Lido ("centralized" staking services) -[is considerable](https://dune.com/queries/2394100/3928083) and continues +Ethereum has grown into a [major economic +force](https://www.grayscale.com/research/reports/the-battle-for-value-in-smart-contract-platforms); +between its native asset Ether (ETH), the smart contract ecosystem +this supports, and the Layer-2 blockchains, a conservative valuation +might be half a trillion dollars. At the core of Ethereum's "brand", +distinguishing it from other smart contract platforms, is the promise +of decentralized governance: via its [consensus layer]() no centtral +authority can censor a transaction, freeze the native asset of a user, +etc. This depends in turn on a sufficient diversity of validators +staking ETH to participate in consensus. + +The share of Ether staked by "centralized" staking services, such as +exchanges and Liquid Staking Providers (LSTs) [is +considerable](https://dune.com/queries/2394100/3928083), and continues to grow. This has provoked [concerns](https://issuance.wtf/), among -Ethereum researchers, and which we share, that the future of Ethereum -might involve (1) all of its native asset being staked, such that (2) -the de facto currency will be controlled by a confederation of -centralized entities wiith less transparent govrnance. +Ethereum researchers that the future of Ethereum might involve (1) +nearly all Ether staked, such that (2) the de facto currencies used by +most users are controlled by a confederation of centralized services +with much less transparent govrnance, and no practical alternative. ## Lookahead -In this blog post we address runaway staking $s\to1$, the first of -these concerns using a "stock and flow" macroecnomics model built with -guidance from dynamical system theory. We find inflation plays a -positive but temporary role in moderating runaway staking. In the -second post, we look more closely at governance centralization and -discuss a means for evaluating macroeconomic interventions inspred by -bifurcation theory. Briefly, we are not optimstic that reducing -issuance will prevent governance centralization. +First, a warning! In an act of hubris, but not without +reasons[^reasons], we have chosen $S$ to refer to Staked ETH, while +others have at times used $S$ for "circulating (S)upply", which we +call instead accessible ETH supply $A$, so $s=S/A$. [Here]() is a +python script for converting between our variables and those most +commonly used on [issuance.wtf](https://issuance.wtf). + +In this blog post we address the first of these concerns "runaway +staking" using a "stock and flow" macroecnomics model built with +guidance from dynamical system theory. In contrast with other +research we find inflation playing a positive but temporary role in +moderating runaway staking, but most likely $$s\to1$ eventually, +regardless of what we do. In the second post, we look more closely at +governance centralization and discuss a means for evaluating +macroeconomic interventions inspred by bifurcation theory. Briefly, +we are not optimstic that reducing issuance will prevent governance +centralization, either. In both posts, we provide a few code examples using `ethode` a thin units-aware wrapper we built around `scipy.odeint` to streamline model @@ -30,13 +50,6 @@ simulations, or learn some dynamical systems are recommended to look at our ethode [guide](), and references therein. It is certainly a work in progress, but should have enough to get you going. -Finally, a warning! In an act of hubris, but not without -reasons[^reasons], we have chosen $S$ to refer to Staked ETH, while -others have at times used $S$ for "circulating (S)upply", which we -call instead accessible ETH supply $A$, so $s=S/A$. [Here]() is a -python script for converting between our variables and those most -commonly used on [issuance.wtf](https://issuance.wtf). - ## For The Impatient! We use our macroeconmics model to identify a "low inflation; even @@ -60,7 +73,7 @@ fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. The bad news for opponents of runaway staking is that long $$t\to\infty$$ term, $$\alpha^\star$$ approaches zero, suggesting $$s^\star\to1$$ anyway. Ethereum would undergo cycles of -inflation/deflation or obtain the "L2 future" that is sometimes +inflation/deflation or obtain an "L2 future" that is sometimes described where most Ether is staked, with the majority used for settlement of L2 rollups. @@ -70,7 +83,8 @@ very problems we seek to avoid. ## Modeling an Open Zeppelin[^humor] -![Ethereum as a balloon with compartments.](../assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg) +![Ethereum as a balloon with compartments.]( + ../assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg) ### Stocks @@ -92,7 +106,7 @@ equilibrated $\dot{V}\approx0$ etc. This gives rise to $Q_\pm,R$ below. Stocks grow or shrink based on flows which add to or subtract from their derivatives. Here all flows are positive real numbers with -units ETH/yr. Flows have a domain (where its coming from) and a +units \[ETH/yr\]. Flows have a domain (where its coming from) and a codomain (where is going to).[^cats] They also obey (in)equalities, usually as a fraction of the source, but sometimes as a fraction of other flows.[^flowfrac] @@ -105,17 +119,16 @@ the [asymptotic behavior]() in the limit of interest $S\to A$; see below. ### Flows and Equations -These intensives are not generally constant, we are supressing their -dependence for readability. Instead the intensives are assumed to be a +We are not assuming these intensive parameters are constant, rather we +are supressing their dependence for readability. Except in cases (like +$y$) where the dependence is known, the intensives are assumed to be a function of the dynamical variables and time, so the burn: -$b(A,S,t)=B/F$. We can often use the dependence on $t$ to smuggle in -any forces, like market panics, etc. that we neglected to include as -dynamical variables. If not, we must add a dynamical variable. +$b(A,S,t)=B/F$.[^time] | Name | Symbol | Source$\to$Target(s) | Constraint | Intensive | Range \[Units\] | | :-- | :-- | :-: | :-- | :-- | :-- | | Tx Fees | $F$ | $U\to\cancel{O},SS$ | $B+P=F$ | $f:=F/U$ | $f\in(\varepsilon_b,1)$ [1/yr] | -| Base Fees[^aves] | $B$ | $U\to\cancel{O}$ | .. | $b:=B/F$ | $b\in[\varepsilon_b,1) \[1\] | +| Base Fees[^aves] | $B$ | $U\to\cancel{O}$ | .. | $b:=B/F$ | $b\in[\varepsilon_b,1) \[1\] | | Priority Fees | $P$ | $U\to S$ | .. | $1-b=P/F | $1-b\in(0,1)$ \[1\] | | Issuance[^aves] | $I$ | $\cdot\to V$ | $yS\leq I$ | $\imath:=I/S= y-\kappa$ | $y\in(0,1)$\[1/yr\] | | Slashing | $J$ | $S\to\cancel{O}$ | $J\leq S$ | $j:=J/S$ | $j\in(0,1)$ \[1/yr\] | @@ -139,7 +152,7 @@ $$ Into one that is defined in its own dynamical and intensve variables, a *dynamical system*. -$$\dislaystyle +$$\displaystyle \begn{array}{rcrlcrl} \dot{S} &=& (ry-\jmath-q_-) & S & + & \left(q_++r(1-b)f\right) & U\\ \dot{U} &=& \left((1-r)y+q_-\right) & S & - & \left(rf+(1-r)bf+q_+\right) & U\\ @@ -554,6 +567,11 @@ trajectories are bounded surrounding a fixed point with at least one eigenvalue with positive real part. We won't rule it out, but look first for stable fixed points. +[^time]: We can often use the dependence on $t$ to smuggle in +any forces, like market panics, etc. that we neglected to include as +dynamical variables. If not, we must add a dynamical variable. + + # Open Questions From 77bf2c85f146622443a12f91cc17e56debda9295 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Sun, 5 Jan 2025 18:13:46 -0600 Subject: [PATCH 28/70] added conclusion --- _posts/2025-01-10-issuance-dynamics.md | 401 ++++++++++++++++++++----- 1 file changed, 320 insertions(+), 81 deletions(-) diff --git a/_posts/2025-01-10-issuance-dynamics.md b/_posts/2025-01-10-issuance-dynamics.md index 547933a..e6d8205 100644 --- a/_posts/2025-01-10-issuance-dynamics.md +++ b/_posts/2025-01-10-issuance-dynamics.md @@ -8,7 +8,7 @@ between its native asset Ether (ETH), the smart contract ecosystem this supports, and the Layer-2 blockchains, a conservative valuation might be half a trillion dollars. At the core of Ethereum's "brand", distinguishing it from other smart contract platforms, is the promise -of decentralized governance: via its [consensus layer]() no centtral +of decentralized governance: via its [consensus mechanism]() no centtral authority can censor a transaction, freeze the native asset of a user, etc. This depends in turn on a sufficient diversity of validators staking ETH to participate in consensus. @@ -20,7 +20,7 @@ to grow. This has provoked [concerns](https://issuance.wtf/), among Ethereum researchers that the future of Ethereum might involve (1) nearly all Ether staked, such that (2) the de facto currencies used by most users are controlled by a confederation of centralized services -with much less transparent govrnance, and no practical alternative. +with much less transparent governance, and no practical alternative. ## Lookahead @@ -36,10 +36,10 @@ staking" using a "stock and flow" macroecnomics model built with guidance from dynamical system theory. In contrast with other research we find inflation playing a positive but temporary role in moderating runaway staking, but most likely $$s\to1$ eventually, -regardless of what we do. In the second post, we look more closely at -governance centralization and discuss a means for evaluating -macroeconomic interventions inspred by bifurcation theory. Briefly, -we are not optimstic that reducing issuance will prevent governance +regardless. In the second post, we look more closely at governance +centralization and discuss a means for evaluating macroeconomic +interventions inspred by bifurcation theory. Briefly, we are not +optimstic that reducing issuance will prevent governance centralization, either. In both posts, we provide a few code examples using `ethode` a thin @@ -47,8 +47,8 @@ units-aware wrapper we built around `scipy.odeint` to streamline model evaluation. Readers desiring to follow our derivations, dive into technical mathematical points not covered here, run their own simulations, or learn some dynamical systems are recommended to look -at our ethode [guide](), and references therein. It is certainly a -work in progress, but should have enough to get you going. +at our ethode [guide](), and references therein. It is certainly +incomplete, but should have enough to get you going. ## For The Impatient! @@ -78,52 +78,115 @@ described where most Ether is staked, with the majority used for settlement of L2 rollups. Given all the above, we advise caution. Intervening to reduce the -issuance yield curve seems quite capable of exaccerbating the -very problems we seek to avoid. +issuance yield curve seems quite capable of exaccerbating the very +problems we seek to avoid, or causing even worse problems; if you +think 100\% staking is bad, have you considered 0\%? ## Modeling an Open Zeppelin[^humor] ![Ethereum as a balloon with compartments.]( ../assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg) -### Stocks - -Consider a balloon with variable internal compartments. The averaged +Consider a balloon with variable internal compartments. The average size of each is measured by *stocks* - ($S$)taked Ether (participating in [consensus]()) is a compartment, as is - ($U$)nstaked unburnt Ether, - - containing the ($V$)alidator reward queue, while -- ($\cancel{O}$) is all irrecoverable (burned, lost, etc.) Ether, and call + - containing the ($V$)alidator reward queue. +- ($\cancel{O}$) is all irrecoverable (burned, lost, etc.) Ether, and - ($A$)ccessible Ether, $A=S+U\approx120.4\times10^6$ in Dec 2024. +- $\mathcal{Q}_\pm$ the Ether in the staking ($+$) and unstaking $-$) queues The net change in time of a stock is written using a dot, such as $\frac{dA}{dt}:=\dot{A}$[^partial], the net change in accessible -Ether. The staking $\mathcal{Q}_+$, unstaking $\nathcal{Q}_-$, and -reward $V$ queues could also have been treated as stocks. By -quarterly averaging[^aves] we *approximate* these queues as -equilibrated $\dot{V}\approx0$ etc. This gives rise to $Q_\pm,R$ below. - -Stocks grow or shrink based on flows which add to or subtract from -their derivatives. Here all flows are positive real numbers with -units \[ETH/yr\]. Flows have a domain (where its coming from) and a -codomain (where is going to).[^cats] They also obey (in)equalities, usually as -a fraction of the source, but sometimes as a fraction of other -flows.[^flowfrac] - -We convert these inequalities; for each uppercase *extensive* flow -$J,F,R\ldots$ we define a lowercase *intensive variable* $j,f,r$: the -fractions \[1\] and fractional rates \[1/yr\]. In forming these, -the ideal is to apply the tightest available bounds that still capture -the [asymptotic behavior]() in the limit of interest $S\to A$; see below. - -### Flows and Equations - -We are not assuming these intensive parameters are constant, rather we -are supressing their dependence for readability. Except in cases (like -$y$) where the dependence is known, the intensives are assumed to be a -function of the dynamical variables and time, so the burn: -$b(A,S,t)=B/F$.[^time] +Ether. Stocks grow or shrink based on flows which add to or subtract +from their derivatives. Here all flows are positive real numbers with +units \[ETH/yr\]. By averaging over "long" timescales (at least +quarterly)[^aves] we approximate the staking and unstaking queues as +equilibrated; $\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$ + +So, our conceptual model: + +$$\dislaystyle +\begn{array}{rcl} +\dot{A} &=& I - B - J\\ +\dot{V} &=& I + P - R - K +\dot{U} - \dot{V} &=& K + Q_- - Q_+ - F\\ +\dot{S} &=& R + Q_+ - Q_- - J\\ +$$ + +| Flow Name | Symbol | Domain$\to$Codomain[^cats] | Constraint | +| :-- | :-- | :-: | :-- | +| Tx Fees | $F$ | $U\to\cancel{O},SS$ | $B+P=F$ | +| Base Fees[^aves] | $B$ | $U\to\cancel{O}$ | .. | +| Priority Fees | $P$ | $U\to S$ | .. | +| Issuance[^aves] | $I$ | $\cdot\to V$ | $yS\leq I$ | +| Slashing | $J$ | $S\to\cancel{O}$ | $J+Q_-\leq S$ | +| Unstaking | $Q_-$ | $S\to U$ | .. | +| New Staking | $Q_+$ | $U-V\to S$ | $Q_++R\leq U$ | +| Reinvestment[^whyr] | $R$ | $V\to S$ | $R+K+\dot{V}=I+P$ | +| Costs & Profits | $K$ | $V\to U$ | .. | + +Flows $(B,J,Q_-,\ldots)$ have a "domain" $(U,S,S,\ldots)$, where its +coming from, and a "codomain" $(\cancel{O},\cancel{O},U,\ldots)$, where +its going to.[^cats] Flows obey constraints, often expressed as +(in)equalities relating a flow to its source. A few deserve further +comment: $I$ and $(R,Q_+,K)$. + +### Bounding Issuance $I$ + +All of these stocks and flows, $(I,S,\ldots)$, are moving +time-averages over *spot values* $(I^\bullet,S^\bullet,\ldots)$ +defined at a given block; $S:=\tau^{-1}\int_{t-\tau}^t +S^\bullet(t')dt'$, etc. In the case of issuance, we +can express spot issuance as a known function of the yield curve +$I^\bullet = y^\bullet S^\bullet$. We'll assume that + +1. Issuance is sublinear $\boxed{I\approx yS\ll S}$,[^asym] +sufficient for [discouragement](), and that +2. the large-stake scaling exponent is not substantially altered by time averaging +$\frac{\partial{d\log~y}}{\partial{d\log~S}}\approx +\frac{\partial{d\log~y^\bullet}}{\partial{d\log~S^\bullet}}$. + +The first is almost certainly an overestimate,[^ycov] +which we deem the correct direction to err in light of our results +concerning (the lack of) runaway inflation. + +### Bounding Reinvestment $R$ + +Staking reward reinvestment is a feedback loop +$S\overset{+}{\rightsquigarrow}S$ quite evidently related to the +potential positive-feedback loop between staking and inflation that +people are so concerned about. To express this concept succinctly in +one flow variable, we require that the averaging timescale $\tau$ be +adjusted upward until most validators claim and reinvest the bulk of +their staking rewards within. That is $\dot{V}=0$, so $R+K=1+P$ so +$R\leq I+P$. The quantity $r=R/(I+P)$ is one of the distinguishing +feature of our model, and also why we have split the staking queue +flow $R+Q_+$. These are our motivations: + +1. Modelling $r$ is necessary to model LSTs.[^rlst] +2. We want to separate the *transient* externally-driven dynamics $Q_+$ from the +long-term endogenous feedback $R$,[^rdyn] and +3. $r$ should be measurable with onchain data. + +If the $\tau$ required to acheive $r=R/(I+P)<1$ in practice becomes +too large, one might revisit the approximations used to model +issuance, or use data to better model $\dot{V}$. In neither case do we expect +this to make a huge qualitative difference, but please, prove us wrong! + +## Intensive Flows yield Dynamical Systems + +Flows obey inequalities, usually as a fraction of the source, but +sometimes as a fraction of other flows.[^flowfrac] We convert these +inequalities; for each uppercase *extensive* flow $(J,F,B,\ldots)$ we +define a lowercase *intensive variable* $(j,f,b,\ldots)$: the fractions \[1\] +and fractional rates \[1/yr\]. In forming these, the ideal is to +apply the tightest available bounds that still capture the [asymptotic +behavior]() in the limit of interest $S\to A$. We do not assume the +intensive parameters are constant, but suppress their dependence for +readability. Unless otherwise stated, the intensives are functions of +the dynamical variables and time, so the burn: $b(A,S,t)=B/F$.[^time] | Name | Symbol | Source$\to$Target(s) | Constraint | Intensive | Range \[Units\] | | :-- | :-- | :-: | :-- | :-- | :-- | @@ -132,25 +195,38 @@ $b(A,S,t)=B/F$.[^time] | Priority Fees | $P$ | $U\to S$ | .. | $1-b=P/F | $1-b\in(0,1)$ \[1\] | | Issuance[^aves] | $I$ | $\cdot\to V$ | $yS\leq I$ | $\imath:=I/S= y-\kappa$ | $y\in(0,1)$\[1/yr\] | | Slashing | $J$ | $S\to\cancel{O}$ | $J\leq S$ | $j:=J/S$ | $j\in(0,1)$ \[1/yr\] | -| Reinvestment[^whencer] | $R$ | $V\to S$ | $R+K\approx I+P$ | $r:=R/(I+P)$ \[1\] | $r\in(0,1)$ \[1\] | +| Reinvestment[^whyr] | $R$ | $V\to S$ | $R+K\approx I+P$ | $r:=R/(I+P)$ \[1\] | $r\in(0,1)$ \[1\] | | Costs & Profit | $K$ | $V\to U$ | .. | $1-r=K/(I+P)$ | $1-r\in(0,1)$ \[1\] | | New Staking | $Q_+$ | $U\to S$ | $Q_+\leq U-V$ | $q_+:=Q_+/U | $q_+\in(0,1)$ \[1/yr\] | | Unstaking | $Q_-$ | $S\to U$ | $Q_-\leq S$ | $q_-:=Q_-/S | $q_-\in(0,1)$ \[1/yr\] | -The result of all these decisions, about which you can read more in -our [modeling guide](), is to create and transform this conceptual -model +The use of intensive variable parameters and the apoproximation +$\dot{V}\approx0$ allows us to reshape our conceptual model into one +that is defined in its own dynamical and intensve variables, a +*dynamical system*. We'll build these up in several stages of complexity in what follows -$$\dislaystyle -\begn{array}{rcl} -\dot{A} &=& I - B - J\\ -\dot{S} &=& R + \Delta{Q}_\pm - J\\ -\dot{V} &=& I + P - R - K -\dot{U} - \dot{V} &=& K - F - \Delta{Q}_\pm -$$ +### One-Dimensional + +#### Staking Fraction + +use the eq without alpha -- same behavior + +### Two Dimensional -Into one that is defined in its own dynamical and intensve variables, -a *dynamical system*. +#### Supply growth, Crudely + +use Adot holding s const. + +#### As system + +### General + +#### Aas system + +### Reflexivity and Inflation-Response + +the conceptual source of potential oscillations can be considered by +modelling the SU system: $$\displaystyle \begn{array}{rcrlcrl} @@ -159,6 +235,34 @@ $$\displaystyle \end{array} $$ + +- we have not emphasized the oscillation because we expect market +participants can profit from it without having to coordinate their +behavior; oscillations in supply will die down if they are big enough +to arbitrage: buy toward the end of an inflationary phase, sell toward +the end of a deflationary phase. + +- contrast this to runaway staking... there's no obvious mechanism to + profit from an increase in staking fraction, other than by joining + in. So this is a challenge: can you, the reader, design a + cryptoeconomic mechanism by which runaway staking is moderated, in a + way that allows individuals to profit? Can you prove that is impossible? + +- we have not explicitly modelled this kind of reflexivity; it is swept into + the partial derivatives. + +Could these same partial derivatives be large enough to magnify +inflation-concerns into runaway staking even at intermediate +timescales? This is absolutely possible, and would require the following: + + + + + + + + + #### Simulating the $(S,U)$ Model You can explore some behaviors of this model using a simulation; see @@ -251,32 +355,29 @@ modeling directly accessible supply, inflation, and staking $(A,\alpha,s)$. But first we need to emphasize what, in our view, makes our model different from others. -#### Reinvestment; Only F@%$ Up the limits when it doesn't matter! - -When forming intensve variables it is *critical* that the modeler's -choices reflect the correct asymptotic behavior in the limits of -concern, in his case $U\to 0$. - -If for instance, instead of assuming $R\leq I+P$, we had assumed $R$ -is bounded by unstaked ETH $U$ (which is necessary but not -sufficient), and so defined $r_{bad}=R/U$ we get a *very* different -model, but this is wrong in a way that matters. It is wrong because -validator rewards $I+P$ can go to zero *independently* of the size of -$U$. It matters because reinvestment -$S\overset{+}{\rightsquigarrow}S$ is quite evidently related to the -positive-feedback loop between staking and issuance that people are so -worried about. Other examples abound.[^elowex] - -Concerns over staking-issuance feedback are in fact exactly why we -have split staking into new investment $Q_+$ and reinvestment $R$. -The quantity $r=R/(I+P)$ is one of the distinguishing feature of our model, and -important for staking fraction under "low inflation; even lower fee" -conditions, see below. - -1. Modelling $r$ is necessary to model LSTs,[^rlst] and -2. we want to separate the transient external forcing $Q_+$ from the -long-term feedback $R$,[^rdyn] and. -3. $r$ should be measurable with onchain data. + + +; Only F@%$ Up the limits when it doesn't matter! + +When forming intensve variables it is critical that our choices as +model-builders reflect the correct asymptotic behavior in the limits +of concern, in his case $U\to 0$. If for instance, instead of +assuming $R\leq I+P$, we had assumed reinvestment $R$ is bounded by +unstaked ETH $R\leq U$ we would get a qualitatively different model. +But this is wrong in a way that matters! It is wrong because +validator rewards $I+P$ can vary or in principal go to zero +independently $U$. It matters because the reinvestment of rewards +represents a sustainable feedback loop. + +both an +exceedingly large supply and dangerously low staking could shrink +$I+P$ to the same value, yet $U$ looks very different in those cases. + + + +Other examples abound.[^elowex] + + ## Infation and Staking Fraction @@ -436,9 +537,126 @@ some considerations. + # Conclusions -Why does this happen, though? The reinvestment of issuance rewards +Why does this happen, though? + +* Short $Q_+$ vs. Long Term $R$ Investment +* The Quotient Rule $\dot{S}=\dot{S}/A-s\dot{A}/A$ + +Short Term vs. Long Term. Novel investment in staking $Q_+$ is driven +largely by speculation, and new users encountering Ethereum. $Q_+$ +acts to increase staking fraction, as seen above, and indeed the glut +in $Q_+$ since the Merge may have been the source for all of the alarm +that prompted this study. Novel speculative investment must eventually dry up, +and be replaced by long-term investmemnt $R$, because (1) everyone +with money who wants to stake eventually will, so will be counted in +$R$ not $Q_+$, and any business that wants to stay in business cannot +consistently reinvest more than its revenue $R\leq I+P$. Of the long +term signal $R$, only the issuance portion of reinvestment, that is +the part that contributes to inflation, can moderate $s$, which brings +us to... + +The Quotient Rule $\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}. What +increases $s$ is any increase in staked Ether $S$, but also any net +decrease in $A=S+U$. Inflation $\alpha$ increases +*both* $U$ and $S$, because some of that increase is used to meet +costs and take profit $(1-r)\alpha$, which increases $U$ relative to +$S$. In contrast, the reinvestment of transaction fees can only ever +increase $S$ at the expense of $U$. Thus transaction fees always act +to increase staked fraction, while the effect of inflation depends on +the relative values of reinvestment ratio and staking fraction. + +So, while we could certainly model reinvestment differently, and there +are lags we are blithely integrating over, we think that these market +forces will still act as described above in a different model. It is +possible that even during sustained inflation, these effects will be +unable to prevent the upward creep in $s$, because $r$ is too large, +or the sensitivity of $r$ to inflation at equilibrium is too great, a +condition which we mathematized above. In fact we expect that every +argument about inflation effects driving increased staking, overpaying +for security, etc. could (perhaps should) be rephrased in terms of +reinvestment of staking rewards. + +Regarding the possible effects of reflexivity. We have not emphasized +the potential for oscillations very much, even thoughh a dynamical +systems person who looks at the $(S,U)$-model is likely to find this +its most striking potential. The reason is reflexivity. +Specifically, if the oscillations in inflation are sufficient to not +be drowned out by the externalities and sources of noise, then market +participants will try to profit off of them. The simplest strategy, +to buy toward the end of an inflationary phase, and sell toward the +end of a deflationary phase, should over time reduce any oscillations +that do arise. Notably though, we only expect this to happen because +it does not require the coordination of market participants: each +individual blindnly pursuing their own utility should help en masse +control these oscillations, or they were never very great to begin +with. + +Could there be a similar effect with staking fraction or inflation? +In short we have no idea, but put it as a challenge to the reader. +Can you devise a cryptoeconomic protocol or trading strategy that +forestalls the seemingly inevitable $s^\star\to1$? If not, can you +prove this is impossible? Regardless, planning for a low-inflation +high staked-fraction future is highly reommended. + +So, finally... should the Ethereum community reduce issuance? + +* Glib answer: + +Nope. + +* Short answer: + +If you don't like the medium term future of inflation or you want to +slow down the transition to high staking, please consider and study +downward adjusting the constants in EIP 7153, adopted during the Deneb +upgrade. EIP 7153 already directly limits $R+Q_+$, but was very +nicely designed to not interfere with existing staking flows. EIP +7153 does not solve, and does not claim to solve, the long term +problems, but we have been forced to conclude that reducing issuance +doesn't solve them either! + +* Long answer: + +Ask the users, especially the validators, especially the LSTs. Model +user preferences so that the demand curve becomes semi-empirical +instead of theorized. Near-100\% staking seems to be baked-in long +term pretty much no matter what we do, and high inflation cannot +sustain under an issuance yield curve designed to avoid +discouragement. So the question becomes essentially "How bad will it +get in the meantime?" This is actually a question about user +preferences. Austrian School devotees may be so inflation-averse that +they are already staking all their previously-liquid ETH at +$\alpha\approx0.5$\%/yr. In contrast users who were content to grow up +with fiat currencies during periods of $\approx$3\% inflation or even +worse might not care, or would just stake in Compound or buy stETH. + +For users seeking to passively preserve wealth, staking in Compound +(or Aave, or whatever) is ideal in terms of raw Ether demand, of +course. Staking in LSTs, though concerning from a governance angle, +may present less of an issue than some have feared. LSTs must share +some yield with users to be in business, but if they want to profit +themselves must maintain $r<1$. + +So how high can $r_{LST}$ go before LSTs find the loss of gross +profits unacceptable? Great question. We will see next time that as +many have recognized, the entity/group that can maintain the highest +$r$ wins the race eventually. It's worth noting however that if you +apply price uncertainty via a risk-discounting rate of, well anything +really, you will see that the "maximize $r$" strategy is far from the +most profitable, risk-adjusted. Of course, a speculative investor +seeking to profit from our analysis would find our study woefully +short of details. So, since you have insisted on reading the "long +answer", we will end with the classic and cowardly refrain of +academics and academic-adjacents everywhere "it requires more +research!" + + + + +The reinvestment of issuance rewards adds to both staked ETH $$S$$ and circulating ETH $$C$$, so the effect on staking fraction depends on the relative size of these two. However, the reinvestment of priority fees plus MEV is simply a net @@ -535,7 +753,7 @@ More importantly $S$ is a *dynamical variable*, so $b=B(A-S)$ is more appropriate here. The function $B$ might do all kinds of complicated nonsense, but it can never go negative and it can never exceed $U$. -[^whencer]: Intensives expressed as fractions of flows such as $R/(I+P)$), instead +[^whyr]: Intensives expressed as fractions of flows such as $R/(I+P)$), instead of fractional rates of sources (like $J/S$ or $Q_-/S$) occur when the source dynamical variable, here $V$, is assumed to equilibrate $\dot{V}\approx0$. Then the outging flows $R+K$ must equal the @@ -571,6 +789,27 @@ first for stable fixed points. any forces, like market panics, etc. that we neglected to include as dynamical variables. If not, we must add a dynamical variable. +[^asym]: Specifically $I\ll S$ means that \lim_{S\to\infty}(I/S)=0, +which according to Buterin's [annotated spec]() is satisfied even +without the limit; see [guide](). + +[^ycov]: The relation between quarterly averaged issuance and +the yield curve $y^\bullet$ is: +$$\displaystyle +I:=\int_{t-\tau}^t y^\bullet(S^\bullet)S^\bullet(t')dt'\approx yS-|COV(y^\bullet,S^\bullet)| +$$ +where we approximate $y,S$ as constant on the interval $(t-\tau,t)$ +and the covariance is negative +$0>COV(y^\bullet,S^\bullet)=\tau^{-2}\int_{t-\tau}^t(y^\bullet - +y)(S^\bullet-S)dt'$. The use of covariance here may seem a bit silly; +$y^\bullet(S^\bullet)$ is a deterministic relationship! +We do this because the community is discussing changing the yield curve, and we want to discuss +generic features of Ethereum. +So we have used approximations that we felt err in a conservative direction without explicit +depenedence on the present-day yield curve $y^\bullet_0(S^\bullet)=y_0(1)\sqrt{S_1/S}$ where +$y_0(1)\approx166.3$/yr and $S_1=1$ETH. + + # Open Questions From d68db4512125733d67b649be02dd91578b28dac6 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 6 Jan 2025 04:52:49 -0600 Subject: [PATCH 29/70] added 1d fp and basic supply growth --- _posts/2025-01-10-issuance-dynamics.md | 160 +++++++++++++++++-------- 1 file changed, 108 insertions(+), 52 deletions(-) diff --git a/_posts/2025-01-10-issuance-dynamics.md b/_posts/2025-01-10-issuance-dynamics.md index e6d8205..3276bf5 100644 --- a/_posts/2025-01-10-issuance-dynamics.md +++ b/_posts/2025-01-10-issuance-dynamics.md @@ -7,11 +7,12 @@ force](https://www.grayscale.com/research/reports/the-battle-for-value-in-smart- between its native asset Ether (ETH), the smart contract ecosystem this supports, and the Layer-2 blockchains, a conservative valuation might be half a trillion dollars. At the core of Ethereum's "brand", -distinguishing it from other smart contract platforms, is the promise -of decentralized governance: via its [consensus mechanism]() no centtral -authority can censor a transaction, freeze the native asset of a user, -etc. This depends in turn on a sufficient diversity of validators -staking ETH to participate in consensus. +distinguishing it from other smart contract platforms, is the +consistent effort put into decentralized governance. Via its +[consensus mechanism]() no centtral authority can censor a +transaction, freeze the native asset of a user, etc. This depends in +turn on a sufficient diversity of validators staking ETH to +participate in consensus. The share of Ether staked by "centralized" staking services, such as exchanges and Liquid Staking Providers (LSTs) [is @@ -35,8 +36,8 @@ In this blog post we address the first of these concerns "runaway staking" using a "stock and flow" macroecnomics model built with guidance from dynamical system theory. In contrast with other research we find inflation playing a positive but temporary role in -moderating runaway staking, but most likely $$s\to1$ eventually, -regardless. In the second post, we look more closely at governance +moderating runaway staking, though most likely $$s\to1$ eventually. +In the second post, we look more closely at governance centralization and discuss a means for evaluating macroeconomic interventions inspred by bifurcation theory. Briefly, we are not optimstic that reducing issuance will prevent governance @@ -54,10 +55,11 @@ incomplete, but should have enough to get you going. We use our macroeconmics model to identify a "low inflation; even lower fees" regime (LI;ELF). Outside of LI;ELF convergence to a -desirable staking future without runaway staking is not possible; high -deflation pushes staking toward 0 or 100\%, while under zero or low -deflation, the tendency toward runaway staking can only be moderated -by very high churn. +desirable staking future without runaway staking is not possible. +Strong deflation likely corresponds to unstable dynamics, where it is +difficult to predict wether staking fraction approaches 0 or 100\%. +Under zero or low deflation, the tendency toward runaway staking can +only be moderated by very high churn and/or slashing. In contrast within LI;ELF staked ETH fraction approaches near the reinvestment ratio. Thus, runaway staking can be avoided only while @@ -71,30 +73,30 @@ $$\frac{dr}{d\alpha}|^\star\leq0$$ (no news here) but simultaneously fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. The bad news for opponents of runaway staking is that long -$$t\to\infty$$ term, $$\alpha^\star$$ approaches zero, suggesting -$$s^\star\to1$$ anyway. Ethereum would undergo cycles of -inflation/deflation or obtain an "L2 future" that is sometimes -described where most Ether is staked, with the majority used for -settlement of L2 rollups. +$$t\to\infty$$ term, $$\alpha^\star$$ approaches zero, so +$$s^\star\to1$$. Ethereum would undergo cycles of inflation/deflation +which we expect to die down, until an "L2 future" is reached. This is +the scenario, recognized by many others where most Ether is staked, +with the majority used for settlement of L2 rollups. Given all the above, we advise caution. Intervening to reduce the issuance yield curve seems quite capable of exaccerbating the very problems we seek to avoid, or causing even worse problems; if you -think 100\% staking is bad, have you considered 0\%? +think Near-100\% staking is bad, have you tried Near-0\%? ## Modeling an Open Zeppelin[^humor] ![Ethereum as a balloon with compartments.]( ../assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg) -Consider a balloon with variable internal compartments. The average +Consider a "balloon" with variable internal compartments. The average size of each is measured by *stocks* - ($S$)taked Ether (participating in [consensus]()) is a compartment, as is - ($U$)nstaked unburnt Ether, - - containing the ($V$)alidator reward queue. + -- containing the ($V$)alidator reward queue. - ($\cancel{O}$) is all irrecoverable (burned, lost, etc.) Ether, and -- ($A$)ccessible Ether, $A=S+U\approx120.4\times10^6$ in Dec 2024. +- ($A$)ccessible Ether supply, $A=S+U\approx120.4\times10^6$ in Dec 2024. - $\mathcal{Q}_\pm$ the Ether in the staking ($+$) and unstaking $-$) queues The net change in time of a stock is written using a dot, such as @@ -103,9 +105,8 @@ Ether. Stocks grow or shrink based on flows which add to or subtract from their derivatives. Here all flows are positive real numbers with units \[ETH/yr\]. By averaging over "long" timescales (at least quarterly)[^aves] we approximate the staking and unstaking queues as -equilibrated; $\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$ - -So, our conceptual model: +equilibrated; $\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$. So, our +conceptual model: $$\dislaystyle \begn{array}{rcl} @@ -127,11 +128,12 @@ $$ | Reinvestment[^whyr] | $R$ | $V\to S$ | $R+K+\dot{V}=I+P$ | | Costs & Profits | $K$ | $V\to U$ | .. | -Flows $(B,J,Q_-,\ldots)$ have a "domain" $(U,S,S,\ldots)$, where its -coming from, and a "codomain" $(\cancel{O},\cancel{O},U,\ldots)$, where -its going to.[^cats] Flows obey constraints, often expressed as -(in)equalities relating a flow to its source. A few deserve further -comment: $I$ and $(R,Q_+,K)$. +Flows $(B,J,Q_-,\ldots)$ have a "domain" $(U,S,S,\ldots)$, where the +flow is coming from, and a "codomain" +$(\cancel{O},\cancel{O},U,\ldots)$, where the flow is going to.[^cats] +Flows obey constraints, often expressed as (in)equalities relating a +flow to its source. A few deserve further comment: $I$ and +$(R,Q_+,K)$. ### Bounding Issuance $I$ @@ -161,9 +163,10 @@ people are so concerned about. To express this concept succinctly in one flow variable, we require that the averaging timescale $\tau$ be adjusted upward until most validators claim and reinvest the bulk of their staking rewards within. That is $\dot{V}=0$, so $R+K=1+P$ so -$R\leq I+P$. The quantity $r=R/(I+P)$ is one of the distinguishing -feature of our model, and also why we have split the staking queue -flow $R+Q_+$. These are our motivations: +$R\leq I+P$. The quantity $r=R/(I+P)$, the ratio of staking rewards +reinvestment over issuance and priority fees is one of the +distinguishing features of our model, and also why we have split the +staking queue flow $R+Q_+$. These are our motivations: 1. Modelling $r$ is necessary to model LSTs.[^rlst] 2. We want to separate the *transient* externally-driven dynamics $Q_+$ from the @@ -175,18 +178,20 @@ too large, one might revisit the approximations used to model issuance, or use data to better model $\dot{V}$. In neither case do we expect this to make a huge qualitative difference, but please, prove us wrong! -## Intensive Flows yield Dynamical Systems +## Intensive Flows give Dynamical Systems + +Flows obey inequalities, usually as a fraction of the +source.[^flowfrac] We convert these inequalities; for each uppercase +*extensive* flow $(J,F,B,\ldots)$ we define a lowercase *intensive +variable* $(j,f,b,\ldots)$: the fractions \[1\] and fractional rates +\[1/yr\]. In forming these, the ideal is to apply the tightest +available bounds that still capture the [asymptotic behavior]() in the +limit of interest $S\to A$. We do not assume the intensive parameters +are constant, but suppress their dependence for readability. Unless +otherwise stated, the intensives are functions of the dynamical +variables and time, so the burn: $b(A,S,t)=B/F$.[^time] -Flows obey inequalities, usually as a fraction of the source, but -sometimes as a fraction of other flows.[^flowfrac] We convert these -inequalities; for each uppercase *extensive* flow $(J,F,B,\ldots)$ we -define a lowercase *intensive variable* $(j,f,b,\ldots)$: the fractions \[1\] -and fractional rates \[1/yr\]. In forming these, the ideal is to -apply the tightest available bounds that still capture the [asymptotic -behavior]() in the limit of interest $S\to A$. We do not assume the -intensive parameters are constant, but suppress their dependence for -readability. Unless otherwise stated, the intensives are functions of -the dynamical variables and time, so the burn: $b(A,S,t)=B/F$.[^time] +### Table of Flows | Name | Symbol | Source$\to$Target(s) | Constraint | Intensive | Range \[Units\] | | :-- | :-- | :-: | :-- | :-- | :-- | @@ -200,26 +205,77 @@ the dynamical variables and time, so the burn: $b(A,S,t)=B/F$.[^time] | New Staking | $Q_+$ | $U\to S$ | $Q_+\leq U-V$ | $q_+:=Q_+/U | $q_+\in(0,1)$ \[1/yr\] | | Unstaking | $Q_-$ | $S\to U$ | $Q_-\leq S$ | $q_-:=Q_-/S | $q_-\in(0,1)$ \[1/yr\] | -The use of intensive variable parameters and the apoproximation +The use of intensive variable parameters and the approximation $\dot{V}\approx0$ allows us to reshape our conceptual model into one that is defined in its own dynamical and intensve variables, a -*dynamical system*. We'll build these up in several stages of complexity in what follows +*dynamical system*. We'll build this up one step at a time. + +### Supply growth, Crudely in One Dimension + +Consider $\dot{A}=I-B-J$ in light of the [above +table](#table-of-flows), $\dot{A} = y(S)S - bf(A-S) - jS$. To +understand the dynamics of circulating supply, we need to understand +that of staked ETH. As a crude approximation, lets temporaruly hold +staking fraction $s=S/A$ constant. + +$$\displaystyle +\dot{A}=\left(y(sA)-bf(1-s)-js\right)A=y_0(1)\sqrt{sA}-[bf(1-s)+js]A +$$ + +ETHODE + +Pretending $s=s^\star$ constant, we obtain the *fixed point* $A^\star$ +at which $\dot{A}=0$. The trivial fixed point $A=0$ is unstable, and +the non-trivial fixed point is -### One-Dimensional +$$\displaystyle +A^\star = \left( +\frac{y(1)}{b^\star f^\star(1-s^\star)+j^\star s^\star} +\right)^2 s^\star +$$ + +Where $f^\star=f(A^\star,s^\star,\ldots)$ represents tx fee fractional +rate $f$ evaluated at the fixed point. A fixed point represents a +market equilibrium when it is stable. A fixed point is locally stable +just when small changes (*perturbations*) shrink. This is governed by +linearizing about the fixed point, and evaluating the sign of +$$\left.\frac{\partial\dot{A}}{\partial A}\right|^\star$, visualized below -#### Staking Fraction +![1D Stability Conditions](../assetsPosts/1d-stab.png) -use the eq without alpha -- same behavior +This reduces after some calculus and algebra to a sufficient condition +on the loss term: -### Two Dimensional +$$\displaystyle +\left.\frac{\partial\log~B}{\partial\log~A}\right|^\star,~ +\left.\frac{\partial\log~J}{\partial\log~A}\right|^\star~~>~~\frac{1}{2}. +$$ -#### Supply growth, Crudely +If either is satisfied, the fixed point is stable, and $A^\star$ +represents the equilibrium supply value. Note that constant $bf,j$ +correspond to scaling exponent $1$, so satisfy. Essentially some +component of the loss term must increase faster than $\sqrt{A}$ around +the fixed point. -use Adot holding s const. +If we think of the burn as a representation of economic activity on +Ethereum, this raises a disturbing possibility, in which supply $A$ +keeps growing but economic activity does not scale at least as fast as +$\sqrt{A}$. That is inflation, not just in the sense of supply +expansion, but the demand not growing at pace with the supply. This +is similar to the concern raised by [Elowsson 2020](). -#### As system +However, we have held $s$ consant; this is not *really* a one +dimensional flow. To really understand the dynamics we need to model +both $A,s$ as dynamical variables. + +### Two Dimensional $(A,s)$ System + +After an algebraic massage, we can obtain for staking fraction + +$$\displaystyle +\dot{s} = -### General +### General $(A,\alpha,s)$ System #### Aas system From 21faffdbd648db1b993a151639238266534c44ea Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 9 Jan 2025 07:00:54 -0600 Subject: [PATCH 30/70] updated date and synced --- _posts/2024-12-19-issuance-basics.md | 110 ---- _posts/2024-12-20-issuance-fundamentals.md | 471 ------------------ ...ics.md => 2025-01-15-issuance-dynamics.md} | 421 ++++++++-------- .../YCharts-alpha.jpg | Bin .../YCharts-x.jpg | Bin .../eth-balloon.jpg | Bin .../staking-fixpoint.png | Bin 7 files changed, 203 insertions(+), 799 deletions(-) delete mode 100644 _posts/2024-12-19-issuance-basics.md delete mode 100644 _posts/2024-12-20-issuance-fundamentals.md rename _posts/{2025-01-10-issuance-dynamics.md => 2025-01-15-issuance-dynamics.md} (82%) rename assetsPosts/{2024-12-30-issuance-dynamics => 2025-01-15-issuance-dynamics}/YCharts-alpha.jpg (100%) rename assetsPosts/{2024-12-30-issuance-dynamics => 2025-01-15-issuance-dynamics}/YCharts-x.jpg (100%) rename assetsPosts/{2024-12-30-issuance-dynamics => 2025-01-15-issuance-dynamics}/eth-balloon.jpg (100%) rename assetsPosts/{2024-12-30-issuance-dynamics => 2025-01-15-issuance-dynamics}/staking-fixpoint.png (100%) diff --git a/_posts/2024-12-19-issuance-basics.md b/_posts/2024-12-19-issuance-basics.md deleted file mode 100644 index b173287..0000000 --- a/_posts/2024-12-19-issuance-basics.md +++ /dev/null @@ -1,110 +0,0 @@ -±s±--- -layout: post -title: Basics of applying dynamics to Ethereum Issuance -author: Eric Downes -categories: [ethereum, issuance] -excerpt: This is a pedagogical blog post on differential equtions for ethereum in a series summarizing our research funded by EF GRANT INFO. -usemathjax: true -thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Danieli Palombi and Philipp Zahn. ---- - -## The Problem - -The share of Ether staked by exchanges such as Coinbase and Liquid -Staking Providers (LSTs) such as Lido ("centralized" staking services) -continues to grow. This has provoked -[concerns](https://issuance.wtf/), first raised by Ethereum -researchers, and which we share, that the future of Ethereum might -involve (1) all of its native asset being staked, such that (2) the de -facto liquid Ether is controlled by a confederation of centralized -un-transparent govrnance. In this blog post we setup some basics for -readers who can read python but are unfamiliar with differential -equations. - -We use "stock and flow" differential equation models to study Ethereum -macroeconomics, specifically how changing issuance impacts these -questions. We will publish several blog posts on this topic: - -0. (This post) Basics of applying Dynamics to Ethereum -1. [Danger in reducing issuance to avoid Runaway Staking](./2024-12-05-issuance-fundamentals.md) -2. Will reducing issuance avoid Governance Centralization? -3. Other Levers besides Issuance, and a means of evaluating levers. -4. Some tools to help the community study and resolve policy debates. - -We wrapped the `scipy.odesim` tool and provided some examples -[here](https://github.com/20squares/issuance-fundamentals) for -researchers wishing to simulate their own models. We will return to -this tool in greater depth later, and demonstrate some basic usage -below. - -# TLDR for the impatient - -* Under the present yield curve, assuming weak Ethereum supply should eventually - stabilize: nearly all Ether is staked with zero inflation; issuance - matches the burn. - --- include styuff from notion.so - - -# Why Diff Eq - -On the face of it, differential equations, such as the reader may have -suffered through in highschool or college does not seem up to the task -of saying anything useful about Ethereum. --- Everything in Ethereum is discrete, not continuous --- Diferential Equations are cookbook recipes solving very limited solutions. - -As often, everything is a matter of perspective. Let's illustrate by -modeling the burn rate from [EIP -1559](https://ethereum.github.io/abm1559/notebooks/eip1559.html). -This EIP split Ethereum transaction fees into base and priority fees, -where the base fee is destroyed forever, and determined dynamically by -the amount of block congestion, using gas price as proxy. The idea is -that traders can still compete for priority in blocks but they must -pay a deflationary penalty when "excessive" competition for -blockspace occurs. - -$$\displaystyle -\beta_{t+\tau} = \left(1+\frac{1}{8} -\left(\frac{g_{t}}{g^\star}-1\right)\right)\beta_t -$$ - -Where: -- $\tau$ is the blocktime between successive blocks $$\approx12$$s/block -- $\beta_t$$ is the burn rate in ETH/block at time $t$ -- $g_t$$ is the gas price in Ether at time $$t$$ -- $g^\star$ is the target gas price $$15\times10^6$$ETH/block. - -The above equation is a "difference equation", solved by - - - -But, if we are -studying policy questions like issuance, we are looking at much longer -times, such as quarters. - - -# A Toy Model - -Let’s start with two assumptions about Ethereum - -1. All Ether ever issued can be split into “Staked Ether” $S$ “Unstaked Ether” $U$ - - $(1)~~~~~E = S + U$ - - $E$ — Total Ether in existence - - $S$ — Ether staked as per the Shanghai hard-fork - - $U$ — Everything else; burned Ether, liquid Ether, Ether locked in smart contracts, etc. - - $U=C+B$ circulating and burned - - -1. Ethereum is Proof-of-Stake with a protocol-level yield curve. - - The existing yield curve, post-Shanghai hardfork, $y_0(S)$ determines issuance of new Ether, and EIP-1559 determines Ether burned “lost” $\ell$ from the system. We count burned Ether as part of total Ether $E$ so: - - $(2)~~~~~\dot{E}:=\frac{dE}{dt}=y(S)S$ - - These staking rewards occur directly to $U$ and may be reinvested into $S$ at a reinvestment rate $r$. diff --git a/_posts/2024-12-20-issuance-fundamentals.md b/_posts/2024-12-20-issuance-fundamentals.md deleted file mode 100644 index da2234c..0000000 --- a/_posts/2024-12-20-issuance-fundamentals.md +++ /dev/null @@ -1,471 +0,0 @@ ---- -layout: post -title: Some danger in reducing Issuance to avoid Runaway Staking. -author: Eric Downes -categories: [ethereum, issuance] -excerpt: This is a blog post on Runaway Staking in a series summarizing our issuance research for the Ethereum Foundation. -usemathjax: true -thanks: I am grateful for useful discussions with Eric Siu, Andrew Sudbury, and the the 20 Squares team; especially Daniele Palombi and Philipp Zahn. We are grateful to the EF for funding this research. ---- - -## The Problem - -The share of Ether staked by exchanges such as Coinbase and Liquid -Staking Providers (LSTs) such as Lido ("centralized" staking services) -[is considerable](https://dune.com/queries/2394100/3928083) continues -to grow. This has provoked [concerns](https://issuance.wtf/), first -raised by Ethereum researchers, and which we share, that the future of -Ethereum might involve (1) all of its native asset being staked, such -that (2) the de facto currency is controlled by a confederation of -centralized entities wiith less transparent govrnance. In this blog -post we address the first of these concerns. - -We use "stock and flow" differential equation models to study Ethereum -macroeconomics, specifically how changing issuance impacts these -questions. We will publish several blog posts on this topic: - -0. Basics of applying Dynamics to Ethereum -1. (This post) Will reducing issuance avoid Runaway Staking? -2. Will reducing issuance avoid Governance Centralization? -3. Other Levers besides Issuance, and a means of evaluating levers. -4. Conclusions, tools to study and resolve policy debates. - -Even if runaway staking does not directly concern you, this post is -essential to understand our view of the interaction between inflation -and staking, a picture in tension with other research. - - -## TLDR - -### For the extraordinarily impatient reader, our conclusions. - -* We identify a "low inflation; even lower fees" regime (LI;ELF) in - which there the amount of Ether staked is stable and (potentially) - less than 100%; $$s^\star<1$$. - -* Staking rewards do *not* simply enter circulation. So long as LSTs - exist some fraction of these are necessarily "reinvested" in - staking at a non-zero ratio $$r$$. - -- When LI;ELF persists, the fraction of Ether staked approaches the - reinvestment ratio from above. - -- Reinvestment can be estimated with onchain data, and its - sensitivity to inflation measured with - validator surveys. - -* Outside of LI;ELF, convergence to a reasonable staking future is not - possible. - -- Under high deflation, Ethereum faces a choice between zero staking - $$s^\star\to0$$ or runaway staking $$s^\star\to1$$. - -- Under no growth or low deflation, runaway staking *is inevitable*. - -* Within LI;ELF, runaway staking may still occur. If it does *not*, - this is because - -- (1) inflation is held low enough, that concerns over inflation - do not dominate the reinvestment of profits by staking businesses - at equilibrium, $$d_\alpha{r}|^\star\leq0$$, but simultaneously - -- (2) inflation is held high enough at equilibrium to numerically - dominate priority fees and MEV, as a fraction of circulating - Ether; $$\alpha^\star\gg f^\star$$. - -* The medium term future of ethereum we see is one in which inflation - $$\alpha$$, while higher than that prefered by inflation hawks, - plays a *positive* role to mediate staking fraction and - centralization. In long $$t\to\infty$$ term, though $$\alpha\to0$$ forcing - $$s\to1$$. We will revisit the consequences of this for Ethereum in - a future blog post. - -* Given all the above, we advise caution. Intervening to reeduce the - issuance yield curve seems dangerously capable of exaccerbating the - very problems we seek to avoid. - -### For the moderately impatient reader. - -#### Dynamics Supporting Market Equilibrium - -In some more detail, a sketch of the reasons behind our conclusions. -The quantity of most importance to this debate is the staked ETH -fraction $$s$$, as of Dec 2024 is [roughly -.28](https://dune.com/queries/1933048/3188490). Staking fraction is -calculated $$s=S/A$$ where $$S$$ is all staked ETH, $$C$$ is -"circulating" (unstaked, unburnt) ETH, and $$A=S+C$$ is the total -"accessible" (unburnt) ETH. Differential changes, such as changes in -time $$\frac{ds}{dt}:=\dot{s}$$ are given by the quotient rule -$$\dot{s}=\dot{S}/A-s\dot{A}/A$$. The quantity $$\alpha = \dot{A}/A$$ -is the average on-paper inflation rate (supply expansion APY) averaged -on at-least-quarterly timescales. - -That is, an increase in staking fraction can be driven by more people -staking, and/or it can be driven by a reduction of the inflation rate. -The latter can be acheived in principle by a reduction of issuance -relative to the base fee "burn rate". Because of this quotient rule -tradeoff, low-but-positive inflation actually plays a positive almost -"infrastructure" role in moderating staking fraction. - -This positive role for inflation can be seen in the contours of the -market equilibrium staking fraction $$s^\star$$ corresponding to -$$\dot{s}=0$$, shown in [figure -1](../assetsPosts/2024-12-05-issuance-fundamentals/staking-fixpoint.png). -The equation graphed is as follows: - -$$\displaystyle -s^\star = r^\star\frac{\alpha^\star/f^\star + 1}{\alpha^\star/f^\star + r^\star} -$$ - -The fraction $$0\leq r\leq 1$$ is the ratio of profits reinvested -quarterly by validators, $$0\leq f\leq 1$$ is the fraction of unstaked -ETH spent on transaction fees (base and priority) quarterly, and as -above $$\alpha$$ is inflation. Here $$r^\star$$ etc. means the -function $$r(s,\alpha,\ldots)$$ at the equilibrium coordinates -$$(s^\star,\alpha^\star,\ldots)$$. To find the equilibrium values -$$(\alpha^\star/f^\star,\,r^\star)$$ necessary to acheive a desired -staking fraction $$x^\star$$, simply pick a colored contour in the -figure: these are the values of constant $$x^\star$$. For every point -on this curve, the equilibrium inflation:fee ratio -$$\alpha^\star/f^\star$$ is the x-coordinate, and the equilibrium -reinvestment ratio $$r^\star$$ is the y-value. - -The two extremes are $$\alpha^\star\ll f^\star$$ fees dominate and -$$\alpha^\star\gg f^\star$$ inflation dominates. In the former, -staking fraction becomes insensitive to reinvestment, raising -$$x^\star$$. Equivalntly, for any value of positive inflation, -$$r^\star$$ is a lower bound for the equilibrium staking fraction we -should expect. - -Approximate present values from YCharts are very roughly -$$f\approx.001$$/year, $$\alpha\approx.005$$/year, $$r\in(.5,.75)$$. -How these transient values $$(\alpha/f,\,r)$$ relate to their -equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ depends on -some considerations. - -### Maintaining LI;ELF while changing Issuance - -Under current market conditions, low inflation and even lower fees -(LI;ELF), the long-term equilibrium staking fraction approaches the -average ratio $$r$$ at which validators reinvest their staking -rewards. This makes staking fraction responsive to economic -interventions that affect the reinvestment decisions of participants. - -So if LI;ELF conditions persist, we expect the staking -ratio converges to no more than +10\% of the the equilibrium -reinvestment ratio. Good news: if issuance can be reduced such that -inflation is reduced by half, while fees etc. maintain, equilibrium is -still around $$r$$ + 15\%, so $$s^\star\in(.58,.8)$$, at present values. - -If the yield curve is reduced from $$y_0=\frac{k}{\sqrt{S}}$$ to -$$y'=\frac{y_0}{1+k'S}$$ as proposed in ----------, what is likely to -happen? - - - -#### Macroeconomics at the Equilibrium Point - -Per the arguments of Ethereum researchers, high inflation could lead -to runaway staking. This has motivated the drive to reduce issuance -and thus inflation. Our model offers a perspective here as well. - -In our model the net effect of inflation on staking fraction at -equilibrium is reflected by taking the derivative -$$0<\left.\frac{ds^\star}{d\alpha}\right|^\star$$ assuming $$r,f$$ are -implicit functions of $$\alpha$$. That is, the necessary condition -for inflation to push the market equilibrium $$s^\star$$ itself into -runaway staking is (see below for explanation): - -$$\displaystyle -1 ~~ < ~~ -\left.\frac{\partial\log\ r}{\partial\log\ \alpha}\right|^\star -\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} ~~ + ~~ -\left.\frac{\partial\log\ f}{\partial\log\ \alpha}\right|^\star -$$ - -We hope that this work can be built upon to focus inflationary -pressure arguments into empirically measurable assertions that can be -tracked as a metric for Ethereum health. If inflation pressures are -indeed the dominant consideration for Ethereum users considering -staking, this should emerge from microeconomic surveys of validators' -reinvestment sentiments. A semi-empirical measurement of correlations -between transaction fees and inflation, using our model or similar to -control for staking queue flows, etc. should also shed light on wether -the above condition is satisfied or not. - -## Modelling Staking - -Now some more details! - -### Splitting Up A Growing Pie - -Consider the quantity $$E$$ of all Ether in existence, averaged -qaurterly. Let the issuance of new Ether as per The Merge be -quantified by $$I$$. Then the change in time of total Ether is -$$\dot{E}:=dE/dt=I$$. - -Now, let us separate $$E$$ into three smaller boxes; -$$S+C+\cancel{O}=E$$, representing (S)taked, (C)icrulating, and burnt -Ether $$\cancel{O}$$, respectively. By considering each quantity a -quarterly average we can plausibly ignore the detailed dynamics of the -staking, unstaking, and withdrawal queues. - -### How Pie is Pushed Around The Plate - -This is our model: - -$$ -\displaystyle -\begin{array}{rcl} -\dot{E} &=& I\\ -\dot{S} &=& R+Q_+-Q_-\\ -\dot{A} &=& I-B\\ -\end{array} -$$ - -Where all quantities are assesed quarterly: -* $$I$$: Total Issuance in a certain quarter -* $$R$$: Total Reinvestment of Validator Yields, in that quarter -* $$Q_\pm$$: Total (Un)Staking Queue Flows, in that quarter -* $$B$$: Total Base Fee (burn) assed, in that quarter. - -Transaction fees $$F+B$$ are split into the base fee $$B$$ which is -burned, and priority fees plus MEV $$F$$ which go to validators via -the wihdrwal queue. Fees obey the inequality $$0\leq F+B\leq C$$. We -ae interested in the limit $$C\to0$$ and pull this dependence out by -defining $$f$$ such that $$fC = F+B$$. The fraction $$f$$ need not be -a constant, but whatever values it takes, $$f$$ must be a fraction: -$$00$$, the staking fraction receives a boost when -it is less than reinvestment rate $$sr$$. -Under deflation $$\dot{A}/A<0$$ the effects are reversed, though see -below. The second term combines new staking $$q_+$$ and the -reinvestment of priority fees and MEV $$rf$$, and is always -non-negative. The third term represents unstaking and is always -non-positive. - -### Equilibrium - -#### Fixed Point - -We now wish to study the fixed point $$s^\star$$ given by solving -$$0=\dot{s}=\frac{\dot{A}}{A}(r-s)+(rf+q_+)(1-s)-q_-s$$. Recall from -our last post that since the Merge it seems -$$|\dot{\alpha}|\ll|\dot{s}|$$ and at medium timescales there is -support for a stable slowly changing value of $$\alpha^\star$$, -supporting a study of the fixed point $$x^\star$$ *given* a value for -$$\alpha^\star$$. - - - -This yields the solution - -$$s^\star = \frac{r^\star (\alpha^\star + f^\star) + {q_+}^\star}{ -\alpha^\star + r^\star f^\star - {q_-}^\star}$$ - -We find that if $\dot{A}=0=\alpha$ (no inflation nor deflation) then -an interior market equilibrium $$s^\star<1$$ is impossible. We reason -as follows. In the absence of in/de-flation, an interior fixed point -$$s^\star<1$$ would require a persistent unstaking/capitulation of -existing validators $$q_->0$$. This in turn either requires "churn", -a persistent supply of new validators to take their place -$$q_+^\star>0$$, or it is only a transient and $$q_-^\star\approx0$$; -recall that reinvestment by existing validators is not counted in -$$q_+$$. - -Net unstaking $$q_->0$$ could only describe a market equilibrium if -one group of stakers was actively capitulating and withdrawing their -stake, while another group with a higher $$r$$ were aggressively -reinvesting in their business, and their reinvestment of fees and MEV -offset the unstaking, adjusted for inflation. This cannot maintain -forever: eventually there will be no new Capitulators left, and -$$s^\star$$ must once again grow as required by the Reinvestors' -higher $$r$$, so $$s^\star$$ was not a fixed point at all. Similarly, -at some point everyone who wants to stake should have staked. If we -judge the quarterly fluxes due to the issuance of new humans and the -burn rate of legacy humans to be small and/or likely to take over -existing businesses, additional validators count overwhelmingly toward -$$r$$, so $$q_+^\star\approx0$$. Thus, the fixed point $$s^\star$$ -simplifies to - -$$\displaystyle -s^\star ~~ = ~~ r^\star ~ -\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} -$$ - -A calculation is illustrative under the current regime of positive -inflation. If inflation dominates fees, $$\alpha\gg f$$ then -$$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll -f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f -\approx .001<.005\approx\alpha$$ so to within 10\% error above, -$$s^\star\approx r$$ over the range of $$r$$ inferred from Lido yield -rate. - -So if these conditions persist at long times, and to be clear there is -no reason to be certain they will, we should expect $$s^\star\approx -r\in(.55,.77)$$. This places the lower range of $$s^\star$$ near the -50% staking target proposed by Elowsson. This is hopeful! However -ittle weight can be attached to such back-of-envelope extrapolations -as these. To make reliable projections we need systematic measurement -of both the statistics of individual validator reinvestment and -metrics or at least surveys reflecting validator sentiment and -(in)capacity to absorb reduced revenue. - -### Market Equilibrium: Stability - -The fixed-point $$x^\star$$ represents "the" market equilibrium when it -is both unqiue and stable. - -Non-uniqueness requires another solution to $$\dot{s}=0$$, thus a -shared and repeated root among all terms $$\alpha,rf+q_+,q_-$$. That -is, each economic flow would have to nearly disappear at a particular -value of staking fraction. Lacking any supporting data or a mechanism -for this, we cannot really analyse such an alternate equilibria, but -intuitively it seems unlikely, so for now we move on, assuming -uniqueness. - -Stability requires that small perturbations shrink; -$$\left.\frac{\partial\dot{x}}{\partial x}\right|^\star<0$$, where -$$\left.\frac{d}{d\xi}\right|^\star$$ denotes a derivative with -respect to $$\xi$$ at the fixed point $$\xi^\star$$. The full -stability condition is - -$$\displaystyle -1 + \left.\frac{r^\star}{\alpha^\star/f^\star} -~~ > ~~ -\frac{\partial\log\ r}{\partial\log\ s}\right|^\star + -\left(1 - \frac{r^\star}{s^\star}\right) -\left.\frac{\partial\log(\alpha/f)}{\partial\log\ s}\right|^\star -$$ - -Where presently, we estimate the dimensionless quantity -$$\alpha/f\in(2,5)$$ while its derivative depends on the slope of the -yield curve $$y$$. - -*If* the derivatives are all small in comparison (and there is no a -priori gaurantee of this), then we can say something useful. -Stability is automatic under inflation so long as -$$\alpha^\star+r^\star f^\star>0$$. If again derivatives are all -small but $$\alpha<0,~rf<|\alpha|0$$. The strong deflation -$$\alpha<-f<0$$ fixed point appears interior unstable, predicting -either runaway staking or a catastrophic loss of staking, depending on -initial considitions. Our interpretation of the latter behavior is -that if raw Ether is appreciating in value so quickly, and there has -been insufficient interest in staking thusfar, then why bother staking at -all? But we must emphasize that what instability *really* means is -that "market externalities take over", rather than supporting any -specific trajectory due to endogenous dynamics. - -We feel it is extremely important that policy interventions, such as -changes to issuance, not disrupt desirable stable fixed points. -Otherwise staking fraction is in danger of no longer influenced by -equilibrium arguments and becomes much harder to predict or control. -Potential for instability would manifest as large groups of validators -and other market participants taking very different bets on the future -of Ethereum. Which future manifests would probably depend on details -not observable from macroeconomics alone. In what follows we will -assume stability for the sake of discussion, but must stress that -while we judge it likely under current conditions, this matter is not -settled without further and more careful study. - -### Inflation Concerns - -As we have seen, reducing issuance in a vacuum decreases inflation, -and increases the relative strength of the $$rf(1-s)$$ term. This -leads to an increase in staking fraction if $$r$$ etc. do not reduce -sufficiently, in response. Very likely $$x_{now}1$$; and there is no -interior fixed point. - -Why does this happen, though? The reinvestment of issuance rewards -adds to both staked ETH $$S$$ and circulating ETH $$C$$, so the effect -on staking fraction depends on the relative size of these two. -However, the reinvestment of priority fees plus MEV is simply a net -value transfer from $$C$$ to $$S$$, which can only ever increase -staking fraction. - -A moderate reduction of inflation to half or even a third of its -present value would still result in a comfortably positive inflation -rate, $$\alpha>f$$ and maintain the dominance of reinvestment in the -long term $$s^\star\to r^\star$$. - -The effect of $$\alpha$$ on $$r$$ then becomes quite salient, and this -is where previous market equilibrium arguemnts come into play. We -agree with issuane hawks that there must be *some* $$(s,\alpha)$$ past -which fears of runaway inflation will drive $$r\to1$$ and therefore -$$s^\star\to1$$. How does our model account for such effects? - -The derivative of staking fraction by inflation near equilibrium -$$\left.\frac{\partial x^\star}{\partial\alpha}\right|^\star$$ shows the response to -small endogenous changes, but not to a market crash, or protocol -hardfork. The condition that gently rising inflation causes an -*increase* in staking is that this derivative should be positive. -Assuming $$r,f$$ are (unknown) implicit funcions of $$\alpha$$ the -positive derivative condition becomes after some manipulation - -$$ -\displaystyle -1 ~~ < ~~ \left.\frac{\partial\log\ r}{\partial\log\ \alpha}\right|^\star -\cdot\frac{1+\alpha^\star/f^\star}{1-r^\star} -~~ + ~~ -\left.\frac{\partial\log\ f}{\partial\log\ \alpha}\right|^\star -$$ - -This gives us a condition expressing how the macroeconomic effects of -inflation on reinvestment and tx fees impact the staking fraction -equilibrium. If reinvestment is steady with small increases in -inflation, then tx fees must respond superlinearly. If tx fees -respond only weakly, then reinvestment must increase with inflation. -The smaller reinvestment has been and the more inflation dominates -fees, the weaker the increase of reinvestment needs to be to satisfy -this conditions. - -Note that $$x^\star/r^\star-1=(1-r^\star)/(1+\alpha^\star/f^\star)$$ -the same quantity that appears in the stability condition, suggesting -a link between the boundary of the stable manifold and the conditions -for inflation-led runaway-staking. - -## In Conclusion - diff --git a/_posts/2025-01-10-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md similarity index 82% rename from _posts/2025-01-10-issuance-dynamics.md rename to _posts/2025-01-15-issuance-dynamics.md index 3276bf5..6f67fcb 100644 --- a/_posts/2025-01-10-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -9,7 +9,7 @@ this supports, and the Layer-2 blockchains, a conservative valuation might be half a trillion dollars. At the core of Ethereum's "brand", distinguishing it from other smart contract platforms, is the consistent effort put into decentralized governance. Via its -[consensus mechanism]() no centtral authority can censor a +[consensus mechanism]() no central authority can censor a transaction, freeze the native asset of a user, etc. This depends in turn on a sufficient diversity of validators staking ETH to participate in consensus. @@ -33,17 +33,16 @@ python script for converting between our variables and those most commonly used on [issuance.wtf](https://issuance.wtf). In this blog post we address the first of these concerns "runaway -staking" using a "stock and flow" macroecnomics model built with -guidance from dynamical system theory. In contrast with other -research we find inflation playing a positive but temporary role in -moderating runaway staking, though most likely $$s\to1$ eventually. -In the second post, we look more closely at governance -centralization and discuss a means for evaluating macroeconomic -interventions inspred by bifurcation theory. Briefly, we are not -optimstic that reducing issuance will prevent governance -centralization, either. - -In both posts, we provide a few code examples using `ethode` a thin +(near 100\%) staking" using a "stock and flow" macroecnomics model +built with guidance from dynamical system theory. In contrast with +other research we find inflation playing a positive but temporary role +in moderating runaway staking, though most likely $$s\to1$ eventually. +In the second post, we look more closely at governance centralization +and discuss a means for evaluating macroeconomic interventions inspred +by bifurcation theory. Briefly, we are not optimstic that reducing +issuance will prevent governance centralization, either. + +In both posts, we provide a few code examples using [ethode]() a thin units-aware wrapper we built around `scipy.odeint` to streamline model evaluation. Readers desiring to follow our derivations, dive into technical mathematical points not covered here, run their own @@ -73,16 +72,16 @@ $$\frac{dr}{d\alpha}|^\star\leq0$$ (no news here) but simultaneously fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. The bad news for opponents of runaway staking is that long -$$t\to\infty$$ term, $$\alpha^\star$$ approaches zero, so -$$s^\star\to1$$. Ethereum would undergo cycles of inflation/deflation -which we expect to die down, until an "L2 future" is reached. This is -the scenario, recognized by many others where most Ether is staked, -with the majority used for settlement of L2 rollups. +$$t\to\infty$$ term, the inflation fixed point $$\alpha^\star$$ +approaches zero, so $$s^\star\to1$$. Ethereum would undergo cycles of +inflation/deflation which we expect to die down, until an "L2 future" +is reached. This is the scenario, recognized by many others where +most Ether is staked, with the majority used for settlement of L2 +rollups. Given all the above, we advise caution. Intervening to reduce the issuance yield curve seems quite capable of exaccerbating the very -problems we seek to avoid, or causing even worse problems; if you -think Near-100\% staking is bad, have you tried Near-0\%? +problems we seek to avoid, or causing even worse problems. ## Modeling an Open Zeppelin[^humor] @@ -144,9 +143,10 @@ S^\bullet(t')dt'$, etc. In the case of issuance, we can express spot issuance as a known function of the yield curve $I^\bullet = y^\bullet S^\bullet$. We'll assume that -1. Issuance is sublinear $\boxed{I\approx yS\ll S}$,[^asym] -sufficient for [discouragement](), and that -2. the large-stake scaling exponent is not substantially altered by time averaging +1. Issuance is sublinear $\boxed{I\approx yS\ll S}$[^asym] for +[discouragement](), and that +2. the large-stake scaling of yield (like, on a log-log plot) is not +substantially altered by time averaging $\frac{\partial{d\log~y}}{\partial{d\log~S}}\approx \frac{\partial{d\log~y^\bullet}}{\partial{d\log~S^\bullet}}$. @@ -206,10 +206,14 @@ variables and time, so the burn: $b(A,S,t)=B/F$.[^time] | Unstaking | $Q_-$ | $S\to U$ | $Q_-\leq S$ | $q_-:=Q_-/S | $q_-\in(0,1)$ \[1/yr\] | The use of intensive variable parameters and the approximation -$\dot{V}\approx0$ allows us to reshape our conceptual model into one +$\dot{V}\approx0$[^flowfrac] allows us to reshape our conceptual model into one that is defined in its own dynamical and intensve variables, a *dynamical system*. We'll build this up one step at a time. + + +## Infation and Staking Fraction + ### Supply growth, Crudely in One Dimension Consider $\dot{A}=I-B-J$ in light of the [above @@ -219,247 +223,176 @@ that of staked ETH. As a crude approximation, lets temporaruly hold staking fraction $s=S/A$ constant. $$\displaystyle -\dot{A}=\left(y(sA)-bf(1-s)-js\right)A=y_0(1)\sqrt{sA}-[bf(1-s)+js]A +\dot{A}=\left(y(sA)-\beta(1-s)-js\right)A=y_0(1)\sqrt{sA}-[bf(1-s)+js]A $$ +You can explore this system using the following code. + +```python +class Acrude(ODESim): + beta: One/Yr + j: One/Yr + y1: One/Yr = 166.3 / Yr + def yield(S:ETH, *args) -> 1/Yr: return y1 / math.sqrt(S) + @staticmethod + def func(A:ETH, t:Yr, p:Params) -> ETH/Yr: + gain = yield(p.s * A) + loss = p.beta * (1 - p.s) + p.j * p.s + return yield(p.s * A) * v - p.beta * ETHODE +``` + +If you have an opinion about $j,\beta$ you can replace the parameter entries +with functions, like what we did for `yield`. + +```python +``` + +$A$ grows when the right hand side (RHS) is positive, and shrinks +otherwise. A *fixed point* is a point at which $\dot{A}=0$. -Pretending $s=s^\star$ constant, we obtain the *fixed point* $A^\star$ -at which $\dot{A}=0$. The trivial fixed point $A=0$ is unstable, and -the non-trivial fixed point is +Pretending $s=s^\star$ constant, we obtain the *fixed points* $A^\star$ +at which $\dot{A}=0$. + +Because $0<\beta,j<1$ these variable parameters cannot contribute +fixed points themselves, only influence where they occur[^params] Let +$\beta^\star=f(A^\star,s^\star,\ldots)$ represent the burn fractional +rate $B/U$ evaluated at the fixed point. There are two trivial fixed +points, trivial $A^\circ=0$, and a non-trivial fixed point $$\displaystyle A^\star = \left( -\frac{y(1)}{b^\star f^\star(1-s^\star)+j^\star s^\star} +\frac{y(1)}{\beta^\star(1-s^\star)+j^\star s^\star} \right)^2 s^\star $$ -Where $f^\star=f(A^\star,s^\star,\ldots)$ represents tx fee fractional -rate $f$ evaluated at the fixed point. A fixed point represents a -market equilibrium when it is stable. A fixed point is locally stable -just when small changes (*perturbations*) shrink. This is governed by -linearizing about the fixed point, and evaluating the sign of -$$\left.\frac{\partial\dot{A}}{\partial A}\right|^\star$, visualized below +In general the *stability* of a fixed point for a one-dimensional flow is +determined by the derivative of the RHS at the fixed point: if its +negative, then small perturbations shrink and the fixed point is +stable. Otherwise, it is unstable.[^centers] ![1D Stability Conditions](../assetsPosts/1d-stab.png) -This reduces after some calculus and algebra to a sufficient condition -on the loss term: +Specifically, the sign of $\left.\frac{\partial\dot{A}}{\partial +A}\right|^\star$ determines wether $A^\star$ is (un)stable. After +simne manipulations, a *sufficient*[^if-vs-iff] condition is $$\displaystyle -\left.\frac{\partial\log~B}{\partial\log~A}\right|^\star,~ -\left.\frac{\partial\log~J}{\partial\log~A}\right|^\star~~>~~\frac{1}{2}. +\left.\frac{\partial\log~B}{\partial\log~A}\right|^\star~~\mathrm{or}~~ +\left.\frac{\partial\log~J}{\partial\log~A}\right|^\star~~~>~~~\frac{1}{2}. $$ If either is satisfied, the fixed point is stable, and $A^\star$ -represents the equilibrium supply value. Note that constant $bf,j$ -correspond to scaling exponent $1$, so satisfy. Essentially some -component of the loss term must increase faster than $\sqrt{A}$ around -the fixed point. - -If we think of the burn as a representation of economic activity on -Ethereum, this raises a disturbing possibility, in which supply $A$ -keeps growing but economic activity does not scale at least as fast as -$\sqrt{A}$. That is inflation, not just in the sense of supply -expansion, but the demand not growing at pace with the supply. This -is similar to the concern raised by [Elowsson 2020](). - -However, we have held $s$ consant; this is not *really* a one -dimensional flow. To really understand the dynamics we need to model -both $A,s$ as dynamical variables. - -### Two Dimensional $(A,s)$ System - -After an algebraic massage, we can obtain for staking fraction - -$$\displaystyle -\dot{s} = - -### General $(A,\alpha,s)$ System +represents the equilibrium supply value, at which Ether acheives +zero-inflation. Note that constant $\beta,j$ correspond to scaling +exponent $1$, so satisfy. + +Essentially the leading component of the loss term must increase +faster than $\sqrt{A}$ near the fixed point. Assuming negligible +slashing, there must be sufficient economic activity so that the burn +scales at least as the square-root of supply, or else $A^\star$ is +unbounded. That is true inflation, not just in the sense of supply +expansion, but the *demand not growing sufficiently with the supply*. +This is different but similar to the concern raised by [Elowsson +2020](). We address such concerns here. -#### Aas system +### Positive Inflation cannot maintain -### Reflexivity and Inflation-Response +What happens to equilibrium supply $A^\star$ then, if $s\to1$, slashing +is negligible $A^\star\to\infty$? -the conceptual source of potential oscillations can be considered by -modelling the SU system: +Inflation is used to refer to many things, but here we mean +specifically the quarterly fractional change in accessible +Ether. Let's express inflation $\alpha:=\frac{d\log~A}{dt}$ in +terms of $s$ and the intensives $$\displaystyle -\begn{array}{rcrlcrl} -\dot{S} &=& (ry-\jmath-q_-) & S & + & \left(q_++r(1-b)f\right) & U\\ -\dot{U} &=& \left((1-r)y+q_-\right) & S & - & \left(rf+(1-r)bf+q_+\right) & U\\ -\end{array} +\begin{array}{rcl} +\dot{A} &=& \alpha A\\ +\alpha &=& (I-B-J)/A \approx y(sA)s - bf(1-s) - js $$ -- we have not emphasized the oscillation because we expect market -participants can profit from it without having to coordinate their -behavior; oscillations in supply will die down if they are big enough -to arbitrage: buy toward the end of an inflationary phase, sell toward -the end of a deflationary phase. - -- contrast this to runaway staking... there's no obvious mechanism to - profit from an increase in staking fraction, other than by joining - in. So this is a challenge: can you, the reader, design a - cryptoeconomic mechanism by which runaway staking is moderated, in a - way that allows individuals to profit? Can you prove that is impossible? - -- we have not explicitly modelled this kind of reflexivity; it is swept into - the partial derivatives. - -Could these same partial derivatives be large enough to magnify -inflation-concerns into runaway staking even at intermediate -timescales? This is absolutely possible, and would require the following: - - +A key feature of $\dot{A} concerns discouragement [Buterin](), which +requires sublinear issuance $I\lesssim S$ to avoid instability to +certain attacks. Because of this, the positive term continues to get +smaller as Accessible ETH supply grows; consistent with the +$(S,U)$-model above, eventually average inflation tends to zero. +In contrast, while slashing can go to zero, so long as blocks are +produced, these is some minimal burn. $A$ could get quite large; for +$s=1-\epsilon$ and the current yield curve we have $A^\star\sim +(y(1)/b^\star f^\star)^2\epsilon^{-2}$; as $\epsilon$ is very small +$\epsilon^{-2}$ is very large. Nonetheless, we should expect +$\alpha^\star=0$ to be the inflation fixed point as $t\to\infty$. +- tighten alpha -> 0 argument: y(A * s) >= alpha(A,s) forall A,s + So A(t) is bounded above by Y = int( y(A,s) ): otherwise they must cross + at which point A(t), s(t) must equal the surface Y(t,s), and + the derivative of Adot must be greater at whatever instantaneous value + of s is reached, which contradicts y(A * s) >= alpha(A,s) + if Ydot is to sustain a positive inflation rate its leading + term must be at least constant (exponential growth), and it is not, + so A(t,s) does not cross, is bounded above by a subexponential function + for all time. +In such models noise generally causes the value to bounce around near +the fixed point.[^goldenfeld] -#### Simulating the $(S,U)$ Model -You can explore some behaviors of this model using a simulation; see -[guide](). Here is an unrealistically simplified example where all -coefficients are held constant that still extracts some broad -features. -```python -from ethode import * -class Toy(ODESim): - cSS: 1/Yr - cSU: 1/Yr & Pos - cUS: 1/Yr & Pos - cSS: 1/Yr & Pos - @staticmethod - def func(v:UnitTuple(ETH, 2), t:Yr, p:Params) -> UnitTuple(ETH/Yr, 2): - S, U = v - dS = p.cSS * S + p.cSU * U - dU = p.cUS * S - p.cUU * U - return dS, dU -Toy(1, 1, 1, 1).sim() -Toy(-1, 1, 1, 1).sim() -``` +### Two Dimensional $(A,s)$ System -Specifically, so long as $ry>\jmath+q_-$ staked ETH $S$ just continues -growing and growing, while $U$ cannot get too big, before its own loss -term $-\left(rf+(1-r)bf+q_+\right)\cdot U$ dominates. At some point -$S$ becomes big enough that $ry<\jmath+q_-$ and the system is capable -of oscillation, depending on a zoo of partial derivatves. Here is a -less trivial code example where all intensves other than issuance yield are -held constant. +So far this is a purely *mathematical* exercise. We know well that +$s$ is *not constant*. We already have equations for +$\dot{A},\dot{S}$, what about $\dot{s}=d(S/A)/dt$? Using the quotient +rule $\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}$, and after an +algebraic massage, we obtain for staking fraction -```python -from math import sqrt -from ethode import * -class SUconst(ODESim): - y1: 1/Yr & Pos - f: 1/Yr & PosFrac - b: NoDim & PosFrac - r: NoDim & PosFrac - qu: 1/Yr & PosFrac - qs: 1/Yr & PosFrac - j: 1/Yr & PosFrac - S1: ETH = 1 * ETH - def y(self, S:ETH, *args) -> ETH/Yr: - return self.y1 * sqrt(self.S1 / S) - @staticmethod - def func(v:UnitTuple(ETH,2), t:Yr, p:Params) -> UnitTuple(EPY, 2): - x = {'S': (S = v[0]), - 'U': (U = v[1])} - y = p.y(**x) - beta = p.b * p.f - rf = p.r * p.f - dS = (p.r * y - p.j - p.qu) * S + (rf * (1 - p.b) + p.qs) * U - dU = ((1 - p.r) * y + p.qu) * S - (rf + (1 - p.r) * beta + p.qs) * U - return dS, dU -su = SUConst.sim(y1 = 166.3 / Yr, f = .002 / Yr, \ - b = .8, r = .5, \ - qu = 1e-4 / Yr, qs = 1e-2 / Yr, j = 1e-6 / Yr) -su.verify() -su.sim() -``` +$$\displaystyle +\begin{array}{rcl} +\dot{s} &=& y(sA)\cdot(r-s) + \\ + && \left[q_++f(1-s)\left(bs +(1-b)r\right)\right]\cdot(1-s) + \\ + && \left[j(1-s+r)+q_-\right]\cdot(0-s). +\end{array} +$$ -If you have an opinion about how, for instance $r$ should depend on -$(S,U)$, you can define, instead of `r: NoDim` your own function -within the class declaration. +The coefficients of $(r-s),~(1-s),~(0-s)$ are variable but *positive*. +Recalling how $s$ increases just when $\dot{s}>0$, these terms draw +$s$ toward respective points $r,1,0$. We emphasize that the action of +yield $y$ is $x\to r$, *not* $x\to1$, unless $r\geq1$. -```python - def rnvst(self, S:ETH, U:ETH, *args) -> NoDim: - return S * U / (S + U) ** 2 / 2 -``` +This is quite a complicated equation, but we finally have a dynamical +system. -Of course the variables we really care about are inflaion and staking -fraction. You can designate these as outputs of the model withn -the class declaration. +You can simulate it! You can simulate it, adding inflation +as an output as follows: ```python - @output - def alpha(self, S:ETH, U:ETH, *args) -> 1/Yr: - return p.y(S) * S - p.b * p.f * U - p.j * S - @output - def sfrac(self, S:ETH, U:ETH, *args) -> 1/Yr: - return S / (S + U) - ... -su.sim(graph_outputs = ('alpha', 'sfrac')) ``` -Below we transform from staked/unstaked variables $(S,U)$ into -modeling directly accessible supply, inflation, and staking $(A,\alpha,s)$. -But first we need to emphasize what, in our view, makes our model -different from others. - - - -; Only F@%$ Up the limits when it doesn't matter! - -When forming intensve variables it is critical that our choices as -model-builders reflect the correct asymptotic behavior in the limits -of concern, in his case $U\to 0$. If for instance, instead of -assuming $R\leq I+P$, we had assumed reinvestment $R$ is bounded by -unstaked ETH $R\leq U$ we would get a qualitatively different model. -But this is wrong in a way that matters! It is wrong because -validator rewards $I+P$ can vary or in principal go to zero -independently $U$. It matters because the reinvestment of rewards -represents a sustainable feedback loop. - -both an -exceedingly large supply and dangerously low staking could shrink -$I+P$ to the same value, yet $U$ looks very different in those cases. - - - -Other examples abound.[^elowex] - - - -## Infation and Staking Fraction +### General $(A,\alpha,s)$ System -Inflation is used to refer to many things, but here we mean -specifically the quarterly fractional change in accessible -Ether. Let's express inflation $\alpha:=\frac{d\log~A}{dt}$ in -terms of $s$ and the intensives +If we know $alpha=\dot{A}/A$ and a starting value, we know $A$. There +are many reasons, especially in the context of the world economy, to +care about total circulating supply $A$. Recent discussions however +have focused on inflation. It also turns out that modeling +inflation $\alpha$ directly simplifies our analysis considerably. $$\displaystyle -\alpha = (I-B-J)/A \approx y(sA)s - bf(1-s) - js +\begin{array}{rcl} +\dot{A} &=& \alpha A\\ +\dot{s} &=& \alpha(r-s) + (rf+q_+)(1-s) + (q_-+(1-r)j)(0-s) +\dot{\alpha} &=& \pm\xi\dot{s} - \gamma\alpha s + \chi +\end{array} $$ -### Positive Inflation cannot maintain - -A key feature of $\dot{A} concerns discouragement [Buterin](), which -requires sublinear issuance $I\lesssim S$ to avoid instability to -certain attacks. Because of this, the positive term continues to get -smaller as Accessible ETH supply grows; consistent with the -$(S,U)$-model above, eventually average inflation tends to zero. - -In contrast, while slashing can go to zero, so long as blocks are -produced, these is some minimal burn. $A$ could get quite large; for -$s=1-\epsilon$ and the current yield curve we have $A^\star\sim -(y(1)/b^\star f^\star)^2\epsilon^{-2}$; as $\epsilon$ is very small -$\epsilon^{-2}$ is very large. Nonetheless, we should expect -$\alpha^\star=0$ to be the inflation fixed point as $t\to\infty$. +xi -- +gamma -- +chi -- ### Changing Variables @@ -497,8 +430,10 @@ $$ * $X= ...$ represents externallities to the $(A,\alpha,s)$ system * $\zeta = ... $ -At intremediate times we believe inflation can be treated as a parameter instead of as its own dynamic -variable +#### Separation of Timescales + +For short periods at intremediate times we believe inflation can be +treated as a parameter instead of as its own dynamic variable $\dot{s}\ll\dot{\alpha}$ @@ -514,7 +449,9 @@ the added precision of a more sophisticated treatment. We will proceed assuming $$\dot{\alpha}\approx0$$ and treating $$\alpha$$ as a parameter. -So, let us examine the fixed point $x^\star$ +#### s^\star + +So, let us examine the fixed point $s^\star$ We find that if $\dot{A}=0=\alpha$ (no inflation nor deflation) then an interior market equilibrium $$s^\star<1$$ requires ver high slashing. We reason @@ -590,6 +527,48 @@ How these transient values $$(\alpha/f,\,r)$$ relate to their equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ depends on some considerations. +### Reflexivity and Inflation-Response + +the conceptual source of potential oscillations can be considered by +modelling the SU system: + +$$\displaystyle +\begn{array}{rcrlcrl} +\dot{S} &=& (ry-\jmath-q_-) & S & + & \left(q_++r(1-b)f\right) & U\\ +\dot{U} &=& \left((1-r)y+q_-\right) & S & - & \left(rf+(1-r)bf+q_+\right) & U\\ +\end{array} +$$ + + +- we have not emphasized the oscillation because we expect market +participants can profit from it without having to coordinate their +behavior; oscillations in supply will die down if they are big enough +to arbitrage: buy toward the end of an inflationary phase, sell toward +the end of a deflationary phase. + +- contrast this to runaway staking... there's no obvious mechanism to + profit from an increase in staking fraction, other than by joining + in. So this is a challenge: can you, the reader, design a + cryptoeconomic mechanism by which runaway staking is moderated, in a + way that allows individuals to profit? Can you prove that is impossible? + +- we have not explicitly modelled this kind of reflexivity; it is swept into + the partial derivatives. + + +Could these same partial derivatives be large enough to magnify +inflation-concerns into runaway staking even at intermediate +timescales? This is absolutely possible, and would require the following: + + + + + + + + + + @@ -865,6 +844,12 @@ So we have used approximations that we felt err in a conservative direction with depenedence on the present-day yield curve $y^\bullet_0(S^\bullet)=y_0(1)\sqrt{S_1/S}$ where $y_0(1)\approx166.3$/yr and $S_1=1$ETH. +[^params]: Variable parameters that are positive fractions cannot +contribute fixed-points themselves, but they can strongly influence +*where* a fixed point is. Example: as $s\to1$, if the leading terms +were $\beta\sim(1-s),j\sim(1-s)^2$ this gives increasingly larger +$A^\star$ as $s\to1$. + diff --git a/assetsPosts/2024-12-30-issuance-dynamics/YCharts-alpha.jpg b/assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg similarity index 100% rename from assetsPosts/2024-12-30-issuance-dynamics/YCharts-alpha.jpg rename to assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg diff --git a/assetsPosts/2024-12-30-issuance-dynamics/YCharts-x.jpg b/assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg similarity index 100% rename from assetsPosts/2024-12-30-issuance-dynamics/YCharts-x.jpg rename to assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg diff --git a/assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg b/assetsPosts/2025-01-15-issuance-dynamics/eth-balloon.jpg similarity index 100% rename from assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg rename to assetsPosts/2025-01-15-issuance-dynamics/eth-balloon.jpg diff --git a/assetsPosts/2024-12-30-issuance-dynamics/staking-fixpoint.png b/assetsPosts/2025-01-15-issuance-dynamics/staking-fixpoint.png similarity index 100% rename from assetsPosts/2024-12-30-issuance-dynamics/staking-fixpoint.png rename to assetsPosts/2025-01-15-issuance-dynamics/staking-fixpoint.png From 624f1c9b4210842af7c4ad88f7e936c5c792f39b Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Fri, 10 Jan 2025 16:59:19 -0600 Subject: [PATCH 31/70] fianl ordering --- _posts/2025-01-15-issuance-dynamics.md | 646 ++++++++++++++----------- 1 file changed, 366 insertions(+), 280 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 6f67fcb..2ea723c 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -42,12 +42,12 @@ and discuss a means for evaluating macroeconomic interventions inspred by bifurcation theory. Briefly, we are not optimstic that reducing issuance will prevent governance centralization, either. -In both posts, we provide a few code examples using [ethode]() a thin +In both posts, we provide a few code examples using [ethode](https://github.com/20squares/ethode/) a thin units-aware wrapper we built around `scipy.odeint` to streamline model evaluation. Readers desiring to follow our derivations, dive into technical mathematical points not covered here, run their own simulations, or learn some dynamical systems are recommended to look -at our ethode [guide](), and references therein. It is certainly +at our ethode [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md), and references therein. It is certainly incomplete, but should have enough to get you going. ## For The Impatient! @@ -75,7 +75,8 @@ The bad news for opponents of runaway staking is that long $$t\to\infty$$ term, the inflation fixed point $$\alpha^\star$$ approaches zero, so $$s^\star\to1$$. Ethereum would undergo cycles of inflation/deflation which we expect to die down, until an "L2 future" -is reached. This is the scenario, recognized by many others where +is reached, which we'll discuss next time. +This is the scenario, recognized by many others where most Ether is staked, with the majority used for settlement of L2 rollups. @@ -86,7 +87,7 @@ problems we seek to avoid, or causing even worse problems. ## Modeling an Open Zeppelin[^humor] ![Ethereum as a balloon with compartments.]( - ../assetsPosts/2024-12-30-issuance-dynamics/eth-balloon.jpg) + ../assetsPosts/2025-01-15-issuance-dynamics/eth-balloon.jpg) Consider a "balloon" with variable internal compartments. The average size of each is measured by *stocks* @@ -117,13 +118,13 @@ $$ | Flow Name | Symbol | Domain$\to$Codomain[^cats] | Constraint | | :-- | :-- | :-: | :-- | -| Tx Fees | $F$ | $U\to\cancel{O},SS$ | $B+P=F$ | +| Tx Fees | $F$ | $U\to\cancel{O},V$ | $B+P=F 1/Yr: return y1 / math.sqrt(S) + s: One = .5 * One + def yield(self, S:ETH) -> One/Yr: return self.y1 / math.sqrt(S) @staticmethod def func(A:ETH, t:Yr, p:Params) -> ETH/Yr: - gain = yield(p.s * A) + gain = p.yield(p.s * A) loss = p.beta * (1 - p.s) + p.j * p.s - return yield(p.s * A) * v - p.beta * -ETHODE + return (gain - loss) * A +model = ACrude(beta = 8e-3/Yr, j = 1e-6/Yr, y1 = 166.3/Yr, s = .3 * One) +model.sim() ``` -If you have an opinion about $j,\beta$ you can replace the parameter entries -with functions, like what we did for `yield`. +If you have an opinion about $j(A),\beta(A)$ you can replace the parameter +entries with functions, like what we did for `yield`: replace `pass` +with your desired behavior, using `self.b` etc. to access parameters. +Within `func` use `p.burn` instead of `self.burn`; `ODESim` takes care +of the plumbing. ```python +class ALessCrude(ACrude): + def slash(self, A:ETH) -> One/Yr: + pass # Your insights here + def burn(self, A:ETH) -> One/Yr: + pass # Your insights here + @staticmethod + def func(A:ETH, t:Yr, p:Params) -> ETH/Yr: + gain = p.yield(p.s * A) * A + loss = p.slash(A) + p.burn(A) + return gain - loss ``` $A$ grows when the right hand side (RHS) is positive, and shrinks -otherwise. A *fixed point* is a point at which $\dot{A}=0$. - -Pretending $s=s^\star$ constant, we obtain the *fixed points* $A^\star$ -at which $\dot{A}=0$. - -Because $0<\beta,j<1$ these variable parameters cannot contribute -fixed points themselves, only influence where they occur[^params] Let -$\beta^\star=f(A^\star,s^\star,\ldots)$ represent the burn fractional -rate $B/U$ evaluated at the fixed point. There are two trivial fixed -points, trivial $A^\circ=0$, and a non-trivial fixed point +otherwise. Pretending $s=s^\star$ constant, we obtain the *fixed +points* $A^\star$ at which $\dot{A}=0$. Because $0<\beta,j<1$ the +parameters cannot contribute fixed points themselves, only influence +where they occur.[^params] Let $\beta^\star=f(A^\star,s^\star,\ldots)$ +represent the burn fractional rate $\beta=bf=B/U$ evaluated at the fixed point. +There are two trivial fixed points, trivial $A^\circ=0$, and a +non-trivial fixed point $$\displaystyle A^\star = \left( -\frac{y(1)}{\beta^\star(1-s^\star)+j^\star s^\star} -\right)^2 s^\star +\frac{y(1)\sqrt{s^\star}}{\beta^\star(1-s^\star)+j^\star s^\star} +\right)^2 $$ +### Stability in One Dimension + In general the *stability* of a fixed point for a one-dimensional flow is determined by the derivative of the RHS at the fixed point: if its negative, then small perturbations shrink and the fixed point is stable. Otherwise, it is unstable.[^centers] ![1D Stability Conditions](../assetsPosts/1d-stab.png) - Specifically, the sign of $\left.\frac{\partial\dot{A}}{\partial A}\right|^\star$ determines wether $A^\star$ is (un)stable. After -simne manipulations, a *sufficient*[^if-vs-iff] condition is +some manipulations, a *sufficient*[^if-vs-iff] condition for +stability is $$\displaystyle \left.\frac{\partial\log~B}{\partial\log~A}\right|^\star~~\mathrm{or}~~ @@ -283,23 +310,23 @@ $$\displaystyle $$ If either is satisfied, the fixed point is stable, and $A^\star$ -represents the equilibrium supply value, at which Ether acheives -zero-inflation. Note that constant $\beta,j$ correspond to scaling -exponent $1$, so satisfy. - -Essentially the leading component of the loss term must increase -faster than $\sqrt{A}$ near the fixed point. Assuming negligible -slashing, there must be sufficient economic activity so that the burn -scales at least as the square-root of supply, or else $A^\star$ is -unbounded. That is true inflation, not just in the sense of supply -expansion, but the *demand not growing sufficiently with the supply*. -This is different but similar to the concern raised by [Elowsson -2020](). We address such concerns here. +represents the equilibrium supply value, at which Ether acheives zero +growth. Essentially the leading component of the loss term must +increase faster than $\sqrt{A}$ near the fixed point. Note that +constant $\beta,j$ correspond to scaling exponent $1$, so satisfy. + +To simplify, assume negligible slashing; there must be sufficient +economic activity so that the burn scales at least as the square-root +of supply, or else $A^\star$ is unbounded. That is true inflation, +not just in the sense of supply expansion, but the *demand not growing +sufficiently with the supply*. This is similar to the concern raised +by [Elowsson 2020](). ### Positive Inflation cannot maintain -What happens to equilibrium supply $A^\star$ then, if $s\to1$, slashing -is negligible $A^\star\to\infty$? +What happens to equilibrium supply $A^\star$ then, if $s\to1$ and +slashing is negligible, or the partials $\partial_AB,\partial_AJ$ are +insufficient such that $A^\star\to\infty$? Inflation is used to refer to many things, but here we mean specifically the quarterly fractional change in accessible @@ -310,44 +337,29 @@ $$\displaystyle \begin{array}{rcl} \dot{A} &=& \alpha A\\ \alpha &=& (I-B-J)/A \approx y(sA)s - bf(1-s) - js +\end{array} $$ - - -A key feature of $\dot{A} concerns discouragement [Buterin](), which -requires sublinear issuance $I\lesssim S$ to avoid instability to -certain attacks. Because of this, the positive term continues to get -smaller as Accessible ETH supply grows; consistent with the -$(S,U)$-model above, eventually average inflation tends to zero. - -In contrast, while slashing can go to zero, so long as blocks are -produced, these is some minimal burn. $A$ could get quite large; for -$s=1-\epsilon$ and the current yield curve we have $A^\star\sim -(y(1)/b^\star f^\star)^2\epsilon^{-2}$; as $\epsilon$ is very small -$\epsilon^{-2}$ is very large. Nonetheless, we should expect -$\alpha^\star=0$ to be the inflation fixed point as $t\to\infty$. - -- tighten alpha -> 0 argument: y(A * s) >= alpha(A,s) forall A,s - So A(t) is bounded above by Y = int( y(A,s) ): otherwise they must cross - at which point A(t), s(t) must equal the surface Y(t,s), and - the derivative of Adot must be greater at whatever instantaneous value - of s is reached, which contradicts y(A * s) >= alpha(A,s) - if Ydot is to sustain a positive inflation rate its leading - term must be at least constant (exponential growth), and it is not, - so A(t,s) does not cross, is bounded above by a subexponential function - for all time. - -In such models noise generally causes the value to bounce around near -the fixed point.[^goldenfeld] - - - - +A key feature of $\dot{A} under the current yield curve is sublinear +issuance $I\leq yS\lesssim S$, chosen to avoid [discouragement +attacks](https://raw.githubusercontent.com/ethereum/research/master/papers/discouragement/discouragement.pdf). +Because of this, the positive term continues to get smaller as +Accessible ETH supply grows; $\dot{A}\leq y(As)A$, under the current +yield curve $\dot{A}(s,A)\leq y_0(1)\sqrt{sA}$. So, at any point, +given supply $A$, the inflation rate $\alpha = \dot{A}/A$ cannot be +larger than $\alpha_{max}=y_0(1)\sqrt{s/A}$. Constant positive +inflation corresponds to $A$ growing, and for +any positive constant $\alpha=\epsilon$ there is some $A$ at which +$\epsilon=\alpha_{max}(A)$. Total circulating suopply $A$ is +bounded above by a subexponential function for all time. + +That is, inflation must eventually return toward zero, even in the +pressence of very weak burn, slashing, and very high staking fraction. +This is not emphasized enough in our discussions. ### Two Dimensional $(A,s)$ System -So far this is a purely *mathematical* exercise. We know well that -$s$ is *not constant*. We already have equations for +We know well that $s$ is *not constant*. We have equations for $\dot{A},\dot{S}$, what about $\dot{s}=d(S/A)/dt$? Using the quotient rule $\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}$, and after an algebraic massage, we obtain for staking fraction @@ -365,103 +377,146 @@ Recalling how $s$ increases just when $\dot{s}>0$, these terms draw $s$ toward respective points $r,1,0$. We emphasize that the action of yield $y$ is $x\to r$, *not* $x\to1$, unless $r\geq1$. -This is quite a complicated equation, but we finally have a dynamical -system. +As expressed by the quotient rule, an increase in staking fraction +can be driven by more people staking, and/or it can be driven by a +reduction of the inflation rate. The latter can be acheived in +principle by a reduction of issuance relative to the base fee "burn +rate". Because of this quotient rule tradeoff, issuance plays a +beneficial "infrastructure" role in moderating staking fraction, +drawing itb toward $r$, which may be less than one. -You can simulate it! You can simulate it, adding inflation -as an output as follows: +Other aspects are harder to read, as this is quite a complicated +equation, but we finally have a dynamical system. You can simulate a +general version of it, filling in your desired behavior for `pass` and +adding inflation as an output, as follows: ```python +class AsSys(ODESim): + def yield(self, S:ETH) -> One/Yr: pass + def burnfrac(self, A:ETH, s:One) -> One: pass + def feefracrate(self, A:ETH, s:One) -> One/Yr: pass + def slashfracrate(self, A:ETH, s:One) -> One/Yr: pass + def rnvstfrac(self, A:ETH, s:One) -> One: pass + def qstake(self, A:ETH, s:One) -> One/Yr: pass + def qunstake(self, A:ETH, s:One) -> One/Yr: pass + def _hlpr(self, A:ETH, s:One) -> tuple[One/Yr,One/Yr,One,One/Yr]: + beta = (b := self.burnfrac(A,s)) * (f := self.feefracrate(A,s)) + return s * self.slashfracrate(A,s), beta * (1 - s), b, f + @output + def inflation(self, A:ETH, s:One) -> One/Yr: + slash, burn = self._hlpr(A,s)[:2] + return self.yield(A * s) - burn - slash + @staticmethod + def func(v:tuple[ETH,One], t:Yr, p:Params) -> tuple[ETH/Yr,ETH/Yr]: + x = {'A': (A := v[0]), 's': (s := v[1])} + slash, burn, b, f = p._hlpr(A, s) + to0 = j * (1 - s - (r := p.rnvstfrac(**x))) + p.qunstake(**x) + to1 = p.qstake(**x) + f * (1-s) * (b * s + (1-b) * r) + dA = p.inflation(A,s) * A + ds = yld * (r - s) + to1 * (1 - s) - to0 * s + return dA, ds ``` -### General $(A,\alpha,s)$ System +# The $(A,\alpha,s)$ Dynamical System + +There are many reasons, especially in the context of the world +economy, to care about total circulating supply $A$. Recent +discussions however have focused most on inflation. It also turns out +that modeling inflation $\alpha$ directly simplifies our analysis +considerably. -If we know $alpha=\dot{A}/A$ and a starting value, we know $A$. There -are many reasons, especially in the context of the world economy, to -care about total circulating supply $A$. Recent discussions however -have focused on inflation. It also turns out that modeling -inflation $\alpha$ directly simplifies our analysis considerably. +Let subscripts denote partial derivatives +$x_y:=\frac{\partial{x}}{\partial{y}}$. Using the product rule +$\dot{xy}=\dot{x}y+x\dot{y}$ and the correct partial derivative +relations for variables $(A,\alpha,s,t)$[^partial] we have $$\displaystyle \begin{array}{rcl} -\dot{A} &=& \alpha A\\ -\dot{s} &=& \alpha(r-s) + (rf+q_+)(1-s) + (q_-+(1-r)j)(0-s) -\dot{\alpha} &=& \pm\xi\dot{s} - \gamma\alpha s + \chi +\dot{A} &=& \pm\alpha A\\ +\dot{s} &=& \pm\alpha(r-s) + (rf+q_+)(1-s) - (q_-+(1-r)j)s +\dot{\alpha} &=& \pm\frac{\xi}{1+\mu}\dot{s} - \frac{\gamma}{1+\mu}\alpha s \pm \frac{\chi}{1+\mu} \end{array} $$ -xi -- -gamma -- -chi -- +Where the new greek letters are fractional rates, defined below, and +$y':=\frac{dy}{dS}$. -### Changing Variables +* $\mu:=\beta_\alpha(1-s)+\jmath_\alpha s$ is the implicit + sensitivity of the inflation loss term to increases in inflation. + We judge $0\leq\mu$; if anything inflation increases burn and + slashing fractional rates.[^mu] -So, our new dynamical variables are $(A,\alpha,s)$. Two important -points now require mention. +* $\xi:=y+y'A+\beta-\beta_s(1-s)-j-j_s$ is the net + correlation between changes in $s$ and changes in $\alpha$, it can + be of either sign. Under the current yield curve + $y_0+\frac{dy}{dS}A=y_0(sA)(1-1/(2s))$. -* The quotient rule $\dot{s}=\frac{\dot{S}{A}-s\alpha$ emphasizes that - changes in a fraction occur from *both* changes in the numerator $S$ - and its denominator $A$. - -* Making conclusions about market equilibria requires us to look at *all* - of the dynamical variables, and perform a [local stability analysis](). - - Solve $\dot{x}:=(\dot{A},\dot{\alpha},\dot{s})=(0,0,0)$, obtaining a fixed point - $x^\star=(A^\star,\alpha^\star,s^\star)$. - - This $x^\star$ is a market equilibrium when it is (possibly meta-)stable, and - *the* market equiibrium just when it is unique.[^metastable] +* $\gamma:=\left(j_As+\beta_A(1-s)+s|y'|\right)A$ is a positive + coefficient expressing how quickly $\alpha\to\alpha^\star$, and the + partials are constant when initial supply is known.[^ics] We have + extracted the sign from the final term because sublinear issuance + implies $y'<0$; under the current yield curve the term + $sA|y'|=\frac{1}{2}y_0(sA)$. -That is, an increase in staking fraction can be driven by more people -staking, and/or it can be driven by a reduction of the inflation rate. -The latter can be acheived in principle by a reduction of issuance -relative to the base fee "burn rate". Because of this quotient rule -tradeoff, low-but-positive inflation actually plays a positive almost -"infrastructure" role in moderating staking fraction. +* $\chi:=-j_ts-\beta_t(1-s)$ represents externalities affecting the + inflation loss term encoded as explicit time-dependencies. -After some calculus, and a lot of algebra, we obtain the following +In what follows we will obtain and asses the nature of the long-term +externality-free fixed point cooridnate $(\alpha^\star=0,s^\star=1)$. +We will observe that it is not hard for $|\dot{\alpha}|\ll|\dot{s}|$ +to obtaon, leading to a seperation of timescales where an $\alpha>0$ +may change slowly enough that we can still use equilibrium arguments +at intermediate times. We extract the behavior of +$x^\star(\alpha~\mathrm{const.},\ldots)$ and break down the fixed +point stability into LI;ELF vs. deflation. We close with a dicussion +of the potential for oscillations in our model vs. actual market data. + +## Local Stability of $\alpha^\star=0$ and Oscillations -$$\displaystyle -\begin{array}{rcl} -\dot{A} &=& \alpha A\\ -\dot{\alpha} &=& \zeta\dot{s}-\gamma\alpha s + X\\ -\dot{s} &=& \alpha\cdot(r-s)~+~(rf+q_+)\cdot(1-s)~-~\eft(q_-+(1-r)j\right)\cdot s\\ -\end{array} -$$ -* $X= ...$ represents externallities to the $(A,\alpha,s)$ system -* $\zeta = ... $ -#### Separation of Timescales -For short periods at intremediate times we believe inflation can be -treated as a parameter instead of as its own dynamic variable -$\dot{s}\ll\dot{\alpha}$ +## Separation of Timescales and $\alpha>0$ -For a sanity-check, a quick look at -YCharts since The Merge shows that $$s,\dot{s}$$ vary over a much -greater range than $$(\ln{A},\alpha)$$. +Observe that every term in $(\xi,\gamma)$ are fractions, fractional +rates, or derivatives thereof, so if $|\xi|,\gamma\ll1$ then the +derivatives obey $|\dot{\alpha}|\ll|\dot{s}|$: a [seperation of +timescales](). For durations when this obtains, short periods at +intremediate times, inflation could be treated as a parameter instead +of its own dynamic variable, with staking fraction equilibrating to +$s^\star$ more quickly than $\alpha^\star$. -![The staking fraction from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-x.jpg) -![The inflation rate from YCharts](../assetsPosts/2024-12-05-issuance-fundamentals/YCharts-alpha.jpg) +Does this hold presently? For a sanity-check, a quick look at +YCharts since The Merge shows that $$s,\dot{s}$$ do indeed vary over a much +greater range than $$(\log{A},\alpha)$$. -At present we judge the lack of empirical data on $$r$$ would obviate -the added precision of a more sophisticated treatment. We will -proceed assuming $$\dot{\alpha}\approx0$$ and treating $$\alpha$$ as a -parameter. +![The staking fraction from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg) +![The inflation rate from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg) -#### s^\star +## The Intermediate s^\star -So, let us examine the fixed point $s^\star$ +So, let us examine the fixed point $s^\star$ with $\alpha^\star$ treated as +a parameter. -We find that if $\dot{A}=0=\alpha$ (no inflation nor deflation) then -an interior market equilibrium $$s^\star<1$$ requires ver high slashing. We reason -as follows. In the absence of in/de-flation, an interior fixed point -$$s^\star<1$$ would require a persistent unstaking/capitulation of -existing validators $$q_->0$$. This in turn either requires "churn", -a persistent supply of new validators to take their place -$$q_+^\star>0$$, or it is only a transient and $$q_-^\star\approx0$$; -recall that reinvestment by existing validators is not counted in -$$q_+$$. +$$\displaystyle +s^\star = \frac{ + r^\star(\alpha^\star + f^\star) + q_+^\star +}{ + \alpha^\star + r^\star f^\star + q_+^\star + q_-^\star + \jmath^\star +} +$$ + +If $\dot{A}=0=\alpha$ (no inflation nor deflation) then an interior +market equilibrium $s^\star<1$ requires very high slashing. + +We reason as follows. In the absence of in/de-flation, an interior +fixed point $s^\star<1$ would require a persistent +unstaking/capitulation of existing validators $q_->0$. This in turn +either requires churn, a persistent +supply of new validators to take their place $q_+^\star>0$, or it is +only a transient and $q_-^\star\approx0$; recall that reinvestment by +existing validators is not counted in $q_+$. Net unstaking $$q_->0$$ could only describe a market equilibrium if one group of stakers was actively capitulating and withdrawing their @@ -479,34 +534,25 @@ $$r$$, so $$q_+^\star\approx0$$. Thus, the fixed point $$s^\star$$ simplifies to $$\displaystyle -s^\star ~~ = ~~ r^\star ~ -\frac{\alpha^\star + f^\star}{\alpha^\star + r^\star f^\star} +s^\star = r^\star \frac{ + \alpha^\star + f^\star +}{ + \alpha^\star + r^\star f^\star + \jmath^\star +} $$ -A calculation is illustrative under the current regime of positive -inflation. If inflation dominates fees, $$\alpha\gg f$$ then -$$s^\star\sim r^\star<1$$, while if fees dominate inflaton $$\alpha\ll -f$$ and $$s^\star\to1$$. For a numerical comparison, at present $$f -\approx .001<.005\approx\alpha$$ so to within 10\% error above, -$$s^\star\approx r$$ over the range of $$r$$ inferred from Lido yield -rate. +#### Stability of $x^\star$ -This positive role for inflation can be seen in the contours of the + + +### LI;ELF Conditions + +A positive role for inflation can be seen in the contours of the market equilibrium staking fraction $$s^\star$$ corresponding to $$\dot{s}=0$$, shown in [figure -1](../assetsPosts/2024-12-05-issuance-fundamentals/staking-fixpoint.png). -The equation graphed is as follows: +1](../assetsPosts/2025-01-15-issuance-fundamentals/staking-fixpoint.png). -$$\displaystyle -s^\star = r^\star\frac{\alpha^\star/f^\star + 1}{\alpha^\star/f^\star + r^\star} -$$ - -The fraction $$0\leq r\leq 1$$ is the ratio of profits reinvested -quarterly by validators, $$0\leq f\leq 1$$ is the fraction of unstaked -ETH spent on transaction fees (base and priority) quarterly, and as -above $$\alpha$$ is inflation. Here $$r^\star$$ etc. means the -function $$r(s,\alpha,\ldots)$$ at the equilibrium coordinates -$$(s^\star,\alpha^\star,\ldots)$$. To find the equilibrium values +To find the equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ necessary to acheive a desired staking fraction $$x^\star$$, simply pick a colored contour in the figure: these are the values of constant $$x^\star$$. For every point @@ -514,23 +560,49 @@ on this curve, the equilibrium inflation:fee ratio $$\alpha^\star/f^\star$$ is the x-coordinate, and the equilibrium reinvestment ratio $$r^\star$$ is the y-value. -The two extremes are $$\alpha^\star\ll f^\star$$ fees dominate and -$$\alpha^\star\gg f^\star$$ inflation dominates. In the former, -staking fraction becomes insensitive to reinvestment, raising -$$x^\star$$. Equivalntly, for any value of positive inflation, -$$r^\star$$ is a lower bound for the equilibrium staking fraction we -should expect. +A breakdown of limitng behaviors is illustrative under positive +inflation. For any value of non-negative inflation, $$r^\star$$ is a +lower bound for the equilibrium staking fraction we should expect. If +inflation dominates fees, $$\alpha\gg f$$ then $s^\star$ is larger by +a small amount than $r^\star$, while if fees dominate inflation +$$\alpha\ll f$$ and $s^\star$ becomes insensitive to non-zero +reinvestment ratio and $$s^\star\to1$$. For a numerical comparison, +eyeballing charts (so *extremely* rough approximations here) $$f +\approx .002<.005\approx\alpha$$ so to within 10\% error above, +$$s^\star\approx r^\star$$ over the range of $$r\in(.5,.75)$$ inferred +from the Lido yield rate. + +How these transient values $$(\alpha/f,r)$$ relate to their +equilibrium values $$(\alpha^\star/f^\star,r^\star)$$ depends on +some considerations: + +* If indeed churn dies down and slashing stays relatively rare, then + $r$ increases to reflect the growing share of businesses that + reinvest the most; $r^\star\approx r_{max}$, where $r_{max}$ is + assesed over all staking pools with at least 10\% of $S$. + +* As we have seen from $\dot{\alpha}$ above, once $\dot{s}\to0$, + inflation decays toward zero unless externalities intervene, but + with a small relaxation term. So *very roughly* + $\alpha_{now}\approx\alpha^\star$ as an overestimate rule of thumb, + that more detailed simulation work could improve. + +#### Runaway $r$ + +Could the sensitivity $r_\alpha$ be sufficient such that even at +intermediate timescales we see $s\to1$? This is certainly possible. -Approximate present values from YCharts are very roughly -$$f\approx.001$$/year, $$\alpha\approx.005$$/year, $$r\in(.5,.75)$$. -How these transient values $$(\alpha/f,\,r)$$ relate to their -equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ depends on -some considerations. +### Zero-growth and Deflation -### Reflexivity and Inflation-Response +### Reflexivity and Oscillations -the conceptual source of potential oscillations can be considered by -modelling the SU system: +One puzzle is that unless $s^\star\to1$ the fixed point of the +intermediate dynamics $0<\alpha\approx$const. is incompatiable with +$\alpha=0$ the externality-free fixed point from $\dot{alpha}$. This +again indicates the endogenous dynamics are oscillation prone. + +The simplest conceptual source of oscillations can be found by +modelling the $(S,U)$ system: $$\displaystyle \begn{array}{rcrlcrl} @@ -539,28 +611,31 @@ $$\displaystyle \end{array} $$ +Specifically, so long as $ry>\jmath+q_-$ staked ETH $S$ just continues +growing and growing, while $U$ cannot get too big, before its own loss +term $-\left(rf+(1-r)bf+q_+\right)\cdot U$ dominates. At some point +$S$ becomes big enough that $ry<\jmath+q_-$ and the system is capable +of oscillation, depending on a zoo of partial derivatves. Readers +wishing for more detail are encouraged to use the two dimensional +local stability criterion to solve for the condition of eigenvalues +with an imaginary part. -- we have not emphasized the oscillation because we expect market -participants can profit from it without having to coordinate their -behavior; oscillations in supply will die down if they are big enough -to arbitrage: buy toward the end of an inflationary phase, sell toward -the end of a deflationary phase. - -- contrast this to runaway staking... there's no obvious mechanism to - profit from an increase in staking fraction, other than by joining - in. So this is a challenge: can you, the reader, design a - cryptoeconomic mechanism by which runaway staking is moderated, in a - way that allows individuals to profit? Can you prove that is impossible? - -- we have not explicitly modelled this kind of reflexivity; it is swept into - the partial derivatives. - - -Could these same partial derivatives be large enough to magnify -inflation-concerns into runaway staking even at intermediate -timescales? This is absolutely possible, and would require the following: - +We have not emphasized the oscillation in the introduction because we +expect market participants can profit from them without having to +coordinate their behavior. Oscillations in supply will die down if +they are big enough to arbitrage: buy "toward the end" of an +inflationary phase, sell "toward the end" of a deflationary phase. +This behavior would enter our model via slow oscillations arising in +the parameters which will eventually sync with the endogenous dynamics +so as to dampen these cycles. Fourier methods and good data +collection are recommended if this potential business-opportunity +interests you. +Contrast this to runaway staking... there's no obvious mechanism to +profit from an increase in staking fraction, other than by joining in. +So this is a challenge: can you, the reader, design a cryptoeconomic +mechanism by which runaway staking is moderated, in a way that allows +individuals to profit? Can you prove that is impossible? @@ -719,7 +794,7 @@ in a bad dynamical systems joke. [^aves]: We use moving quarterly averages, though any timescale $\tau$ sufficiently long that the [fast dynammics]() of [the base]() fee are [integrated out](), and the lags from (un)staking queues are -not appreciable; see [guide]() for details. As we are averaging +not appreciable; see [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md) for details. As we are averaging quarterly, we set the staking $\dot{\mathcal{Q}}_+$, unstaking $\dot{\mathcal{Q}}_-$, and reward $\dot{W}$ queues to zero, including their respective flows ($R+Q_+,Q_-,I+P$) in their target stocks @@ -732,26 +807,30 @@ I=\frac{1}{\tau}\int_{t-\tau}^ty^\bullet S^\bullet dt'=yS-|\kappa| $$ where $$S^\bullet,y^\bullet$ are spot values and the quarterly covariance between staking and yield is $\kappa=\int_{t-\tau}^t(y^\bullet-y)(S^\bullet-S)dt'<0$. See the -[ethode guide]() for more discussion +[ethode guide](https://github.com/20squares/ethode/blob/master/guide/guide.md) for more discussion. [^regex]: This regex script is provided to translate the $\LaTeX$ within the markdown source to (our esimatimate of) terminology more common at [issuance.wtf](https://issuance.wtf) -[^partial]: Sometimes $\dot{x}$ is used e.g. for the partial +[^partial]: Sometimes $\dot{x}=dx/dt$ is used e.g. for the partial derivative $\frac{\partial x}{\partial t}$, *but not here*. For a model -in which you assume dynamical variables $(x,y,z)$, these derivatives -are thus related - -$$\displaystyle \frac{dx}{dt} := \dot{x} = +in which you assume dynamical variables $(A,s,\alpha)$, these derivatives +are related for some parameter $x(A,s,\alpha,t)$ +$$\displaystyle +\frac{dx}{dt} := \dot{x} = \frac{\partial x}{\partial t} + -\frac{\partial x}{\partial y}\dot{y} + -\frac{\partial x}{\partial z}\dot{z} +\frac{\partial x}{\partial A}\dot{A} + +\frac{\partial x}{\partial s}\dot{s} + +\frac{\partial x}{\partial\alpha}\dot{\alpha} =: +x_t + x_A\dot{A} + x_s\dot{s} + x_\alpha\dot{\alpha} $$ - -in which each partial is taken holding the other variables constant, +in which each partial is taken holding all the other variables constant, and $\frac{\partial x}{\partial t}$ is used in practice to capture any -variability from non-dynamical variables. +variability from non-dynamical variables. In principle +$\partial_A{x}$ and $\partial_\alpha{x}$ are distinct; a quantity can +depend on supply (how big ETH market cap is compared to BTC, say) +and inflation independently. [^yield]: For anyone from finance, this is *not* the same as a [bond yield curve](https://www.investopedia.com/terms/y/yieldcurve.asp); @@ -826,7 +905,7 @@ dynamical variables. If not, we must add a dynamical variable. [^asym]: Specifically $I\ll S$ means that \lim_{S\to\infty}(I/S)=0, which according to Buterin's [annotated spec]() is satisfied even -without the limit; see [guide](). +without the limit; see [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md). [^ycov]: The relation between quarterly averaged issuance and the yield curve $y^\bullet$ is: @@ -850,29 +929,36 @@ contribute fixed-points themselves, but they can strongly influence were $\beta\sim(1-s),j\sim(1-s)^2$ this gives increasingly larger $A^\star$ as $s\to1$. - - - -# Open Questions - -Reinvestment within each quarter may not be realistic, and could -introduce interesting oscillations. Some averaging timescale $\tau$, -at least monthly, is necessary to integrate over the short-term burn -dynamics, and (un)staking queue lags. The shorter $\tau$, the better -the approximations $I\approx yS$ and -$\frac{d\log~y}{d\log~S}\approx\frac{d\log~y^\bullet}{d\log~S^\bullet}$. -Yet, $\tau$ must be long enough so that $\dot{V}\approx0$ holds, -allowing us to define reinvestment without a time lag $r:=R/(I+P)$. -Once we have better data on how long the tail of reinvestment is we -could model selling rewards for revenue $K$ and reinvestment $R$ in a -more nuanced fashion, allowing $V$ to be its own dynamical variable. - -We would like to pay more closer attention to how the Deneb hardfork's -limit on staking queue interacts with inflation and reinvestment. We -have ignored this affect in our dynamics, because it does not -presently limit staking flows, and we see inflation decreasing long -term. However for finer questions, it depends on the resources and -patience of the validators. How does a potential race condition -change the composition of the staking queue, in terms of old $R$ -vs. new $Q_+$ and LSTs $L$ vs. everybody else $S-L$. - +[^vitalikp]: Our $\frac{\partial{d\log~y}}{\partial{d\log~S}}$ is +$p-1$ in the [discouragement paper](https://github.com/20squares/ethode/blob/master/guide/guide.md) + +[^deneb]: + + +[^noburn]: While slashing could believably go to zero on quarterly +timescales, no burn $B=0$ implies blocks are empty, at least. +Obviously this isn't really a functioning state for Ethereum, but +think of it as a limiting behavior. A more nuanced treatment of +$s\to1$ can be obtained assuming $s^\star=1-\epsilon$ where +$\epsilon\ll1$, making $A^\star\sim (y(1)/b^\star +f^\star)^2\epsilon^{-2}$; as $\epsilon$ is very small $\epsilon^{-2}$ +is very large. Detailed treatment of the forces using expansions in +$\epsilon$ are useful, and we would need to model churn, slashing, and +burn in light of stochasticity/quantization. One can use difference +equations, but a useful generic behavior of such systems (a "weak +coupling" limit) is that such perturbations move the dynamics away from the +fixed point, apparently randomly, yet the average rate of precession +about $s^\star=1-\epsilon$ is often still given by the imaginary +component of the largest eigenvalue. + +[^mu]: If anything the fractional rates + of slashing and burn are positive with small changes in inflation, + due to either a single ETH potentially being of less real value, or + stimulation of economic activity attracting more validators and + higher average burn. + +[^ics]: $\gamma$ expresses the sensitvity of inflation to supply + initial conditions; the partial $x_A$ always holds $(\alpha,s)$ + consant, but $\dot{A}=\alpha A$, and the only partials in $\gamma$ + are $\jmath_A,\beta_A$. + \ No newline at end of file From 6d30a16ee3bef2fd9ec2806ac3f29a4d6afe503e Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 07:53:05 -0600 Subject: [PATCH 32/70] jesus fing christ almost done for reals --- _posts/2025-01-15-issuance-dynamics.md | 871 ++++++++++++++----------- 1 file changed, 472 insertions(+), 399 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 2ea723c..c3a73db 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -18,19 +18,12 @@ The share of Ether staked by "centralized" staking services, such as exchanges and Liquid Staking Providers (LSTs) [is considerable](https://dune.com/queries/2394100/3928083), and continues to grow. This has provoked [concerns](https://issuance.wtf/), among -Ethereum researchers that the future of Ethereum might involve (1) -nearly all Ether staked, such that (2) the de facto currencies used by -most users are controlled by a confederation of centralized services -with much less transparent governance, and no practical alternative. +Ethereum researchers that the future of Ethereum might involve -## Lookahead +1. Nearly all Ether staked $s\to1$, such that or independently +2. The de facto currency is controlled by a confederation of centralized services. -First, a warning! In an act of hubris, but not without -reasons[^reasons], we have chosen $S$ to refer to Staked ETH, while -others have at times used $S$ for "circulating (S)upply", which we -call instead accessible ETH supply $A$, so $s=S/A$. [Here]() is a -python script for converting between our variables and those most -commonly used on [issuance.wtf](https://issuance.wtf). +## Lookahead In this blog post we address the first of these concerns "runaway (near 100\%) staking" using a "stock and flow" macroecnomics model @@ -42,23 +35,35 @@ and discuss a means for evaluating macroeconomic interventions inspred by bifurcation theory. Briefly, we are not optimstic that reducing issuance will prevent governance centralization, either. -In both posts, we provide a few code examples using [ethode](https://github.com/20squares/ethode/) a thin -units-aware wrapper we built around `scipy.odeint` to streamline model -evaluation. Readers desiring to follow our derivations, dive into -technical mathematical points not covered here, run their own -simulations, or learn some dynamical systems are recommended to look -at our ethode [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md), and references therein. It is certainly -incomplete, but should have enough to get you going. +In both posts, we provide a few code examples using +[ethode](https://github.com/20squares/ethode/) a thin units-aware +wrapper we built around `scipy.odeint` to streamline model evaluation. +Readers desiring to follow our derivations, dive into technical +mathematical points not covered here, run their own simulations, or +learn some dynamical systems are recommended to look at our ethode +[guide](https://github.com/20squares/ethode/blob/master/guide/guide.md), +and references therein. It is certainly a work in progress, but should have +enough to get you going. ## For The Impatient! +Issuance does not all get dumped into liquid unstaked Ether. Some +portion of it is reinvested by staking businesses; indeed this process is +coded into LST smart contracts. + +It is important to distinguish between transient behavior, such as +speculation in staking, and medium/long-term behavior, such as the +reinvestment of staking rewards by staking businesses. When staking +is dominated by reinvestment instead of speculation, inflation +persistently decreases. + We use our macroeconmics model to identify a "low inflation; even lower fees" regime (LI;ELF). Outside of LI;ELF convergence to a -desirable staking future without runaway staking is not possible. -Strong deflation likely corresponds to unstable dynamics, where it is -difficult to predict wether staking fraction approaches 0 or 100\%. -Under zero or low deflation, the tendency toward runaway staking can -only be moderated by very high churn and/or slashing. +desirable staking future without runaway staking is unlikely. Under +strong deflation, in which the magnitude of deflation exceeds the +reinvestment of transaction fees, probably corresponds to unstable +dynamics. Under zero or low deflation, the tendency toward runaway +staking can be moderated only by very high churn and/or slashing. In contrast within LI;ELF staked ETH fraction approaches near the reinvestment ratio. Thus, runaway staking can be avoided only while @@ -66,26 +71,31 @@ inflation is held 1. low enough, that concerns over inflation do not dominate the reinvestment of profits by staking businesses at equilibrium, -$$\frac{dr}{d\alpha}|^\star\leq0$$ (no news here) but simultaneously +$$\frac{dr}{d\alpha}|^\star<0$$ (no news here) but simultaneously 1. high enough to numerically dominate priority fees and MEV, as a fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. -The bad news for opponents of runaway staking is that long -$$t\to\infty$$ term, the inflation fixed point $$\alpha^\star$$ -approaches zero, so $$s^\star\to1$$. Ethereum would undergo cycles of -inflation/deflation which we expect to die down, until an "L2 future" -is reached, which we'll discuss next time. -This is the scenario, recognized by many others where +The importance of this last point in moderatingh staking fraction is +very different than the view on inflation expressed, for instance [in +this podcast](). + +Inflation should *eventually* decay, driving $s\to1$, though it will +take some time. This "L2 future" has been recognized by many others: most Ether is staked, with the majority used for settlement of L2 -rollups. +rollups. We'll discuss it more next time. Given all the above, we advise caution. Intervening to reduce the issuance yield curve seems quite capable of exaccerbating the very -problems we seek to avoid, or causing even worse problems. +problems we seek to avoid. ## Modeling an Open Zeppelin[^humor] +First, a warning! In an act of hubris, and with apologies, but not +without reasons[^reasons], we have chosen $S$ to refer to Staked ETH, +while others have at times used $S$ for "circulating (S)upply", which +we call instead $A$, so $s=S/A$. Please proceed! + ![Ethereum as a balloon with compartments.]( ../assetsPosts/2025-01-15-issuance-dynamics/eth-balloon.jpg) @@ -96,17 +106,21 @@ size of each is measured by *stocks* - ($U$)nstaked unburnt Ether, -- containing the ($V$)alidator reward queue. - ($\cancel{O}$) is all irrecoverable (burned, lost, etc.) Ether, and -- ($A$)ccessible Ether supply, $A=S+U\approx120.4\times10^6$ in Dec 2024. +- ($A$)ccessible/Circul($A$)ting Ether supply, $A=S+U\approx120.4\times10^6$ in Dec 2024. - $\mathcal{Q}_\pm$ the Ether in the staking ($+$) and unstaking $-$) queues The net change in time of a stock is written using a dot, such as -$\frac{dA}{dt}:=\dot{A}$[^partial], the net change in accessible -Ether. Stocks grow or shrink based on flows which add to or subtract +$\frac{dA}{dt}:=\dot{A}$[^partial], the net change in accessible Ether +supply. Stocks grow or shrink based on flows which add to or subtract from their derivatives. Here all flows are positive real numbers with -units \[ETH/yr\]. By averaging over "long" timescales (at least -quarterly)[^aves] we approximate the staking and unstaking queues as -equilibrated; $\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$. So, our -conceptual model: +units \[ETH/yr\]. + +By averaging over "long" timescales (at least quarterly)[^aves] we +approximate the staking and unstaking queues as equilibrated; +$\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$, and average over many +cycles of the erratic base fee oscillations. + +So, our conceptual model: $$\dislaystyle \begn{array}{rcl} @@ -134,16 +148,19 @@ $(\cancel{O},\cancel{O},U,\ldots)$, where the flow is going to.[^cats] Flows obey constraints, often expressed as (in)equalities relating a flow to its source. -In response to the concerns about $s\to1$, the Deneb upgrade -implemented EIP 7153, an upper limit on $R+Q_+$ chosen to not limit -any present flows.[^deneb] We also ignore the pre-existing symmmetric -limits on (un)staking $Q_\pm$. The purpose of our models is to show, -in the absence of such limits, where the dynamics push the system. We -will revisit EIP 7153 in our next post, on staking *composition*. So -the constraints ($Q_- 1/Yr: + return self.y1 * sqrt(self.s1 / S) +``` -### Supply growth, Crudely in One Dimension +### $(S,U)$; not just a UNIX Command! -Consider $\dot{A}=I-B-J$ in light of the [above -table](#table-of-flows), $\dot{A} = y(S)S - bf(A-S) - jS$. To -understand the dynamics of circulating supply, we need to understand -that of staked ETH. As a crude approximation, lets temporarily hold -staking fraction $s=S/A$ constant. +With the above, you should be able to construct the following $(S,U)$ system: $$\displaystyle -\dot{A}=\left(y(sA)-\beta(1-s)-js\right)A=y_0(1)\sqrt{sA}-[bf(1-s)+js]A +\begn{array}{rcrlcrl} +\dot{S} &=& (ry-\jmath-q_-) & S & + & \left(q_++r(1-b)f\right) & U\\ +\dot{U} &=& \left((1-r)y+q_-\right) & S & - & \left(rf+(1-r)bf+q_+\right) & U\\ +\end{array} $$ -You can explore this system using the following code. +These coefficients of $S,U$ are miserably complicated-looking, but as +written they are all (but one) positive, and so we can reason about +this model's evolution. Specifically, so long as $ry(S)>\jmath+q_-$ +staked ETH $S$ just continues growing and growing. In contrast it is +harder for $U$ to get as big, limited by its own loss term +$-\left(rf+(1-r)bf+q_+\right)\cdot U$. At some point in the (far) +future $S$ becomes big enough that $ry(S)<\jmath+q_-$ and the system +becomes *capable* of oscillation, depending on parameters and a zoo of +partial derivatives.[^SU] ```python -class ACrude(ODESim): - beta: One/Yr - j: One/Yr - y1: One/Yr = 166.3 / Yr - s: One = .5 * One - def yield(self, S:ETH) -> One/Yr: return self.y1 / math.sqrt(S) +@dataclass +class SUSimConst(ODESim): + ic: tuple[ETH, ETH] = (120e6 * .3, 120e6 * .7) + tspan: tuple[Yr, Yr] = (0, 100) + params: Params = ConstParams() @staticmethod - def func(A:ETH, t:Yr, p:Params) -> ETH/Yr: - gain = p.yield(p.s * A) - loss = p.beta * (1 - p.s) + p.j * p.s - return (gain - loss) * A -model = ACrude(beta = 8e-3/Yr, j = 1e-6/Yr, y1 = 166.3/Yr, s = .3 * One) -model.sim() + def func(t:Yr, v:tuple[ETH, ETH], p:Params) -> tuple[ETH/Yr, ETH/Yr]: + x = {'S': (S := v[0]), 'U': (U := v[1])} + dS = (p.r * (y := p.yld(S)) - p.j - p.qu) * S + \ + ((rf := p.r * p.f) * (1 - p.b) + p.qs) * U + dU = ((1 - p.r) * y + p.qu) * S - \ + (rf + (1 - p.r) * p.b * p.f + p.qs) * U + return dS, dU +su = SUSim() +su.sim() ``` -If you have an opinion about $j(A),\beta(A)$ you can replace the parameter -entries with functions, like what we did for `yield`: replace `pass` -with your desired behavior, using `self.b` etc. to access parameters. -Within `func` use `p.burn` instead of `self.burn`; `ODESim` takes care -of the plumbing. +These dynamic variables are kind of boring, but critically and unlike +$(A,S)$, there are *no extra conditions* (such as $S One/Yr: - pass # Your insights here - def burn(self, A:ETH) -> One/Yr: - pass # Your insights here - @staticmethod - def func(A:ETH, t:Yr, p:Params) -> ETH/Yr: - gain = p.yield(p.s * A) * A - loss = p.slash(A) + p.burn(A) - return gain - loss -``` +### Inflation -$A$ grows when the right hand side (RHS) is positive, and shrinks -otherwise. Pretending $s=s^\star$ constant, we obtain the *fixed -points* $A^\star$ at which $\dot{A}=0$. Because $0<\beta,j<1$ the -parameters cannot contribute fixed points themselves, only influence -where they occur.[^params] Let $\beta^\star=f(A^\star,s^\star,\ldots)$ -represent the burn fractional rate $\beta=bf=B/U$ evaluated at the fixed point. -There are two trivial fixed points, trivial $A^\circ=0$, and a -non-trivial fixed point +Inflation is used to refer to many things, but here we mean +specifically the quarterly fractional change in accessible Ether. +Consider $\alpha:=\dot{A}/A\approx(I-B-J)/A$ in light of the [above +table](#table-of-flows). In general and under +the existing yield curve we have (where $\beta=bf=B/U$): $$\displaystyle -A^\star = \left( -\frac{y(1)\sqrt{s^\star}}{\beta^\star(1-s^\star)+j^\star s^\star} -\right)^2 +\alpha\approx y(sA)s-\beta(1-s)-\jmath s\right) = y_0(1)\sqrt{s/A}-\beta(1-s)-\jmath s $$ -### Stability in One Dimension +You can explore this by noting `alpha(), sfrac()` as `@output` methods -In general the *stability* of a fixed point for a one-dimensional flow is -determined by the derivative of the RHS at the fixed point: if its -negative, then small perturbations shrink and the fixed point is -stable. Otherwise, it is unstable.[^centers] +```python +@dataclass +class SUaSim(SUSim): + @output + def sfrac(self, S:ETH, U:ETH) -> One: + return S / (S + U) + @output + def alpha(self, S:ETH, U:ETH) -> 1/Yr: + s = self.sfrac(S,U) + return p.yld(S) * s - p.b * p.f * (1 - s) - p.j * s +su_a = SUSim_alpha() +su_a.sim() +``` -![1D Stability Conditions](../assetsPosts/1d-stab.png) -Specifically, the sign of $\left.\frac{\partial\dot{A}}{\partial -A}\right|^\star$ determines wether $A^\star$ is (un)stable. After -some manipulations, a *sufficient*[^if-vs-iff] condition for -stability is +### Persistent inflation cannot maintain + +A key feature of $\dot{A} under the current yield curve $y_0(S)$ is +sublinear issuance $I\leq yS\lesssim S$, chosen to avoid +[discouragement +attacks](https://raw.githubusercontent.com/ethereum/research/master/papers/discouragement/discouragement.pdf). +Because of this, positive inflation cannot maintain indeinitely. We +will demonstrate with the existing yield curve, but the argument is +general. Unusually for this blog post, we will show most of the steps so +the argument is hopefully understood. $$\displaystyle -\left.\frac{\partial\log~B}{\partial\log~A}\right|^\star~~\mathrm{or}~~ -\left.\frac{\partial\log~J}{\partial\log~A}\right|^\star~~~>~~~\frac{1}{2}. +dA = \alpha dt &\leq& \left(ys-\beta(1-s)-\jmath s\right)dt\\ +dA &\leq& ysdt = y_0(1)\sqrt{s/A}dt \leq y_0(1)/\sqrt{A}dt\\ +\sqrt{A}dA &\leq& y_0(1)dt\\ +\int_{A(0)}^{A(t)}\sqrt{A}dA &\leq& \int_0^t y_0(1)dt\\ +\frac{2}{3}x^3\Big|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ +A(t) &\leq& \left(A(0)+\frac{3}{2}y_0(1)t\right)^{2/3}\\ +\therefore A(t) \ll e^{kt} ~\forall ~\mathrm{const.}~k>0 $$ -If either is satisfied, the fixed point is stable, and $A^\star$ -represents the equilibrium supply value, at which Ether acheives zero -growth. Essentially the leading component of the loss term must -increase faster than $\sqrt{A}$ near the fixed point. Note that -constant $\beta,j$ correspond to scaling exponent $1$, so satisfy. +The last line, for those unfamiliar is asymptotic notation. For two +positive functions, $g$ dominates $f$, written $f(t)\ll g(t)$ just +when $\lim_{t\to\infty}[f(t)/g(t)]=0$. Since supply $A(t)$ is +*eventually* less than a powerlaw of $t$, it is *subexponential*. +Thus, no positive inflation rate can maintain. -To simplify, assume negligible slashing; there must be sufficient -economic activity so that the burn scales at least as the square-root -of supply, or else $A^\star$ is unbounded. That is true inflation, -not just in the sense of supply expansion, but the *demand not growing -sufficiently with the supply*. This is similar to the concern raised -by [Elowsson 2020](). +This does not mean we would find every intermediate inflation rate +pleasant. Following surges in $Q_+$, inflation can accelerate quite +alarmingly. A good example will be the 2032 Ethereum staking-mania. +This can be modeled in the following graph showing `qs=.05, qu=.01`; a +constant $q_+=5$\% of unstaked ETH is added to the staking queue each +year. -### Positive Inflation cannot maintain +![US President-Elect Swift Endorses Ethereum!]( + ../assetsPosts/swifties.jpg) -What happens to equilibrium supply $A^\star$ then, if $s\to1$ and -slashing is negligible, or the partials $\partial_AB,\partial_AJ$ are -insufficient such that $A^\star\to\infty$? +Instead we mean to separate concerns. Unpleasantly high inflation in +the medium term, even if that "medium term" lasts decades, is a +*dynamics* problem, not an equilibrium problem, and so dynamical +solutions (like EIP 7514) are perhaps better suited. Unfortunately we +will see that given the above, $s\to1$ is an equilibrium problem. -Inflation is used to refer to many things, but here we mean -specifically the quarterly fractional change in accessible -Ether. Let's express inflation $\alpha:=\frac{d\log~A}{dt}$ in -terms of $s$ and the intensives +### Staking Fraction -$$\displaystyle -\begin{array}{rcl} -\dot{A} &=& \alpha A\\ -\alpha &=& (I-B-J)/A \approx y(sA)s - bf(1-s) - js -\end{array} -$$ - -A key feature of $\dot{A} under the current yield curve is sublinear -issuance $I\leq yS\lesssim S$, chosen to avoid [discouragement -attacks](https://raw.githubusercontent.com/ethereum/research/master/papers/discouragement/discouragement.pdf). -Because of this, the positive term continues to get smaller as -Accessible ETH supply grows; $\dot{A}\leq y(As)A$, under the current -yield curve $\dot{A}(s,A)\leq y_0(1)\sqrt{sA}$. So, at any point, -given supply $A$, the inflation rate $\alpha = \dot{A}/A$ cannot be -larger than $\alpha_{max}=y_0(1)\sqrt{s/A}$. Constant positive -inflation corresponds to $A$ growing, and for -any positive constant $\alpha=\epsilon$ there is some $A$ at which -$\epsilon=\alpha_{max}(A)$. Total circulating suopply $A$ is -bounded above by a subexponential function for all time. - -That is, inflation must eventually return toward zero, even in the -pressence of very weak burn, slashing, and very high staking fraction. -This is not emphasized enough in our discussions. - -### Two Dimensional $(A,s)$ System - -We know well that $s$ is *not constant*. We have equations for -$\dot{A},\dot{S}$, what about $\dot{s}=d(S/A)/dt$? Using the quotient -rule $\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}$, and after an -algebraic massage, we obtain for staking fraction +We have equations for $\dot{A},\dot{S}$, what about +$\dot{s}=d(S/A)/dt$? Using the quotient rule +$\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}$, and after an algebraic +massage, we obtain for staking fraction $$\displaystyle \begin{array}{rcl} @@ -383,58 +412,26 @@ reduction of the inflation rate. The latter can be acheived in principle by a reduction of issuance relative to the base fee "burn rate". Because of this quotient rule tradeoff, issuance plays a beneficial "infrastructure" role in moderating staking fraction, -drawing itb toward $r$, which may be less than one. - -Other aspects are harder to read, as this is quite a complicated -equation, but we finally have a dynamical system. You can simulate a -general version of it, filling in your desired behavior for `pass` and -adding inflation as an output, as follows: +drawing it toward $r$, which can be less than one. -```python -class AsSys(ODESim): - def yield(self, S:ETH) -> One/Yr: pass - def burnfrac(self, A:ETH, s:One) -> One: pass - def feefracrate(self, A:ETH, s:One) -> One/Yr: pass - def slashfracrate(self, A:ETH, s:One) -> One/Yr: pass - def rnvstfrac(self, A:ETH, s:One) -> One: pass - def qstake(self, A:ETH, s:One) -> One/Yr: pass - def qunstake(self, A:ETH, s:One) -> One/Yr: pass - def _hlpr(self, A:ETH, s:One) -> tuple[One/Yr,One/Yr,One,One/Yr]: - beta = (b := self.burnfrac(A,s)) * (f := self.feefracrate(A,s)) - return s * self.slashfracrate(A,s), beta * (1 - s), b, f - @output - def inflation(self, A:ETH, s:One) -> One/Yr: - slash, burn = self._hlpr(A,s)[:2] - return self.yield(A * s) - burn - slash - @staticmethod - def func(v:tuple[ETH,One], t:Yr, p:Params) -> tuple[ETH/Yr,ETH/Yr]: - x = {'A': (A := v[0]), 's': (s := v[1])} - slash, burn, b, f = p._hlpr(A, s) - to0 = j * (1 - s - (r := p.rnvstfrac(**x))) + p.qunstake(**x) - to1 = p.qstake(**x) + f * (1-s) * (b * s + (1-b) * r) - dA = p.inflation(A,s) * A - ds = yld * (r - s) + to1 * (1 - s) - to0 * s - return dA, ds -``` - -# The $(A,\alpha,s)$ Dynamical System +### $(A,\alpha,s)$ Dynamical System There are many reasons, especially in the context of the world economy, to care about total circulating supply $A$. Recent -discussions however have focused most on inflation. It also turns out -that modeling inflation $\alpha$ directly simplifies our analysis -considerably. +discussions however have focused most on inflation $\alpha=\dot{A}/A$, +the growth in supply over time. It also turns out that modeling +inflation $\alpha$ directly simplifies our analysis of the $(A,s)$ +system considerably. Let subscripts denote partial derivatives -$x_y:=\frac{\partial{x}}{\partial{y}}$. Using the product rule -$\dot{xy}=\dot{x}y+x\dot{y}$ and the correct partial derivative -relations for variables $(A,\alpha,s,t)$[^partial] we have +$x_y:=\frac{\partial{x}}{\partial{y}}$. Using the correct partial +derivative relations for variables $(A,\alpha,s,t)$[^partial] we have $$\displaystyle \begin{array}{rcl} \dot{A} &=& \pm\alpha A\\ \dot{s} &=& \pm\alpha(r-s) + (rf+q_+)(1-s) - (q_-+(1-r)j)s -\dot{\alpha} &=& \pm\frac{\xi}{1+\mu}\dot{s} - \frac{\gamma}{1+\mu}\alpha s \pm \frac{\chi}{1+\mu} +\dot{\alpha} &=& \pm\xi\dot{s} - \gamma\alpha s \pm\chi \end{array} $$ @@ -442,110 +439,166 @@ Where the new greek letters are fractional rates, defined below, and $y':=\frac{dy}{dS}$. * $\mu:=\beta_\alpha(1-s)+\jmath_\alpha s$ is the implicit - sensitivity of the inflation loss term to increases in inflation. + sensitivity of the inflation loss-term to increases in inflation. We judge $0\leq\mu$; if anything inflation increases burn and slashing fractional rates.[^mu] -* $\xi:=y+y'A+\beta-\beta_s(1-s)-j-j_s$ is the net - correlation between changes in $s$ and changes in $\alpha$, it can - be of either sign. Under the current yield curve - $y_0+\frac{dy}{dS}A=y_0(sA)(1-1/(2s))$. - -* $\gamma:=\left(j_As+\beta_A(1-s)+s|y'|\right)A$ is a positive - coefficient expressing how quickly $\alpha\to\alpha^\star$, and the - partials are constant when initial supply is known.[^ics] We have - extracted the sign from the final term because sublinear issuance - implies $y'<0$; under the current yield curve the term +* $\xi:=(y+y'A+\beta-\beta_s(1-s)-j-j_s)/(1+\mu)$ is the net + correlation between changes in $s$ and changes in $\alpha$ + normalized by $1+\mu$. $\xi$ can be of either sign. Under the + current yield curve $y_0+\frac{dy}{dS}A=y_0(sA)(1-1/(2s))$, which + changes its sign at 50\% ETH staked. + +* $\gamma:=j_{\log{A}}s+\beta_{\log{A}}(1-s)+s|y'|A$ is a + positive coefficient expressing how quickly $\alpha\to\alpha^\star$, + and the partials are constant when initial supply is known.[^ics] We + have extracted the sign from the final term because sublinear + issuance implies $y'<0$; under the current yield curve the term $sA|y'|=\frac{1}{2}y_0(sA)$. * $\chi:=-j_ts-\beta_t(1-s)$ represents externalities affecting the - inflation loss term encoded as explicit time-dependencies. - -In what follows we will obtain and asses the nature of the long-term -externality-free fixed point cooridnate $(\alpha^\star=0,s^\star=1)$. -We will observe that it is not hard for $|\dot{\alpha}|\ll|\dot{s}|$ -to obtaon, leading to a seperation of timescales where an $\alpha>0$ -may change slowly enough that we can still use equilibrium arguments -at intermediate times. We extract the behavior of -$x^\star(\alpha~\mathrm{const.},\ldots)$ and break down the fixed -point stability into LI;ELF vs. deflation. We close with a dicussion -of the potential for oscillations in our model vs. actual market data. - -## Local Stability of $\alpha^\star=0$ and Oscillations - - - + inflation loss term encoded as explicit time-dependencies. We + neglect externalities $\chi\approx0$ because we have nothing + intelligent to say about them, but you might not want to. + +#### Separation of Timescales and $\alpha>0$ + +Observe that every term in $(\xi,\gamma)$ are small fractions, +fractional rates, or derivatives thereof. Indeed, when +$\alpha\approx0$ and sensityivities are weak, $|\xi|\sim\gamma\sim y$ +obtains. Generally if $|\xi|,\gamma\ll1$ then the derivatives obey +$|\dot{\alpha}|\ll|\dot{s}|$: a [seperation of +timescales](https://en.wikipedia.org/wiki/Method_of_averaging). For +durations when this obtains, short periods at intremediate times, +inflation could be treated as a parameter instead of its own dynamic +variable, with staking fraction equilibrating to $s^\star$ more +quickly than $\alpha^\star$ equilibrates. + +Does this hold presently? For a sanity-check, a quick look at YCharts +since The Merge shows that $$s,\dot{s}$$ do indeed seem to vary over a +much greater range than $$(\log{A},\alpha)$$. +![The staking fraction from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg) +![The inflation rate from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg) -## Separation of Timescales and $\alpha>0$ - -Observe that every term in $(\xi,\gamma)$ are fractions, fractional -rates, or derivatives thereof, so if $|\xi|,\gamma\ll1$ then the -derivatives obey $|\dot{\alpha}|\ll|\dot{s}|$: a [seperation of -timescales](). For durations when this obtains, short periods at -intremediate times, inflation could be treated as a parameter instead -of its own dynamic variable, with staking fraction equilibrating to -$s^\star$ more quickly than $\alpha^\star$. +For the remainder of this post, we will assume this obtains. +Anecdotally, even when it does not, the revealed interplay between +inflation and staking fraction shows up, and is a very important +concept for Ethereum Macroeconomics. -Does this hold presently? For a sanity-check, a quick look at -YCharts since The Merge shows that $$s,\dot{s}$$ do indeed vary over a much -greater range than $$(\log{A},\alpha)$$. +# Staking Fraction in 1D -![The staking fraction from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg) -![The inflation rate from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg) +Recall our approximate equation for the fraction of staked ETH $s$, in +which all coefficients are positive but inflation $\alpha$: -## The Intermediate s^\star +$$\displaystyle +\dot{s} &=& \alpha\cdot(r-s) + (rf+q_+)\cdot(1-s) + (q_-+(1-r)j)\cdot(0-s) +$$ -So, let us examine the fixed point $s^\star$ with $\alpha^\star$ treated as -a parameter. +So assuming $|\dot{\alpha}|\ll|\dot{s}|$, let us examine the fixed +point $s^\star$ during a period in which we may treat inflation as +constant $\alpha=\alpha_{const}$. $$\displaystyle s^\star = \frac{ - r^\star(\alpha^\star + f^\star) + q_+^\star + r^\star(\alpha_0 + f^\star) + q_+^\star }{ - \alpha^\star + r^\star f^\star + q_+^\star + q_-^\star + \jmath^\star + m^\star := (\alpha_0 + r^\star f^\star + q_+^\star + q_-^\star + (1-r)\jmath^\star) } += 1 - \frac{(1-r)\jmath+q_-}{m^star} - \frac{\alpha_0(1-r^\star)}{m^\star} $$ -If $\dot{A}=0=\alpha$ (no inflation nor deflation) then an interior -market equilibrium $s^\star<1$ requires very high slashing. +We will see that without $\alpha>0$ an interior market equillibrium is +very unlikely. First, as a thought experiment, consider $\alpha=0$. + +## Concerning Churn -We reason as follows. In the absence of in/de-flation, an interior -fixed point $s^\star<1$ would require a persistent -unstaking/capitulation of existing validators $q_->0$. This in turn -either requires churn, a persistent -supply of new validators to take their place $q_+^\star>0$, or it is -only a transient and $q_-^\star\approx0$; recall that reinvestment by -existing validators is not counted in $q_+$. +A fixed point $s^\star(\alpha=0)<1$ would require either high churn or +a persistent unstaking/capitulation of existing validators +$q_-+\jmath>0$. Validators try to minimize slashing as intended, but what would a +large unstaking flow $q_-s\gtrsim rf$ at $s^\star$ mean? -Net unstaking $$q_->0$$ could only describe a market equilibrium if +High unstaking requires *churn*, a persistent supply of new validators +to take their place $q_+^\star(1-s)\sim q_-s\gtrsim rf$, or it is only +a transient and $q_-^\star\ll rf$; recall that reinvestment by +existing validators is not counted in $q_+$. Persistently high +unstaking $$q_-\sim rf$$ could only describe a market equilibrium if one group of stakers was actively capitulating and withdrawing their stake, while another group with a higher $$r$$ were aggressively reinvesting in their business, and their reinvestment of fees and MEV -offset the unstaking, adjusted for inflation. This cannot maintain -forever: eventually there will be no new Capitulators left, and -$$s^\star$$ must once again grow as required by the Reinvestors' -higher $$r$$, so $$s^\star$$ was not a fixed point at all. Similarly, -at some point everyone who wants to stake should have staked. If we -judge the quarterly fluxes due to the issuance of new humans and the -burn rate of legacy humans to be small and/or likely to take over -existing businesses, additional validators count overwhelmingly toward -$$r$$, so $$q_+^\star\approx0$$. Thus, the fixed point $$s^\star$$ +offset the unstaking, adjusted for inflation. + +High churn, not counting reinvestment, cannot maintain forever: +eventually there will be no new Capitulators left, and $$s^\star$$ +must once again grow as required by the Reinvestors' higher $$r$$, so +$$s^\star$$ was not a fixed point at all. So $q_-^\star\ll +(rf)^\star$$. + +Similarly, at some point everyone who wants to stake should have +staked. If we judge the $\tau$-averaged flows due to the issuance of +new humans and the burn rate of legacy humans to be small, additional +validators count overwhelmingly toward $$r^\star$$ and +$$q_+^\star\ll(rf)^\star$$. Thus, the fixed point $$s^\star$$ simplifies to $$\displaystyle -s^\star = r^\star \frac{ - \alpha^\star + f^\star -}{ - \alpha^\star + r^\star f^\star + \jmath^\star -} +s^\star \approx r^\star \frac{ + \alpha^\star + f^\star}{ + \alpha^\star + r^\star f^\star + (1-r)\jmath^\star} += 1 - \frac{ + \jmath^\star}{ + \alpha^\star + r^\star f^\star + (1-r)\jmath^\star} $$ -#### Stability of $x^\star$ +We will explore the *stability* of this fixed point below, and based on +the range of $\alpha$ categorize three basic behaviors. + +## Stability in One Dimension + +Fixed-points are market equilibria just when they are stable. Every +stable fixed point in a one-dimensional flow is locally stable. In +general a fixed point is locally stable when small changes +(*perturbations*) shrink over time. For a continuous map like ours, +this concerns the derivative of the RHS at the fixed point: if it is +negative, then small perturbations shrink and the fixed point is a +stable *sink*. If the derivative is zero, the fixed point is a +degenerate *center*, unrealistic outside of physics. If positive, the +fixed point is an unstable *source*. + +![1D Stability Conditions](../assetsPosts/1d-stab.png) + +Specifically, we want the sign of +$\left.\frac{\partial\dot{s}}{\partial s}\right|^\star$ to determine +wether $s^\star$ is (un)stable. We will be ignoring the partial +derivatives ("sensitivities") by assuming they are small in comparison +with their corresponding intensives.[^small-part] + + + +### Weak Deflation + +If $\alpha_{const}\in(-rf-\jmath,0]$ then an interior market equilibrium +$s^\star$ requires high slashing, as we argued before based on +Churn. All other things being equal, it is also less stable. + + + +### Strong Deflation +Consider now $\alpha<-rf-\jmath$. This could happen for instance if +the issuance curve were reduced particularly bluntly, or changes in +fundamentals drove either MEV or the base fee (and thus $f$) to a +persistently higher amount, such that $ys\ll bf(1-s)+\jmath s$. +These conditions cannot maintain of course. You don't need +differential equations to see that $\alpha<0$ shrinks $A$, which +eventually raises $y(sA)$. But as a temporary intervention to tame +runaway staking fraction how would this work? -### LI;ELF Conditions +### Low Inflation; Even Lower Fees + +Well, that was deflating! Let's cheer ourselves up by considering the +behaviors under $\alpha>0$. A positive role for inflation can be seen in the contours of the market equilibrium staking fraction $$s^\star$$ corresponding to @@ -587,96 +640,85 @@ some considerations: $\alpha_{now}\approx\alpha^\star$ as an overestimate rule of thumb, that more detailed simulation work could improve. -#### Runaway $r$ +### Runaway $r$ from Inflation pressure Could the sensitivity $r_\alpha$ be sufficient such that even at -intermediate timescales we see $s\to1$? This is certainly possible. - -### Zero-growth and Deflation - -### Reflexivity and Oscillations +intermediate timescales we see $s\to1$? This is certainly possible; +per the arguments of Ethereum researchers, high inflation could still +lead to runaway staking if $r$ is sensitive enough. -One puzzle is that unless $s^\star\to1$ the fixed point of the -intermediate dynamics $0<\alpha\approx$const. is incompatiable with -$\alpha=0$ the externality-free fixed point from $\dot{alpha}$. This -again indicates the endogenous dynamics are oscillation prone. - -The simplest conceptual source of oscillations can be found by -modelling the $(S,U)$ system: +In our model the net effect of inflation on LI;ELF staking fraction +equilibrium is reflected by taking the derivative +$$0<\left.\frac{ds^\star}{d\alpha}\right|^\star$$ assuming $$r,f$$ are +implicit functions of $$\alpha$$. That is, the necessary condition +for inflation to push the market equilibrium $$s^\star$$ itself into +runaway staking is (see below for explanation): $$\displaystyle -\begn{array}{rcrlcrl} -\dot{S} &=& (ry-\jmath-q_-) & S & + & \left(q_++r(1-b)f\right) & U\\ -\dot{U} &=& \left((1-r)y+q_-\right) & S & - & \left(rf+(1-r)bf+q_+\right) & U\\ -\end{array} +1 ~~ < ~~ +\left.\frac{\partial\log\ r}{\partial\log\ \alpha}\right|^\star +\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} ~~ + ~~ +\left.\frac{\partial\log\ f}{\partial\log\ \alpha}\right|^\star $$ -Specifically, so long as $ry>\jmath+q_-$ staked ETH $S$ just continues -growing and growing, while $U$ cannot get too big, before its own loss -term $-\left(rf+(1-r)bf+q_+\right)\cdot U$ dominates. At some point -$S$ becomes big enough that $ry<\jmath+q_-$ and the system is capable -of oscillation, depending on a zoo of partial derivatves. Readers -wishing for more detail are encouraged to use the two dimensional -local stability criterion to solve for the condition of eigenvalues -with an imaginary part. - -We have not emphasized the oscillation in the introduction because we -expect market participants can profit from them without having to -coordinate their behavior. Oscillations in supply will die down if -they are big enough to arbitrage: buy "toward the end" of an -inflationary phase, sell "toward the end" of a deflationary phase. -This behavior would enter our model via slow oscillations arising in -the parameters which will eventually sync with the endogenous dynamics -so as to dampen these cycles. Fourier methods and good data -collection are recommended if this potential business-opportunity -interests you. - -Contrast this to runaway staking... there's no obvious mechanism to -profit from an increase in staking fraction, other than by joining in. -So this is a challenge: can you, the reader, design a cryptoeconomic -mechanism by which runaway staking is moderated, in a way that allows -individuals to profit? Can you prove that is impossible? - - +In practtice it all depends on what ETH users consider sufficiently +"high" inflation to respond; the above conditions just reflect a +quantification of this preference. We hope that this work can be +built upon to determine what threshold of inflation or business +conditions satisfy the above condition. No mean feat, but it would help focus +inflationary pressure arguments into empirically measurable assertions +that can be tracked for Ethereum health. +# Concluding Discussion +We saw above a few things: +1. Reinvestment $r$ is a lower-bound for $s^\star$. +1. Low (but positive) inflation moderates staking fraction closer to this lower bound + at intermediate timescales +1. Positive inflation cannot maintain indefinately, so eventually $s\to1$. +Conceptually, how can inflation *moderate* staking fraction, though? +Shouldn't more staking lead to more issuance, which leads to more +inflation, etc.? Briefly the reasons are: +* Short $Q_+$ vs. Long Term $R$ Investment +* The Quotient Rule $\dot{S}=\dot{S}/A-s\dot{A}/A$ +## Short Term vs. Long Term. +Novel investment in staking $Q_+$ is driven largely by speculation, +and new users encountering Ethereum. $Q_+$ acts to increase staking +fraction, as seen above, and indeed the glut in $Q_+$ since the Merge +may have been the source for all of the alarm that prompted this +study. Novel speculative investment must eventually dry up, and be +replaced by long-term investmemnt $R$, because +1. everyone with money who wants to stake eventually will, so will be +counted in $R$ not $Q_+$ +2. any business that wants to stay in +business cannot consistently reinvest more than its revenue $R\leq +I+P$. -# Conclusions +Of the long term signal $R$, only the issuance portion of +reinvestment, that is the part that contributes to inflation, can +moderate $s$, which brings us to the quotient rule. -Why does this happen, though? +## The Quotient Rule -* Short $Q_+$ vs. Long Term $R$ Investment -* The Quotient Rule $\dot{S}=\dot{S}/A-s\dot{A}/A$ +$$\displaystyle +\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}=\frac{\dot{S}}{S}-s\alpha +$$ -Short Term vs. Long Term. Novel investment in staking $Q_+$ is driven -largely by speculation, and new users encountering Ethereum. $Q_+$ -acts to increase staking fraction, as seen above, and indeed the glut -in $Q_+$ since the Merge may have been the source for all of the alarm -that prompted this study. Novel speculative investment must eventually dry up, -and be replaced by long-term investmemnt $R$, because (1) everyone -with money who wants to stake eventually will, so will be counted in -$R$ not $Q_+$, and any business that wants to stay in business cannot -consistently reinvest more than its revenue $R\leq I+P$. Of the long -term signal $R$, only the issuance portion of reinvestment, that is -the part that contributes to inflation, can moderate $s$, which brings -us to... - -The Quotient Rule $\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}. What -increases $s$ is any increase in staked Ether $S$, but also any net -decrease in $A=S+U$. Inflation $\alpha$ increases -*both* $U$ and $S$, because some of that increase is used to meet -costs and take profit $(1-r)\alpha$, which increases $U$ relative to -$S$. In contrast, the reinvestment of transaction fees can only ever -increase $S$ at the expense of $U$. Thus transaction fees always act -to increase staked fraction, while the effect of inflation depends on -the relative values of reinvestment ratio and staking fraction. +What increases $s$ is any increase in staked Ether $S$, but also any +net decrease in $A=S+U$. Inflation $\alpha$ increases *both* $U$ and +$S$, because some of that increase is used to meet costs and take +profit $(1-r)\alpha$, which increases $U$ relative to $S$. In +contrast, the reinvestment of transaction fees can only ever increase +$S$ at the expense of $U$. Thus transaction fees always act to +increase staked fraction, while the effect of inflation depends on the +relative values of reinvestment ratio and staking fraction. So, while we could certainly model reinvestment differently, and there are lags we are blithely integrating over, we think that these market @@ -687,18 +729,22 @@ or the sensitivity of $r$ to inflation at equilibrium is too great, a condition which we mathematized above. In fact we expect that every argument about inflation effects driving increased staking, overpaying for security, etc. could (perhaps should) be rephrased in terms of -reinvestment of staking rewards. - -Regarding the possible effects of reflexivity. We have not emphasized -the potential for oscillations very much, even thoughh a dynamical -systems person who looks at the $(S,U)$-model is likely to find this -its most striking potential. The reason is reflexivity. -Specifically, if the oscillations in inflation are sufficient to not -be drowned out by the externalities and sources of noise, then market -participants will try to profit off of them. The simplest strategy, -to buy toward the end of an inflationary phase, and sell toward the -end of a deflationary phase, should over time reduce any oscillations -that do arise. Notably though, we only expect this to happen because +reinvestment of staking rewards. All these critically depend on the +preferences of ETH users for, and thus their behavior in reaction to, +inflation rate etc., which thusfar are not measured. We encourage the +community to rectify this! + +## Can reflexivity prevent $s\to1$? + +Regarding the possible effects of reflexivity. We have neglected even +discussing oscillations in $(S,U)$, even though the model is plainly +capable of such behavior under different parameters or when coupled to +price. Why such negligence? If cycles *do* arise, we expect market +participants, anticipating such cycles, would act to profit off of +these cycles in a way that should reduce them. Buy late in the +inflation cycle, sell, late in the deflation cycle, etc. + +Notably though, we only expect this to happen because it does not require the coordination of market participants: each individual blindnly pursuing their own utility should help en masse control these oscillations, or they were never very great to begin @@ -708,8 +754,9 @@ Could there be a similar effect with staking fraction or inflation? In short we have no idea, but put it as a challenge to the reader. Can you devise a cryptoeconomic protocol or trading strategy that forestalls the seemingly inevitable $s^\star\to1$? If not, can you -prove this is impossible? Regardless, planning for a low-inflation -high staked-fraction future is highly reommended. +prove this is impossible? + +## Should We Change $y$? So, finally... should the Ethereum community reduce issuance? @@ -719,32 +766,32 @@ Nope. * Short answer: -If you don't like the medium term future of inflation or you want to -slow down the transition to high staking, please consider and study -downward adjusting the constants in EIP 7153, adopted during the Deneb -upgrade. EIP 7153 already directly limits $R+Q_+$, but was very -nicely designed to not interfere with existing staking flows. EIP -7153 does not solve, and does not claim to solve, the long term -problems, but we have been forced to conclude that reducing issuance -doesn't solve them either! +If you are very inflation averse or you want to slow down the +transition to high staking, please consider and simulate downward +adjusting the constants in EIP 7514, adopted during the Deneb upgrade. +This already directly limits $R+Q_+$, but was very nicely designed +to not interfere with existing staking flows. EIP 7514 does not +solve, and does not claim to solve, the long term problems, but we +have been forced to conclude that reducing issuance doesn't solve them +either! * Long answer: Ask the users, especially the validators, especially the LSTs. Model user preferences so that the demand curve becomes semi-empirical -instead of theorized. Near-100\% staking seems to be baked-in long -term pretty much no matter what we do, and high inflation cannot -sustain under an issuance yield curve designed to avoid -discouragement. So the question becomes essentially "How bad will it -get in the meantime?" This is actually a question about user -preferences. Austrian School devotees may be so inflation-averse that -they are already staking all their previously-liquid ETH at -$\alpha\approx0.5$\%/yr. In contrast users who were content to grow up -with fiat currencies during periods of $\approx$3\% inflation or even -worse might not care, or would just stake in Compound or buy stETH. +instead of theorized. Near-100\% staking seems to be baked-in +eventually, and high inflation cannot sustain under an issuance yield +curve designed to avoid discouragement. So the question becomes +essentially "How bad will it get in the meantime?" This is actually a +question about user preferences. Austrian School devotees may be so +inflation-averse that they are already staking all their +previously-liquid ETH at $\alpha\approx0.5$\%/yr. In contrast users +who were content to grow up with fiat currencies during periods of +$\approx$3\% inflation or even worse might not care, or would just +stake in Compound or buy stETH. For users seeking to passively preserve wealth, staking in Compound -(or Aave, or whatever) is ideal in terms of raw Ether demand, of +(or Aave, or whatever) is ideal for generating raw Ether demand, of course. Staking in LSTs, though concerning from a governance angle, may present less of an issue than some have feared. LSTs must share some yield with users to be in business, but if they want to profit @@ -758,32 +805,39 @@ apply price uncertainty via a risk-discounting rate of, well anything really, you will see that the "maximize $r$" strategy is far from the most profitable, risk-adjusted. Of course, a speculative investor seeking to profit from our analysis would find our study woefully -short of details. So, since you have insisted on reading the "long -answer", we will end with the classic and cowardly refrain of -academics and academic-adjacents everywhere "it requires more +short of details. Put anoher way, since you have insisted on reading +the "long answer", we will end with the classic and cowardly refrain +of academics and academic-adjacents everywhere "it requires more research!" +## Externalities +There are many things we have not tried to model: price, external +markets, etc. These must enter via the sensitivity of intensives, yet +even here we have done very little. We have not assumed the +intensives $\beta,f,r,\ldots$ constant, but nor do we know enough to +reliably model their dependence on the dynamical variables +$A,s,\alpha$. For finding fixed points this has not mattered +because the intensives did not have zeros by construction. For +stability arguments, however, including how the $(A,s,\alpha)$ system +couples to cyclical forcings, this does matter quite a bit. +Every local stability condition reduces to an expression of intensives +$\betya,f,r$ and their sensitivites (partial derivatives) $\beta_s$ +etc. In making stability judgements, we have assumed that the +magnitude of sensitivities is always dominated by their corresponding +intensives. This may not be true, especially when the intensives get +very small, as they do here! There is a lot of good economics and +interesting mathematics that could go into approximating the forms of +these sensitivities, and how they couple to price, either from first +principles or data. A real understanding of Ethereum macroeconomics +will require this; we encourage you to take these tools and find what +we have missed. -The reinvestment of issuance rewards -adds to both staked ETH $$S$$ and circulating ETH $$C$$, so the effect -on staking fraction depends on the relative size of these two. -However, the reinvestment of priority fees plus MEV is simply a net -value transfer from $$C$$ to $$S$$, which can only ever increase -staking fraction. - -In a future post we turn to the challenges facing governance -centralization, sketch a potential solution, and propose a framework -by which macroeconomics interventions can be assesed. -Briefly, for the impatient: -- LST market is indeed winner take all, as many have described -- canging the yield curve doesn't change this at all -- if decreasng issuance does prevent gov capture by CEXes and LSTs then it is because - solo staers are subsidizing the protocol -- bifurcaton theory quantifies our intuitions +# Dynamical Systems References +# Footnotes [^humor]: [Open Zeppelin]() is an early icon of smart contract best practices, and continues to provide templates and auditing services in @@ -961,4 +1015,23 @@ component of the largest eigenvalue. initial conditions; the partial $x_A$ always holds $(\alpha,s)$ consant, but $\dot{A}=\alpha A$, and the only partials in $\gamma$ are $\jmath_A,\beta_A$. - \ No newline at end of file + +[^small-part]: Small sensitivities wrt intensives is not gauranteed. +Certainly a large magnitude, say $\beta_s>1$ cannot maintain for too +long: $\beta\in(0,1)$ afterall, but locally, a large derivative +$\beta_s>\beta$ is possible. + +[^stability]: Assesing the stability of very long-term +$\alpha^\star=0$ in the $(A,s,\alpha)$ model directly is interesting +but academic. One of the eigenvalues at any fixed point with +$\alpha^\star=0$ must be zero, so higher-order terms matter (the fixed +point is degenerate), and linear-stability analysis is insufficient. +A reader imbued with mathematical athleticism and free time is +encouraged to think of a Lyapunovv function $л(y)$ for issuance yield, +and derive a contraction mapping $\dot{л}(y)<0$. + +[^SU]: Readers wishing for more detail are encouraged to use the two +dimensional local stability criterion (see +[Strogatz](https://www.youtube.com/watch?v=9yh9DmNqdk4)) to solve for +the condition of eigenvalues with an imaginary part. But simulate it +too! From 155d48618a3e22d121a1a2514513b9877c6d1d93 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 08:37:38 -0600 Subject: [PATCH 33/70] fixed some typos added links --- _posts/2025-01-15-issuance-dynamics.md | 154 ++++++++++++------------- 1 file changed, 76 insertions(+), 78 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index c3a73db..f4af1d7 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -1,3 +1,18 @@ +--- +layout: post +title: "Ethereum Macroecon via Dynamics 1/2" +author: Eric Downes +categories: ["ethereum", "macroeconomics", "dynamics", "issuance"] +excerpt: Work on ETH Macro supported by the Ethereumm Foundation +usemathjax: true +thanks: We are deeply grateful to the Ethereum Foundation for their +support of this work. I, the author, feel indebted to the profound +patience, forebearance, and professionalism of Eric Siu, as this is +*months* overdo. This work has benefited significantly from +conversations with Danieli and Philipp, as well as Eric Siu, Andrew +Sudbury, and Ansgar Dietrichs. +--- + # Ethereum Macroeconomics This is the first of two posts on Ethereum macroeconomics. @@ -9,10 +24,11 @@ this supports, and the Layer-2 blockchains, a conservative valuation might be half a trillion dollars. At the core of Ethereum's "brand", distinguishing it from other smart contract platforms, is the consistent effort put into decentralized governance. Via its -[consensus mechanism]() no central authority can censor a -transaction, freeze the native asset of a user, etc. This depends in -turn on a sufficient diversity of validators staking ETH to -participate in consensus. +[consensus +mechanism](https://ethereum.org/en/developers/docs/consensus-mechanisms/pos/gasper/) +no central authority can censor a transaction, freeze the native asset +of a user, etc. This brand commitment depends in turn on a sufficient diversity of +validators staking ETH to participate in consensus. The share of Ether staked by "centralized" staking services, such as exchanges and Liquid Staking Providers (LSTs) [is @@ -76,9 +92,9 @@ $$\frac{dr}{d\alpha}|^\star<0$$ (no news here) but simultaneously 1. high enough to numerically dominate priority fees and MEV, as a fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. -The importance of this last point in moderatingh staking fraction is -very different than the view on inflation expressed, for instance [in -this podcast](). +The importance of this last point in moderating staking fraction feels +quite different to us, than the view on inflation expressed, for instance [in +this video/podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). Inflation should *eventually* decay, driving $s\to1$, though it will take some time. This "L2 future" has been recognized by many others: @@ -128,6 +144,7 @@ $$\dislaystyle \dot{V} &=& I + P - R - K \dot{U} - \dot{V} &=& K + Q_- - Q_+ - F\\ \dot{S} &=& R + Q_+ - Q_- - J\\ +\end{array} $$ | Flow Name | Symbol | Domain$\to$Codomain[^cats] | Constraint | @@ -355,36 +372,41 @@ general. Unusually for this blog post, we will show most of the steps so the argument is hopefully understood. $$\displaystyle +\begin{array}{rcl} dA = \alpha dt &\leq& \left(ys-\beta(1-s)-\jmath s\right)dt\\ dA &\leq& ysdt = y_0(1)\sqrt{s/A}dt \leq y_0(1)/\sqrt{A}dt\\ \sqrt{A}dA &\leq& y_0(1)dt\\ \int_{A(0)}^{A(t)}\sqrt{A}dA &\leq& \int_0^t y_0(1)dt\\ \frac{2}{3}x^3\Big|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ A(t) &\leq& \left(A(0)+\frac{3}{2}y_0(1)t\right)^{2/3}\\ -\therefore A(t) \ll e^{kt} ~\forall ~\mathrm{const.}~k>0 +\therefore A(t) &\ll& e^{kt} ~\forall ~\mathrm{const.}~k>0 +\end{array} $$ -The last line, for those unfamiliar is asymptotic notation. For two -positive functions, $g$ dominates $f$, written $f(t)\ll g(t)$ just -when $\lim_{t\to\infty}[f(t)/g(t)]=0$. Since supply $A(t)$ is -*eventually* less than a powerlaw of $t$, it is *subexponential*. -Thus, no positive inflation rate can maintain. +The last line is asymptotic notation. For two positive functions, $g$ +dominates $f$, written $f(t)\ll g(t)$ just when +$\lim_{t\to\infty}[f(t)/g(t)]=0$. Since supply $A(t)$ is *eventually* +less than a powerlaw of $t$, it is *subexponential*. Thus, no +positive inflation rate can maintain indefinitely. This does not mean we would find every intermediate inflation rate pleasant. Following surges in $Q_+$, inflation can accelerate quite alarmingly. A good example will be the 2032 Ethereum staking-mania. -This can be modeled in the following graph showing `qs=.05, qu=.01`; a -constant $q_+=5$\% of unstaked ETH is added to the staking queue each -year. +This can be modeled in future retrospect by the following graph +showing a simulation with `qs=.05, qu=.01`; a constant $q_+=5$\% of +unstaked ETH is added to the staking queue each year. We aren't +excited to hodl through multiple decades of 10\% inflation, and we +expect you aren't either. ![US President-Elect Swift Endorses Ethereum!]( ../assetsPosts/swifties.jpg) -Instead we mean to separate concerns. Unpleasantly high inflation in -the medium term, even if that "medium term" lasts decades, is a -*dynamics* problem, not an equilibrium problem, and so dynamical -solutions (like EIP 7514) are perhaps better suited. Unfortunately we -will see that given the above, $s\to1$ is an equilibrium problem. +Instead of downplaying fears of inflation, rather we mean to separate +concerns. Unpleasantly high inflation in the medium term, even if +that "medium term" lasts decades, is a *dynamics* problem, not an +equilibrium problem, and so dynamical solutions (like EIP 7514) seem +better suited. Unfortunately we will see that given the above, +$s\to1$ is an *equilibrium* problem. ### Staking Fraction @@ -404,7 +426,7 @@ $$ The coefficients of $(r-s),~(1-s),~(0-s)$ are variable but *positive*. Recalling how $s$ increases just when $\dot{s}>0$, these terms draw $s$ toward respective points $r,1,0$. We emphasize that the action of -yield $y$ is $x\to r$, *not* $x\to1$, unless $r\geq1$. +yield $y$ is $x\to r$, which may not be the same as $x\to1$. As expressed by the quotient rule, an increase in staking fraction can be driven by more people staking, and/or it can be driven by a @@ -469,10 +491,10 @@ $\alpha\approx0$ and sensityivities are weak, $|\xi|\sim\gamma\sim y$ obtains. Generally if $|\xi|,\gamma\ll1$ then the derivatives obey $|\dot{\alpha}|\ll|\dot{s}|$: a [seperation of timescales](https://en.wikipedia.org/wiki/Method_of_averaging). For -durations when this obtains, short periods at intremediate times, -inflation could be treated as a parameter instead of its own dynamic -variable, with staking fraction equilibrating to $s^\star$ more -quickly than $\alpha^\star$ equilibrates. +durations when this obtains, one or more periods at intremediate +times, inflation can be usefully approximated as a parameter instead +of its own dynamic variable, with staking fraction equilibrating to +$s^\star$ more quickly than $\alpha^\star$ equilibrates. Does this hold presently? For a sanity-check, a quick look at YCharts since The Merge shows that $$s,\dot{s}$$ do indeed seem to vary over a @@ -501,9 +523,9 @@ constant $\alpha=\alpha_{const}$. $$\displaystyle s^\star = \frac{ - r^\star(\alpha_0 + f^\star) + q_+^\star + r^\star(\alpha_{const} + f^\star) + q_+^\star }{ - m^\star := (\alpha_0 + r^\star f^\star + q_+^\star + q_-^\star + (1-r)\jmath^\star) + m^\star := (\alpha_{const} + r^\star f^\star + q_+^\star + q_-^\star + (1-r)\jmath^\star) } = 1 - \frac{(1-r)\jmath+q_-}{m^star} - \frac{\alpha_0(1-r^\star)}{m^\star} $$ @@ -543,11 +565,12 @@ simplifies to $$\displaystyle s^\star \approx r^\star \frac{ - \alpha^\star + f^\star}{ - \alpha^\star + r^\star f^\star + (1-r)\jmath^\star} + \alpha_const} + f^\star}{ + \alpha_const} + r^\star f^\star + (1-r)\jmath^\star} = 1 - \frac{ \jmath^\star}{ - \alpha^\star + r^\star f^\star + (1-r)\jmath^\star} + \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} + - \frac{\alpha_{const}(1-2)}{\alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} $$ We will explore the *stability* of this fixed point below, and based on @@ -598,34 +621,34 @@ runaway staking fraction how would this work? ### Low Inflation; Even Lower Fees Well, that was deflating! Let's cheer ourselves up by considering the -behaviors under $\alpha>0$. +behaviors under $\alpha_{const}>0$. A positive role for inflation can +be seen in the contours of the market equilibrium staking fraction +$$s^\star$$ corresponding to $$\dot{s}=0$$, shown here with +$\alpha^\star=\alpha_{const}$. -A positive role for inflation can be seen in the contours of the -market equilibrium staking fraction $$s^\star$$ corresponding to -$$\dot{s}=0$$, shown in [figure -1](../assetsPosts/2025-01-15-issuance-fundamentals/staking-fixpoint.png). +![alpha vs. s Figure](../assetsPosts/2025-01-15-issuance-fundamentals/staking-fixpoint.png). -To find the equilibrium values -$$(\alpha^\star/f^\star,\,r^\star)$$ necessary to acheive a desired -staking fraction $$x^\star$$, simply pick a colored contour in the -figure: these are the values of constant $$x^\star$$. For every point -on this curve, the equilibrium inflation:fee ratio -$$\alpha^\star/f^\star$$ is the x-coordinate, and the equilibrium -reinvestment ratio $$r^\star$$ is the y-value. +To find the equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ +necessary to acheive a desired staking fraction $$x^\star$$, simply +pick a colored contour in the figure: these are the values of constant +$$x^\star$$. For every point on this curve, the equilibrium +inflation:fee ratio $$\alpha^\star/f^\star$$ is the x-coordinate, and +the equilibrium reinvestment ratio $$r^\star$$ is the y-value. A breakdown of limitng behaviors is illustrative under positive inflation. For any value of non-negative inflation, $$r^\star$$ is a lower bound for the equilibrium staking fraction we should expect. If -inflation dominates fees, $$\alpha\gg f$$ then $s^\star$ is larger by -a small amount than $r^\star$, while if fees dominate inflation -$$\alpha\ll f$$ and $s^\star$ becomes insensitive to non-zero -reinvestment ratio and $$s^\star\to1$$. For a numerical comparison, -eyeballing charts (so *extremely* rough approximations here) $$f +inflation dominates fees, $$\alpha_{const}\gg f^\star$$ then $s^\star$ +is larger by a small amount than $r^\star$, while if fees dominate +inflation $$0<\alpha_{const}\ll f^\star$$ then $s^\star$ becomes +insensitive to non-zero reinvestment ratio and $$s^\star\to1$$. For a +numerical comparison, eyeballing charts (so *extremely* rough +approximations here, possibly off by an order of magnitude) $$f \approx .002<.005\approx\alpha$$ so to within 10\% error above, $$s^\star\approx r^\star$$ over the range of $$r\in(.5,.75)$$ inferred from the Lido yield rate. -How these transient values $$(\alpha/f,r)$$ relate to their +How the transient values $$(\alpha_{now}/f,r)$$ relate to the true equilibrium values $$(\alpha^\star/f^\star,r^\star)$$ depends on some considerations: @@ -634,11 +657,10 @@ some considerations: reinvest the most; $r^\star\approx r_{max}$, where $r_{max}$ is assesed over all staking pools with at least 10\% of $S$. -* As we have seen from $\dot{\alpha}$ above, once $\dot{s}\to0$, - inflation decays toward zero unless externalities intervene, but - with a small relaxation term. So *very roughly* - $\alpha_{now}\approx\alpha^\star$ as an overestimate rule of thumb, - that more detailed simulation work could improve. +* We are holding $\alpha_{const}=\alpha^\star$, so + $\alpha_{now}\approx\alpha^\star$ but a more sophisticated approximation + is likely possible keeping within the two-timescale context... maybe + you'll find one! ### Runaway $r$ from Inflation pressure @@ -810,30 +832,6 @@ the "long answer", we will end with the classic and cowardly refrain of academics and academic-adjacents everywhere "it requires more research!" -## Externalities - -There are many things we have not tried to model: price, external -markets, etc. These must enter via the sensitivity of intensives, yet -even here we have done very little. We have not assumed the -intensives $\beta,f,r,\ldots$ constant, but nor do we know enough to -reliably model their dependence on the dynamical variables -$A,s,\alpha$. For finding fixed points this has not mattered -because the intensives did not have zeros by construction. For -stability arguments, however, including how the $(A,s,\alpha)$ system -couples to cyclical forcings, this does matter quite a bit. - -Every local stability condition reduces to an expression of intensives -$\betya,f,r$ and their sensitivites (partial derivatives) $\beta_s$ -etc. In making stability judgements, we have assumed that the -magnitude of sensitivities is always dominated by their corresponding -intensives. This may not be true, especially when the intensives get -very small, as they do here! There is a lot of good economics and -interesting mathematics that could go into approximating the forms of -these sensitivities, and how they couple to price, either from first -principles or data. A real understanding of Ethereum macroeconomics -will require this; we encourage you to take these tools and find what -we have missed. - # Dynamical Systems References From a214234c8d5f8c91675e0adb9d35f47b95411d0b Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 09:51:14 -0600 Subject: [PATCH 34/70] simplified intro --- _posts/2025-01-15-issuance-dynamics.md | 597 +++++++++++++------------ 1 file changed, 300 insertions(+), 297 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index f4af1d7..78d0214 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -1,16 +1,16 @@ --- layout: post -title: "Ethereum Macroecon via Dynamics 1/2" +title: "Ethereum Macroeconomics via Dynamics 1/2" author: Eric Downes categories: ["ethereum", "macroeconomics", "dynamics", "issuance"] excerpt: Work on ETH Macro supported by the Ethereumm Foundation usemathjax: true thanks: We are deeply grateful to the Ethereum Foundation for their support of this work. I, the author, feel indebted to the profound -patience, forebearance, and professionalism of Eric Siu, as this is -*months* overdo. This work has benefited significantly from -conversations with Danieli and Philipp, as well as Eric Siu, Andrew -Sudbury, and Ansgar Dietrichs. +patience and forebearance of Eric Siu and my colleagues at 20 Squares, +as this is *months* overdo. This work has benefited +from conversations with Danieli and Philipp, as well as Eric Siu, +Andrew Sudbury, Ansgar Dietrichs, and Mac Budkowski. --- # Ethereum Macroeconomics @@ -34,18 +34,22 @@ The share of Ether staked by "centralized" staking services, such as exchanges and Liquid Staking Providers (LSTs) [is considerable](https://dune.com/queries/2394100/3928083), and continues to grow. This has provoked [concerns](https://issuance.wtf/), among -Ethereum researchers that the future of Ethereum might involve +Ethereum researchers that the future of Ethereum might involve a +confluence of three interrelated challenges: -1. Nearly all Ether staked $s\to1$, such that or independently -2. The de facto currency is controlled by a confederation of centralized services. +1. Nearly all Ether becomes staked. +1. Inflation becomes excessive. +1. Governance becomes centralized. ## Lookahead In this blog post we address the first of these concerns "runaway -(near 100\%) staking" using a "stock and flow" macroecnomics model -built with guidance from dynamical system theory. In contrast with -other research we find inflation playing a positive but temporary role -in moderating runaway staking, though most likely $$s\to1$ eventually. +(near 100\%) staking" $$s\to1$$ and how it relates to the second, using a "stock +and flow" macroecnomics model built with guidance from dynamical +system theory. In contrast with other research we find inflation +playing a positive but temporary role in moderating runaway staking, +though most likely $$s\to1$$ eventually. + In the second post, we look more closely at governance centralization and discuss a means for evaluating macroeconomic interventions inspred by bifurcation theory. Briefly, we are not optimstic that reducing @@ -53,13 +57,14 @@ issuance will prevent governance centralization, either. In both posts, we provide a few code examples using [ethode](https://github.com/20squares/ethode/) a thin units-aware -wrapper we built around `scipy.odeint` to streamline model evaluation. -Readers desiring to follow our derivations, dive into technical -mathematical points not covered here, run their own simulations, or -learn some dynamical systems are recommended to look at our ethode +wrapper we built around `scipy.integrate.solve_ivp` to streamline +model evaluation. Readers desiring to follow our derivations, dive +into technical mathematical points not covered here, run their own +simulations, or learn some dynamical systems are recommended to look +at our ethode [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md), -and references therein. It is certainly a work in progress, but should have -enough to get you going. +which contains [References](#references) section below. The guide is +certainly a work in progress, but should have enough to get you going. ## For The Impatient! @@ -75,11 +80,11 @@ persistently decreases. We use our macroeconmics model to identify a "low inflation; even lower fees" regime (LI;ELF). Outside of LI;ELF convergence to a -desirable staking future without runaway staking is unlikely. Under -strong deflation, in which the magnitude of deflation exceeds the +desirable staking future without runaway staking is unlikely. Strong +deflation, in which the magnitude of deflation exceeds the reinvestment of transaction fees, probably corresponds to unstable -dynamics. Under zero or low deflation, the tendency toward runaway -staking can be moderated only by very high churn and/or slashing. +dynamics. Under zero or weak deflation, the tendency toward runaway +staking can be moderated only by high churn and/or slashing. In contrast within LI;ELF staked ETH fraction approaches near the reinvestment ratio. Thus, runaway staking can be avoided only while @@ -94,12 +99,13 @@ fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. The importance of this last point in moderating staking fraction feels quite different to us, than the view on inflation expressed, for instance [in -this video/podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). +this helpful review podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). -Inflation should *eventually* decay, driving $s\to1$, though it will -take some time. This "L2 future" has been recognized by many others: -most Ether is staked, with the majority used for settlement of L2 -rollups. We'll discuss it more next time. +Inflation should *eventually* decay, driving $$s\to1$$ if it is not +already there, though it may take some time. This "L2 future" has +been recognized by many others: most Ether is staked, with the +majority used for settlement of L2 rollups. We'll discuss it more +next time. Given all the above, we advise caution. Intervening to reduce the issuance yield curve seems quite capable of exaccerbating the very @@ -108,9 +114,9 @@ problems we seek to avoid. ## Modeling an Open Zeppelin[^humor] First, a warning! In an act of hubris, and with apologies, but not -without reasons[^reasons], we have chosen $S$ to refer to Staked ETH, -while others have at times used $S$ for "circulating (S)upply", which -we call instead $A$, so $s=S/A$. Please proceed! +without reasons[^reasons], we have chosen $$S$$ to refer to Staked ETH, +while others have at times used $$S$$ for "circulating (S)upply", which +we call instead $$A$$, so $$s=S/A$$. Please proceed! ![Ethereum as a balloon with compartments.]( ../assetsPosts/2025-01-15-issuance-dynamics/eth-balloon.jpg) @@ -118,22 +124,22 @@ we call instead $A$, so $s=S/A$. Please proceed! Consider a "balloon" with variable internal compartments. The average size of each is measured by *stocks* -- ($S$)taked Ether (participating in [consensus]()) is a compartment, as is -- ($U$)nstaked unburnt Ether, - -- containing the ($V$)alidator reward queue. -- ($\cancel{O}$) is all irrecoverable (burned, lost, etc.) Ether, and -- ($A$)ccessible/Circul($A$)ting Ether supply, $A=S+U\approx120.4\times10^6$ in Dec 2024. -- $\mathcal{Q}_\pm$ the Ether in the staking ($+$) and unstaking $-$) queues +- ($$S$$)taked Ether (participating in [consensus]()) is a compartment, as is +- ($$U$$)nstaked unburnt Ether, + -- containing the ($$V$$)alidator reward queue. +- ($$\cancel{O}$$) is all irrecoverable (burned, lost, etc.) Ether, and +- ($$A$$)ccessible/Circul($$A$$)ting Ether supply, $$A=S+U\approx120.4\times10^6$$ in Dec 2024. +- $$\mathcal{Q}_\pm$$ the Ether in the staking ($$+$$) and unstaking $$-$$) queues The net change in time of a stock is written using a dot, such as -$\frac{dA}{dt}:=\dot{A}$[^partial], the net change in accessible Ether +$$\frac{dA}{dt}:=\dot{A}$$[^partial], the net change in accessible Ether supply. Stocks grow or shrink based on flows which add to or subtract from their derivatives. Here all flows are positive real numbers with units \[ETH/yr\]. By averaging over "long" timescales (at least quarterly)[^aves] we approximate the staking and unstaking queues as equilibrated; -$\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$, and average over many +$$\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$$, and average over many cycles of the erratic base fee oscillations. So, our conceptual model: @@ -147,88 +153,88 @@ $$\dislaystyle \end{array} $$ -| Flow Name | Symbol | Domain$\to$Codomain[^cats] | Constraint | -| :-- | :-- | :-: | :-- | -| Tx Fees | $F$ | $U\to\cancel{O},V$ | $B+P=F\jmath+q_-$ -staked ETH $S$ just continues growing and growing. In contrast it is -harder for $U$ to get as big, limited by its own loss term -$-\left(rf+(1-r)bf+q_+\right)\cdot U$. At some point in the (far) -future $S$ becomes big enough that $ry(S)<\jmath+q_-$ and the system +this model's evolution. Specifically, so long as $$ry(S)>\jmath+q_-$$ +staked ETH $$S$$ just continues growing and growing. In contrast it is +harder for $$U$$ to get as big, limited by its own loss term +$$-\left(rf+(1-r)bf+q_+\right)\cdot U$$. At some point in the (far) +future $$S$$ becomes big enough that $$ry(S)<\jmath+q_-$$ and the system becomes *capable* of oscillation, depending on parameters and a zoo of partial derivatives.[^SU] @@ -323,22 +329,22 @@ su.sim() ``` These dynamic variables are kind of boring, but critically and unlike -$(A,S)$, there are *no extra conditions* (such as $S0$, these terms draw -$s$ toward respective points $r,1,0$. We emphasize that the action of -yield $y$ is $x\to r$, which may not be the same as $x\to1$. +The coefficients of $$(r-s),~(1-s),~(0-s)$$ are variable but *positive*. +Recalling how $$s$$ increases just when $$\dot{s}>0$$, these terms draw +$$s$$ toward respective points $$r,1,0$$. We emphasize that the action of +yield $$y$$ is $$x\to r$$, which may not be the same as $$x\to1$$. As expressed by the quotient rule, an increase in staking fraction can be driven by more people staking, and/or it can be driven by a @@ -434,20 +440,20 @@ reduction of the inflation rate. The latter can be acheived in principle by a reduction of issuance relative to the base fee "burn rate". Because of this quotient rule tradeoff, issuance plays a beneficial "infrastructure" role in moderating staking fraction, -drawing it toward $r$, which can be less than one. +drawing it toward $$r$$, which can be less than one. -### $(A,\alpha,s)$ Dynamical System +### $$(A,\alpha,s)$$ Dynamical System There are many reasons, especially in the context of the world -economy, to care about total circulating supply $A$. Recent -discussions however have focused most on inflation $\alpha=\dot{A}/A$, +economy, to care about total circulating supply $$A$$. Recent +discussions however have focused most on inflation $$\alpha=\dot{A}/A$$, the growth in supply over time. It also turns out that modeling -inflation $\alpha$ directly simplifies our analysis of the $(A,s)$ +inflation $$\alpha$$ directly simplifies our analysis of the $$(A,s)$$ system considerably. Let subscripts denote partial derivatives -$x_y:=\frac{\partial{x}}{\partial{y}}$. Using the correct partial -derivative relations for variables $(A,\alpha,s,t)$[^partial] we have +$$x_y:=\frac{\partial{x}}{\partial{y}}$$. Using the correct partial +derivative relations for variables $$(A,\alpha,s,t)$$[^partial] we have $$\displaystyle \begin{array}{rcl} @@ -458,43 +464,43 @@ $$\displaystyle $$ Where the new greek letters are fractional rates, defined below, and -$y':=\frac{dy}{dS}$. +$$y':=\frac{dy}{dS}$$. -* $\mu:=\beta_\alpha(1-s)+\jmath_\alpha s$ is the implicit +* $$\mu:=\beta_\alpha(1-s)+\jmath_\alpha s$$ is the implicit sensitivity of the inflation loss-term to increases in inflation. - We judge $0\leq\mu$; if anything inflation increases burn and + We judge $$0\leq\mu$$; if anything inflation increases burn and slashing fractional rates.[^mu] -* $\xi:=(y+y'A+\beta-\beta_s(1-s)-j-j_s)/(1+\mu)$ is the net - correlation between changes in $s$ and changes in $\alpha$ - normalized by $1+\mu$. $\xi$ can be of either sign. Under the - current yield curve $y_0+\frac{dy}{dS}A=y_0(sA)(1-1/(2s))$, which +* $$\xi:=(y+y'A+\beta-\beta_s(1-s)-j-j_s)/(1+\mu)$$ is the net + correlation between changes in $$s$$ and changes in $$\alpha$$ + normalized by $$1+\mu$$. $$\xi$$ can be of either sign. Under the + current yield curve $$y_0+\frac{dy}{dS}A=y_0(sA)(1-1/(2s))$$, which changes its sign at 50\% ETH staked. -* $\gamma:=j_{\log{A}}s+\beta_{\log{A}}(1-s)+s|y'|A$ is a - positive coefficient expressing how quickly $\alpha\to\alpha^\star$, +* $$\gamma:=j_{\log{A}}s+\beta_{\log{A}}(1-s)+s|y'|A$$ is a + positive coefficient expressing how quickly $$\alpha\to\alpha^\star$$, and the partials are constant when initial supply is known.[^ics] We have extracted the sign from the final term because sublinear - issuance implies $y'<0$; under the current yield curve the term - $sA|y'|=\frac{1}{2}y_0(sA)$. + issuance implies $$y'<0$$; under the current yield curve the term + $$sA|y'|=\frac{1}{2}y_0(sA)$$. -* $\chi:=-j_ts-\beta_t(1-s)$ represents externalities affecting the +* $$\chi:=-j_ts-\beta_t(1-s)$$ represents externalities affecting the inflation loss term encoded as explicit time-dependencies. We - neglect externalities $\chi\approx0$ because we have nothing + neglect externalities $$\chi\approx0$$ because we have nothing intelligent to say about them, but you might not want to. -#### Separation of Timescales and $\alpha>0$ +#### Separation of Timescales and $$\alpha>0$$ -Observe that every term in $(\xi,\gamma)$ are small fractions, +Observe that every term in $$(\xi,\gamma)$$ are small fractions, fractional rates, or derivatives thereof. Indeed, when -$\alpha\approx0$ and sensityivities are weak, $|\xi|\sim\gamma\sim y$ -obtains. Generally if $|\xi|,\gamma\ll1$ then the derivatives obey -$|\dot{\alpha}|\ll|\dot{s}|$: a [seperation of +$$\alpha\approx0$$ and sensityivities are weak, $$|\xi|\sim\gamma\sim y$$ +obtains. Generally if $$|\xi|,\gamma\ll1$$ then the derivatives obey +$$|\dot{\alpha}|\ll|\dot{s}|$$: a [seperation of timescales](https://en.wikipedia.org/wiki/Method_of_averaging). For durations when this obtains, one or more periods at intremediate times, inflation can be usefully approximated as a parameter instead of its own dynamic variable, with staking fraction equilibrating to -$s^\star$ more quickly than $\alpha^\star$ equilibrates. +$$s^\star$$ more quickly than $$\alpha^\star$$ equilibrates. Does this hold presently? For a sanity-check, a quick look at YCharts since The Merge shows that $$s,\dot{s}$$ do indeed seem to vary over a @@ -510,16 +516,16 @@ concept for Ethereum Macroeconomics. # Staking Fraction in 1D -Recall our approximate equation for the fraction of staked ETH $s$, in -which all coefficients are positive but inflation $\alpha$: +Recall our approximate equation for the fraction of staked ETH $$s$$, in +which all coefficients are positive but inflation $$\alpha$$: $$\displaystyle \dot{s} &=& \alpha\cdot(r-s) + (rf+q_+)\cdot(1-s) + (q_-+(1-r)j)\cdot(0-s) $$ -So assuming $|\dot{\alpha}|\ll|\dot{s}|$, let us examine the fixed -point $s^\star$ during a period in which we may treat inflation as -constant $\alpha=\alpha_{const}$. +So assuming $$|\dot{\alpha}|\ll|\dot{s}|$$, let us examine the fixed +point $$s^\star$$ during a period in which we may treat inflation as +constant $$\alpha=\alpha_{const}$$. $$\displaystyle s^\star = \frac{ @@ -530,20 +536,20 @@ s^\star = \frac{ = 1 - \frac{(1-r)\jmath+q_-}{m^star} - \frac{\alpha_0(1-r^\star)}{m^\star} $$ -We will see that without $\alpha>0$ an interior market equillibrium is -very unlikely. First, as a thought experiment, consider $\alpha=0$. +We will see that without $$\alpha>0$$ an interior market equillibrium is +very unlikely. First, as a thought experiment, consider $$\alpha=0$$. ## Concerning Churn -A fixed point $s^\star(\alpha=0)<1$ would require either high churn or +A fixed point $$s^\star(\alpha=0)<1$$ would require either high churn or a persistent unstaking/capitulation of existing validators -$q_-+\jmath>0$. Validators try to minimize slashing as intended, but what would a -large unstaking flow $q_-s\gtrsim rf$ at $s^\star$ mean? +$$q_-+\jmath>0$$. Validators try to minimize slashing as intended, but what would a +large unstaking flow $$q_-s\gtrsim rf$$ at $$s^\star$$ mean? High unstaking requires *churn*, a persistent supply of new validators -to take their place $q_+^\star(1-s)\sim q_-s\gtrsim rf$, or it is only -a transient and $q_-^\star\ll rf$; recall that reinvestment by -existing validators is not counted in $q_+$. Persistently high +to take their place $$q_+^\star(1-s)\sim q_-s\gtrsim rf$$, or it is only +a transient and $$q_-^\star\ll rf$$; recall that reinvestment by +existing validators is not counted in $$q_+$$. Persistently high unstaking $$q_-\sim rf$$ could only describe a market equilibrium if one group of stakers was actively capitulating and withdrawing their stake, while another group with a higher $$r$$ were aggressively @@ -553,11 +559,11 @@ offset the unstaking, adjusted for inflation. High churn, not counting reinvestment, cannot maintain forever: eventually there will be no new Capitulators left, and $$s^\star$$ must once again grow as required by the Reinvestors' higher $$r$$, so -$$s^\star$$ was not a fixed point at all. So $q_-^\star\ll +$$s^\star$$ was not a fixed point at all. So $$q_-^\star\ll (rf)^\star$$. Similarly, at some point everyone who wants to stake should have -staked. If we judge the $\tau$-averaged flows due to the issuance of +staked. If we judge the $$\tau$$-averaged flows due to the issuance of new humans and the burn rate of legacy humans to be small, additional validators count overwhelmingly toward $$r^\star$$ and $$q_+^\star\ll(rf)^\star$$. Thus, the fixed point $$s^\star$$ @@ -574,7 +580,7 @@ s^\star \approx r^\star \frac{ $$ We will explore the *stability* of this fixed point below, and based on -the range of $\alpha$ categorize three basic behaviors. +the range of $$\alpha$$ categorize three basic behaviors. ## Stability in One Dimension @@ -591,8 +597,8 @@ fixed point is an unstable *source*. ![1D Stability Conditions](../assetsPosts/1d-stab.png) Specifically, we want the sign of -$\left.\frac{\partial\dot{s}}{\partial s}\right|^\star$ to determine -wether $s^\star$ is (un)stable. We will be ignoring the partial +$$\left.\frac{\partial\dot{s}}{\partial s}\right|^\star$$ to determine +wether $$s^\star$$ is (un)stable. We will be ignoring the partial derivatives ("sensitivities") by assuming they are small in comparison with their corresponding intensives.[^small-part] @@ -600,31 +606,31 @@ with their corresponding intensives.[^small-part] ### Weak Deflation -If $\alpha_{const}\in(-rf-\jmath,0]$ then an interior market equilibrium -$s^\star$ requires high slashing, as we argued before based on +If $$\alpha_{const}\in(-rf-\jmath,0]$$ then an interior market equilibrium +$$s^\star$$ requires high slashing, as we argued before based on Churn. All other things being equal, it is also less stable. ### Strong Deflation -Consider now $\alpha<-rf-\jmath$. This could happen for instance if +Consider now $$\alpha<-rf-\jmath$$. This could happen for instance if the issuance curve were reduced particularly bluntly, or changes in -fundamentals drove either MEV or the base fee (and thus $f$) to a -persistently higher amount, such that $ys\ll bf(1-s)+\jmath s$. +fundamentals drove either MEV or the base fee (and thus $$f$$) to a +persistently higher amount, such that $$ys\ll bf(1-s)+\jmath s$$. These conditions cannot maintain of course. You don't need -differential equations to see that $\alpha<0$ shrinks $A$, which -eventually raises $y(sA)$. But as a temporary intervention to tame +differential equations to see that $$\alpha<0$$ shrinks $$A$$, which +eventually raises $$y(sA)$$. But as a temporary intervention to tame runaway staking fraction how would this work? ### Low Inflation; Even Lower Fees Well, that was deflating! Let's cheer ourselves up by considering the -behaviors under $\alpha_{const}>0$. A positive role for inflation can +behaviors under $$\alpha_{const}>0$$. A positive role for inflation can be seen in the contours of the market equilibrium staking fraction $$s^\star$$ corresponding to $$\dot{s}=0$$, shown here with -$\alpha^\star=\alpha_{const}$. +$$\alpha^\star=\alpha_{const}$$. ![alpha vs. s Figure](../assetsPosts/2025-01-15-issuance-fundamentals/staking-fixpoint.png). @@ -638,9 +644,9 @@ the equilibrium reinvestment ratio $$r^\star$$ is the y-value. A breakdown of limitng behaviors is illustrative under positive inflation. For any value of non-negative inflation, $$r^\star$$ is a lower bound for the equilibrium staking fraction we should expect. If -inflation dominates fees, $$\alpha_{const}\gg f^\star$$ then $s^\star$ -is larger by a small amount than $r^\star$, while if fees dominate -inflation $$0<\alpha_{const}\ll f^\star$$ then $s^\star$ becomes +inflation dominates fees, $$\alpha_{const}\gg f^\star$$ then $$s^\star$$ +is larger by a small amount than $$r^\star$$, while if fees dominate +inflation $$0<\alpha_{const}\ll f^\star$$ then $$s^\star$$ becomes insensitive to non-zero reinvestment ratio and $$s^\star\to1$$. For a numerical comparison, eyeballing charts (so *extremely* rough approximations here, possibly off by an order of magnitude) $$f @@ -653,21 +659,21 @@ equilibrium values $$(\alpha^\star/f^\star,r^\star)$$ depends on some considerations: * If indeed churn dies down and slashing stays relatively rare, then - $r$ increases to reflect the growing share of businesses that - reinvest the most; $r^\star\approx r_{max}$, where $r_{max}$ is - assesed over all staking pools with at least 10\% of $S$. + $$r$$ increases to reflect the growing share of businesses that + reinvest the most; $$r^\star\approx r_{max}$$, where $$r_{max}$$ is + assesed over all staking pools with at least 10\% of $$S$$. -* We are holding $\alpha_{const}=\alpha^\star$, so - $\alpha_{now}\approx\alpha^\star$ but a more sophisticated approximation +* We are holding $$\alpha_{const}=\alpha^\star$$, so + $$\alpha_{now}\approx\alpha^\star$$ but a more sophisticated approximation is likely possible keeping within the two-timescale context... maybe you'll find one! -### Runaway $r$ from Inflation pressure +### Runaway $$r$$ from Inflation pressure -Could the sensitivity $r_\alpha$ be sufficient such that even at -intermediate timescales we see $s\to1$? This is certainly possible; +Could the sensitivity $$r_\alpha$$ be sufficient such that even at +intermediate timescales we see $$s\to1$$? This is certainly possible; per the arguments of Ethereum researchers, high inflation could still -lead to runaway staking if $r$ is sensitive enough. +lead to runaway staking if $$r$$ is sensitive enough. In our model the net effect of inflation on LI;ELF staking fraction equilibrium is reflected by taking the derivative @@ -696,36 +702,36 @@ that can be tracked for Ethereum health. We saw above a few things: -1. Reinvestment $r$ is a lower-bound for $s^\star$. +1. Reinvestment $$r$$ is a lower-bound for $$s^\star$$. 1. Low (but positive) inflation moderates staking fraction closer to this lower bound at intermediate timescales -1. Positive inflation cannot maintain indefinately, so eventually $s\to1$. +1. Positive inflation cannot maintain indefinately, so eventually $$s\to1$$. Conceptually, how can inflation *moderate* staking fraction, though? Shouldn't more staking lead to more issuance, which leads to more inflation, etc.? Briefly the reasons are: -* Short $Q_+$ vs. Long Term $R$ Investment -* The Quotient Rule $\dot{S}=\dot{S}/A-s\dot{A}/A$ +* Short $$Q_+$$ vs. Long Term $$R$$ Investment +* The Quotient Rule $$\dot{S}=\dot{S}/A-s\dot{A}/A$$ ## Short Term vs. Long Term. -Novel investment in staking $Q_+$ is driven largely by speculation, -and new users encountering Ethereum. $Q_+$ acts to increase staking -fraction, as seen above, and indeed the glut in $Q_+$ since the Merge +Novel investment in staking $$Q_+$$ is driven largely by speculation, +and new users encountering Ethereum. $$Q_+$$ acts to increase staking +fraction, as seen above, and indeed the glut in $$Q_+$$ since the Merge may have been the source for all of the alarm that prompted this study. Novel speculative investment must eventually dry up, and be -replaced by long-term investmemnt $R$, because +replaced by long-term investmemnt $$R$$, because 1. everyone with money who wants to stake eventually will, so will be -counted in $R$ not $Q_+$ +counted in $$R$$ not $$Q_+$$ 2. any business that wants to stay in -business cannot consistently reinvest more than its revenue $R\leq -I+P$. +business cannot consistently reinvest more than its revenue $$R\leq +I+P$$. -Of the long term signal $R$, only the issuance portion of +Of the long term signal $$R$$, only the issuance portion of reinvestment, that is the part that contributes to inflation, can -moderate $s$, which brings us to the quotient rule. +moderate $$s$$, which brings us to the quotient rule. ## The Quotient Rule @@ -733,12 +739,12 @@ $$\displaystyle \dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}=\frac{\dot{S}}{S}-s\alpha $$ -What increases $s$ is any increase in staked Ether $S$, but also any -net decrease in $A=S+U$. Inflation $\alpha$ increases *both* $U$ and -$S$, because some of that increase is used to meet costs and take -profit $(1-r)\alpha$, which increases $U$ relative to $S$. In +What increases $$s$$ is any increase in staked Ether $$S$$, but also any +net decrease in $$A=S+U$$. Inflation $$\alpha$$ increases *both* $$U$$ and +$$S$$, because some of that increase is used to meet costs and take +profit $$(1-r)\alpha$$, which increases $$U$$ relative to $$S$$. In contrast, the reinvestment of transaction fees can only ever increase -$S$ at the expense of $U$. Thus transaction fees always act to +$$S$$ at the expense of $$U$$. Thus transaction fees always act to increase staked fraction, while the effect of inflation depends on the relative values of reinvestment ratio and staking fraction. @@ -746,8 +752,8 @@ So, while we could certainly model reinvestment differently, and there are lags we are blithely integrating over, we think that these market forces will still act as described above in a different model. It is possible that even during sustained inflation, these effects will be -unable to prevent the upward creep in $s$, because $r$ is too large, -or the sensitivity of $r$ to inflation at equilibrium is too great, a +unable to prevent the upward creep in $$s$$, because $$r$$ is too large, +or the sensitivity of $$r$$ to inflation at equilibrium is too great, a condition which we mathematized above. In fact we expect that every argument about inflation effects driving increased staking, overpaying for security, etc. could (perhaps should) be rephrased in terms of @@ -756,10 +762,10 @@ preferences of ETH users for, and thus their behavior in reaction to, inflation rate etc., which thusfar are not measured. We encourage the community to rectify this! -## Can reflexivity prevent $s\to1$? +## Can reflexivity prevent $$s\to1$$? Regarding the possible effects of reflexivity. We have neglected even -discussing oscillations in $(S,U)$, even though the model is plainly +discussing oscillations in $$(S,U)$$, even though the model is plainly capable of such behavior under different parameters or when coupled to price. Why such negligence? If cycles *do* arise, we expect market participants, anticipating such cycles, would act to profit off of @@ -775,10 +781,10 @@ with. Could there be a similar effect with staking fraction or inflation? In short we have no idea, but put it as a challenge to the reader. Can you devise a cryptoeconomic protocol or trading strategy that -forestalls the seemingly inevitable $s^\star\to1$? If not, can you +forestalls the seemingly inevitable $$s^\star\to1$$? If not, can you prove this is impossible? -## Should We Change $y$? +## Should We Change $$y$$? So, finally... should the Ethereum community reduce issuance? @@ -791,7 +797,7 @@ Nope. If you are very inflation averse or you want to slow down the transition to high staking, please consider and simulate downward adjusting the constants in EIP 7514, adopted during the Deneb upgrade. -This already directly limits $R+Q_+$, but was very nicely designed +This already directly limits $$R+Q_+$$, but was very nicely designed to not interfere with existing staking flows. EIP 7514 does not solve, and does not claim to solve, the long term problems, but we have been forced to conclude that reducing issuance doesn't solve them @@ -807,9 +813,9 @@ curve designed to avoid discouragement. So the question becomes essentially "How bad will it get in the meantime?" This is actually a question about user preferences. Austrian School devotees may be so inflation-averse that they are already staking all their -previously-liquid ETH at $\alpha\approx0.5$\%/yr. In contrast users +previously-liquid ETH at $$\alpha\approx0.5$$\%/yr. In contrast users who were content to grow up with fiat currencies during periods of -$\approx$3\% inflation or even worse might not care, or would just +$$\approx$$3\% inflation or even worse might not care, or would just stake in Compound or buy stETH. For users seeking to passively preserve wealth, staking in Compound @@ -817,14 +823,14 @@ For users seeking to passively preserve wealth, staking in Compound course. Staking in LSTs, though concerning from a governance angle, may present less of an issue than some have feared. LSTs must share some yield with users to be in business, but if they want to profit -themselves must maintain $r<1$. +themselves must maintain $$r<1$$. -So how high can $r_{LST}$ go before LSTs find the loss of gross +So how high can $$r_{LST}$$ go before LSTs find the loss of gross profits unacceptable? Great question. We will see next time that as many have recognized, the entity/group that can maintain the highest -$r$ wins the race eventually. It's worth noting however that if you +$$r$$ wins the race eventually. It's worth noting however that if you apply price uncertainty via a risk-discounting rate of, well anything -really, you will see that the "maximize $r$" strategy is far from the +really, you will see that the "maximize $$r$$" strategy is far from the most profitable, risk-adjusted. Of course, a speculative investor seeking to profit from our analysis would find our study woefully short of details. Put anoher way, since you have insisted on reading @@ -844,31 +850,31 @@ models, etc. and hopefully they will not sue us for using their name in a bad dynamical systems joke. [^aves]: We use moving quarterly averages, though any timescale -$\tau$ sufficiently long that the [fast dynammics]() of [the base]() +$$\tau$$ sufficiently long that the [fast dynammics]() of [the base]() fee are [integrated out](), and the lags from (un)staking queues are not appreciable; see [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md) for details. As we are averaging -quarterly, we set the staking $\dot{\mathcal{Q}}_+$, unstaking -$\dot{\mathcal{Q}}_-$, and reward $\dot{W}$ queues to zero, including -their respective flows ($R+Q_+,Q_-,I+P$) in their target stocks -($S,U,V$); even if ethereum produces empty blocks, so long as the -reward queue is not empty $U>0$. +quarterly, we set the staking $$\dot{\mathcal{Q}}_+$$, unstaking +$$\dot{\mathcal{Q}}_-$$, and reward $$\dot{W}$$ queues to zero, including +their respective flows ($$R+Q_+,Q_-,I+P$$) in their target stocks +($$S,U,V$$); even if ethereum produces empty blocks, so long as the +reward queue is not empty $$U>0$$. [^IyS]: The approximation is due to the use of quarterly averages. A more precise approximation is $$\displaystyle I=\frac{1}{\tau}\int_{t-\tau}^ty^\bullet S^\bullet dt'=yS-|\kappa| $$ -where $$S^\bullet,y^\bullet$ are spot values and the quarterly +where $$S^\bullet,y^\bullet$$ are spot values and the quarterly covariance between staking and yield is -$\kappa=\int_{t-\tau}^t(y^\bullet-y)(S^\bullet-S)dt'<0$. See the +$$\kappa=\int_{t-\tau}^t(y^\bullet-y)(S^\bullet-S)dt'<0$$. See the [ethode guide](https://github.com/20squares/ethode/blob/master/guide/guide.md) for more discussion. -[^regex]: This regex script is provided to translate the $\LaTeX$ +[^regex]: This regex script is provided to translate the $$\LaTeX$$ within the markdown source to (our esimatimate of) terminology more common at [issuance.wtf](https://issuance.wtf) -[^partial]: Sometimes $\dot{x}=dx/dt$ is used e.g. for the partial -derivative $\frac{\partial x}{\partial t}$, *but not here*. For a model -in which you assume dynamical variables $(A,s,\alpha)$, these derivatives -are related for some parameter $x(A,s,\alpha,t)$ +[^partial]: Sometimes $$\dot{x}=dx/dt$$ is used e.g. for the partial +derivative $$\frac{\partial x}{\partial t}$$, *but not here*. For a model +in which you assume dynamical variables $$(A,s,\alpha)$$, these derivatives +are related for some parameter $$x(A,s,\alpha,t)$$ $$\displaystyle \frac{dx}{dt} := \dot{x} = \frac{\partial x}{\partial t} + @@ -878,9 +884,9 @@ $$\displaystyle x_t + x_A\dot{A} + x_s\dot{s} + x_\alpha\dot{\alpha} $$ in which each partial is taken holding all the other variables constant, -and $\frac{\partial x}{\partial t}$ is used in practice to capture any +and $$\frac{\partial x}{\partial t}$$ is used in practice to capture any variability from non-dynamical variables. In principle -$\partial_A{x}$ and $\partial_\alpha{x}$ are distinct; a quantity can +$$\partial_A{x}$$ and $$\partial_\alpha{x}$$ are distinct; a quantity can depend on supply (how big ETH market cap is compared to BTC, say) and inflation independently. @@ -900,50 +906,50 @@ per dynamical systems. The analogy, whie imprecise is not inappropriate. It is routine to implicitly use associativity to account for fibers of flows through multiple steps; "electricity from wind/nuclear/gas" even though the electrons are indistinguishable. -Flows such as tx fees $U\overset{F}{\longrightarrow}V,\cancel{O}$ -involve a categorical product $V\times\cancel{O}$ in that the -fractional flows $U\overset{B}{\longrightarrow}\cancel{O}$ and must -factor through $V\times\cancel{O}$. For any flow with contributions -to stocks $V,\cancel{O}$ it is not hard imagining a uu2i universal -object $V\times\cancel{O}$ mediating. Similarly the staking queue -$V+(U-V)\overset{R+Q_+}{\longrightarrow}S$ should involve a +Flows such as tx fees $$U\overset{F}{\longrightarrow}V,\cancel{O}$$ +involve a categorical product $$V\times\cancel{O}$$ in that the +fractional flows $$U\overset{B}{\longrightarrow}\cancel{O}$$ and must +factor through $$V\times\cancel{O}$$. For any flow with contributions +to stocks $$V,\cancel{O}$$ it is not hard imagining a uu2i universal +object $$V\times\cancel{O}$$ mediating. Similarly the staking queue +$$V+(U-V)\overset{R+Q_+}{\longrightarrow}S$$ should involve a categorical coproduct. Wether there is content here beyond "flows are -functions in $\sf Set$" is unclear. None of this matters in the least +functions in $$\sf Set$$" is unclear. None of this matters in the least for Ethereum dynamics, of course. If you're reading it consider this an easter egg / attempt to detect a living and alert audience. -[^elowex]: Why not simply choose \"$B=bA$\" as was done more simply (modulo +[^elowex]: Why not simply choose \"$$B=bA$$\" as was done more simply (modulo syntax) in [Elowsson 2020]()?[^syntax] Obviously if there is no unstaked Ether no one can afford tx fees, but this might not matter. -More importantly $S$ is a *dynamical variable*, so $b=B(A-S)$ is more -appropriate here. The function $B$ might do all kinds of complicated -nonsense, but it can never go negative and it can never exceed $U$. - -[^whyr]: Intensives expressed as fractions of flows such as $R/(I+P)$), instead -of fractional rates of sources (like $J/S$ or $Q_-/S$) occur when the -source dynamical variable, here $V$, is assumed to equilibrate -$\dot{V}\approx0$. Then the outging flows $R+K$ must equal the -incoming flows $I+P$, so we choose $R=r(I+P)$. If onchain data -indicates %\approx$70\% reinvestment of staking rewards into $S$ takes +More importantly $$S$$ is a *dynamical variable*, so $$b=B(A-S)$$ is more +appropriate here. The function $$B$$ might do all kinds of complicated +nonsense, but it can never go negative and it can never exceed $$U$$. + +[^whyr]: Intensives expressed as fractions of flows such as $$R/(I+P)$$), instead +of fractional rates of sources (like $$J/S$$ or $$Q_-/S$$) occur when the +source dynamical variable, here $$V$$, is assumed to equilibrate +$$\dot{V}\approx0$$. Then the outging flows $$R+K$$ must equal the +incoming flows $$I+P$$, so we choose $$R=r(I+P)$$. If onchain data +indicates %\approx$$70\% reinvestment of staking rewards into $$S$$ takes a lot longer than a quarter we would revisit this assumption, though we do not expect our qualitative results to change re inflation and staking fraction. -[^rlst]: A non-zero $r=R/(I+P)$ is built into the smart contract of every +[^rlst]: A non-zero $$r=R/(I+P)$$ is built into the smart contract of every Liquid Staking Provider (LSP). Here, token-holders provide Ether and receive a redeemable token (LST) that shares some staking rewards -with them. This fraction of rewards $r_{LST}$ is a lower bound on -the long term behavior $r_{LST}\leq r_{LSP}\leq -R^\star(I^\star+P^\star)$; more on this next time. +with them. This fraction of rewards $$r_{LST}$$ is a lower bound on +the long term behavior $$r_{LST}\leq r_{LSP}\leq +R^\star(I^\star+P^\star)$$; more on this next time. -[^rdyn]: Splitting the staking queue into $R+Q_+$ allows us to +[^rdyn]: Splitting the staking queue into $$R+Q_+$$ allows us to somewhat separate short-term *transient* behavior from long-term dynamics. Speculative investment in staking by venture capitalists and novice stakers is expected to die down eventually -$\lim_{t\to\infty}Q_+\approx0$; either they give up or they run +$$\lim_{t\to\infty}Q_+\approx0$$; either they give up or they run staking like a business where making a profit matters. Every business that wants to stay in business reinvests some portion of its profits, -so $r,R>0$ and is what matters in the long run, once most everyone who +so $$r,R>0$$ and is what matters in the long run, once most everyone who wants to stake is indeed staking. [^metastable]: Metastability involves a region of space where @@ -951,56 +957,53 @@ trajectories are bounded surrounding a fixed point with at least one eigenvalue with positive real part. We won't rule it out, but look first for stable fixed points. -[^time]: We can often use the dependence on $t$ to smuggle in +[^time]: We can often use the dependence on $$t$$ to smuggle in any forces, like market panics, etc. that we neglected to include as dynamical variables. If not, we must add a dynamical variable. -[^asym]: Specifically $I\ll S$ means that \lim_{S\to\infty}(I/S)=0, +[^asym]: Specifically $$I\ll S$$ means that \lim_{S\to\infty}(I/S)=0, which according to Buterin's [annotated spec]() is satisfied even without the limit; see [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md). [^ycov]: The relation between quarterly averaged issuance and -the yield curve $y^\bullet$ is: +the yield curve $$y^\bullet$$ is: $$\displaystyle I:=\int_{t-\tau}^t y^\bullet(S^\bullet)S^\bullet(t')dt'\approx yS-|COV(y^\bullet,S^\bullet)| $$ -where we approximate $y,S$ as constant on the interval $(t-\tau,t)$ +where we approximate $$y,S$$ as constant on the interval $$(t-\tau,t)$$ and the covariance is negative -$0>COV(y^\bullet,S^\bullet)=\tau^{-2}\int_{t-\tau}^t(y^\bullet - -y)(S^\bullet-S)dt'$. The use of covariance here may seem a bit silly; -$y^\bullet(S^\bullet)$ is a deterministic relationship! +$$0>COV(y^\bullet,S^\bullet)=\tau^{-2}\int_{t-\tau}^t(y^\bullet - +y)(S^\bullet-S)dt'$$. The use of covariance here may seem a bit silly; +$$y^\bullet(S^\bullet)$$ is a deterministic relationship! We do this because the community is discussing changing the yield curve, and we want to discuss generic features of Ethereum. So we have used approximations that we felt err in a conservative direction without explicit -depenedence on the present-day yield curve $y^\bullet_0(S^\bullet)=y_0(1)\sqrt{S_1/S}$ where -$y_0(1)\approx166.3$/yr and $S_1=1$ETH. +depenedence on the present-day yield curve $$y^\bullet_0(S^\bullet)=y_0(1)\sqrt{S_1/S}$$ where +$$y_0(1)\approx166.3$$/yr and $$S_1=1$$ETH. [^params]: Variable parameters that are positive fractions cannot contribute fixed-points themselves, but they can strongly influence -*where* a fixed point is. Example: as $s\to1$, if the leading terms -were $\beta\sim(1-s),j\sim(1-s)^2$ this gives increasingly larger -$A^\star$ as $s\to1$. - -[^vitalikp]: Our $\frac{\partial{d\log~y}}{\partial{d\log~S}}$ is -$p-1$ in the [discouragement paper](https://github.com/20squares/ethode/blob/master/guide/guide.md) - -[^deneb]: +*where* a fixed point is. Example: as $$s\to1$$, if the leading terms +were $$\beta\sim(1-s),j\sim(1-s)^2$$ this gives increasingly larger +$$A^\star$$ as $$s\to1$$. +[^vitalikp]: Our $$\frac{\partial{d\log~y}}{\partial{d\log~S}}$$ is +$$p-1$$ in the [discouragement paper](https://github.com/20squares/ethode/blob/master/guide/guide.md) [^noburn]: While slashing could believably go to zero on quarterly -timescales, no burn $B=0$ implies blocks are empty, at least. +timescales, no burn $$B=0$$ implies blocks are empty, at least. Obviously this isn't really a functioning state for Ethereum, but think of it as a limiting behavior. A more nuanced treatment of -$s\to1$ can be obtained assuming $s^\star=1-\epsilon$ where -$\epsilon\ll1$, making $A^\star\sim (y(1)/b^\star -f^\star)^2\epsilon^{-2}$; as $\epsilon$ is very small $\epsilon^{-2}$ +$$s\to1$$ can be obtained assuming $$s^\star=1-\epsilon$$ where +$$\epsilon\ll1$$, making $$A^\star\sim (y(1)/b^\star +f^\star)^2\epsilon^{-2}$$; as $$\epsilon$$ is very small $$\epsilon^{-2}$$ is very large. Detailed treatment of the forces using expansions in -$\epsilon$ are useful, and we would need to model churn, slashing, and +$$\epsilon$$ are useful, and we would need to model churn, slashing, and burn in light of stochasticity/quantization. One can use difference equations, but a useful generic behavior of such systems (a "weak coupling" limit) is that such perturbations move the dynamics away from the fixed point, apparently randomly, yet the average rate of precession -about $s^\star=1-\epsilon$ is often still given by the imaginary +about $$s^\star=1-\epsilon$$ is often still given by the imaginary component of the largest eigenvalue. [^mu]: If anything the fractional rates @@ -1009,24 +1012,24 @@ component of the largest eigenvalue. stimulation of economic activity attracting more validators and higher average burn. -[^ics]: $\gamma$ expresses the sensitvity of inflation to supply - initial conditions; the partial $x_A$ always holds $(\alpha,s)$ - consant, but $\dot{A}=\alpha A$, and the only partials in $\gamma$ - are $\jmath_A,\beta_A$. +[^ics]: $$\gamma$$ expresses the sensitvity of inflation to supply + initial conditions; the partial $$x_A$$ always holds $$(\alpha,s)$$ + consant, but $$\dot{A}=\alpha A$$, and the only partials in $$\gamma$$ + are $$\jmath_A,\beta_A$$. [^small-part]: Small sensitivities wrt intensives is not gauranteed. -Certainly a large magnitude, say $\beta_s>1$ cannot maintain for too -long: $\beta\in(0,1)$ afterall, but locally, a large derivative -$\beta_s>\beta$ is possible. +Certainly a large magnitude, say $$\beta_s>1$$ cannot maintain for too +long: $$\beta\in(0,1)$$ afterall, but locally, a large derivative +$$\beta_s>\beta$$ is possible. [^stability]: Assesing the stability of very long-term -$\alpha^\star=0$ in the $(A,s,\alpha)$ model directly is interesting +$$\alpha^\star=0$$ in the $$(A,s,\alpha)$$ model directly is interesting but academic. One of the eigenvalues at any fixed point with -$\alpha^\star=0$ must be zero, so higher-order terms matter (the fixed +$$\alpha^\star=0$$ must be zero, so higher-order terms matter (the fixed point is degenerate), and linear-stability analysis is insufficient. A reader imbued with mathematical athleticism and free time is -encouraged to think of a Lyapunovv function $л(y)$ for issuance yield, -and derive a contraction mapping $\dot{л}(y)<0$. +encouraged to think of a Lyapunovv function $$л(\alpha=0)\geq0$$ for issuance yield, +and obtain a contraction mapping $$\dot{л}(y)<0$$. [^SU]: Readers wishing for more detail are encouraged to use the two dimensional local stability criterion (see From 52fc76e56f52003b6b6729814400b9161a0d74dd Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 09:57:43 -0600 Subject: [PATCH 35/70] fixed header --- _posts/2025-01-15-issuance-dynamics.md | 31 +++++++++++--------------- 1 file changed, 13 insertions(+), 18 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 78d0214..766a36f 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -5,34 +5,29 @@ author: Eric Downes categories: ["ethereum", "macroeconomics", "dynamics", "issuance"] excerpt: Work on ETH Macro supported by the Ethereumm Foundation usemathjax: true -thanks: We are deeply grateful to the Ethereum Foundation for their -support of this work. I, the author, feel indebted to the profound -patience and forebearance of Eric Siu and my colleagues at 20 Squares, -as this is *months* overdo. This work has benefited -from conversations with Danieli and Philipp, as well as Eric Siu, -Andrew Sudbury, Ansgar Dietrichs, and Mac Budkowski. +thanks: We are deeply grateful to the Ethereum Foundation for their support of this work. I, the author, feel indebted to the profound patience and forebearance of Eric Siu and my colleagues at 20 Squares, as this is *months* overdo. This work has benefited from conversations with Danieli and Philipp, as well as Eric Siu, Andrew Sudbury, Ansgar Dietrichs, and Mac Budkowski. --- # Ethereum Macroeconomics This is the first of two posts on Ethereum macroeconomics. -Ethereum has grown into a [major economic -force](https://www.grayscale.com/research/reports/the-battle-for-value-in-smart-contract-platforms); +Ethereum has grown into a [major economic force]( + https://www.grayscale.com/research/reports/the-battle-for-value-in-smart-contract-platforms); between its native asset Ether (ETH), the smart contract ecosystem this supports, and the Layer-2 blockchains, a conservative valuation might be half a trillion dollars. At the core of Ethereum's "brand", distinguishing it from other smart contract platforms, is the consistent effort put into decentralized governance. Via its -[consensus -mechanism](https://ethereum.org/en/developers/docs/consensus-mechanisms/pos/gasper/) +[consensus mechanism]( + https://ethereum.org/en/developers/docs/consensus-mechanisms/pos/gasper/) no central authority can censor a transaction, freeze the native asset of a user, etc. This brand commitment depends in turn on a sufficient diversity of validators staking ETH to participate in consensus. The share of Ether staked by "centralized" staking services, such as -exchanges and Liquid Staking Providers (LSTs) [is -considerable](https://dune.com/queries/2394100/3928083), and continues +exchanges and Liquid Staking Providers (LSTs) [is considerable]( + https://dune.com/queries/2394100/3928083), and continues to grow. This has provoked [concerns](https://issuance.wtf/), among Ethereum researchers that the future of Ethereum might involve a confluence of three interrelated challenges: @@ -41,6 +36,10 @@ confluence of three interrelated challenges: 1. Inflation becomes excessive. 1. Governance becomes centralized. +The view of inflation emphasized in this work in particular feels +quite different to us, than the views on inflation expressed, for instance in +this [very helpful review podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). + ## Lookahead In this blog post we address the first of these concerns "runaway @@ -97,13 +96,9 @@ $$\frac{dr}{d\alpha}|^\star<0$$ (no news here) but simultaneously 1. high enough to numerically dominate priority fees and MEV, as a fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. -The importance of this last point in moderating staking fraction feels -quite different to us, than the view on inflation expressed, for instance [in -this helpful review podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). - Inflation should *eventually* decay, driving $$s\to1$$ if it is not -already there, though it may take some time. This "L2 future" has -been recognized by many others: most Ether is staked, with the +already there, though it may take considerable time. This "L2 future" +has been recognized by many others: most Ether is staked, with the majority used for settlement of L2 rollups. We'll discuss it more next time. From 5e7551f787b9d0944d6606e7f732d80c25544df4 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 10:58:07 -0600 Subject: [PATCH 36/70] fixed several footnotes --- _posts/2025-01-15-issuance-dynamics.md | 104 +++++++++++++++---------- 1 file changed, 61 insertions(+), 43 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 766a36f..aa2d698 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -37,17 +37,19 @@ confluence of three interrelated challenges: 1. Governance becomes centralized. The view of inflation emphasized in this work in particular feels -quite different to us, than the views on inflation expressed, for instance in -this [very helpful review podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). +quite different to us, than the views expressed, for instance, in this +[very helpful review +podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). ## Lookahead In this blog post we address the first of these concerns "runaway -(near 100\%) staking" $$s\to1$$ and how it relates to the second, using a "stock -and flow" macroecnomics model built with guidance from dynamical -system theory. In contrast with other research we find inflation -playing a positive but temporary role in moderating runaway staking, -though most likely $$s\to1$$ eventually. +(near 100\%) staking" $$s\to1$$ and how it relates to the second, +using a "stock and flow" macroecnomics model built with guidance from +dynamical system theory. In contrast with other research we find +inflation playing a positive role in moderating runaway staking, but +eventually inflation must subside, along with the moderation it +provides. In the second post, we look more closely at governance centralization and discuss a means for evaluating macroeconomic interventions inspred @@ -62,8 +64,9 @@ into technical mathematical points not covered here, run their own simulations, or learn some dynamical systems are recommended to look at our ethode [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md), -which contains [References](#references) section below. The guide is -certainly a work in progress, but should have enough to get you going. +which contains [References](#dynamical-systems-references) section +below. The guide is certainly a work in progress, but should have +enough to get you going. ## For The Impatient! @@ -91,7 +94,7 @@ inflation is held 1. low enough, that concerns over inflation do not dominate the reinvestment of profits by staking businesses at equilibrium, -$$\frac{dr}{d\alpha}|^\star<0$$ (no news here) but simultaneously +$$\left\frac{dr}{d\alpha}\right|^\star<0$$ (no news here) but simultaneously 1. high enough to numerically dominate priority fees and MEV, as a fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. @@ -119,12 +122,12 @@ we call instead $$A$$, so $$s=S/A$$. Please proceed! Consider a "balloon" with variable internal compartments. The average size of each is measured by *stocks* -- ($$S$$)taked Ether (participating in [consensus]()) is a compartment, as is +- ($$S$$)taked Ether (participating in consensus) is a compartment, as is - ($$U$$)nstaked unburnt Ether, -- containing the ($$V$$)alidator reward queue. -- ($$\cancel{O}$$) is all irrecoverable (burned, lost, etc.) Ether, and +- ($$\mathrm{\{O}}$$) is all irrecoverable (burned, lost, etc.) Ether, and - ($$A$$)ccessible/Circul($$A$$)ting Ether supply, $$A=S+U\approx120.4\times10^6$$ in Dec 2024. -- $$\mathcal{Q}_\pm$$ the Ether in the staking ($$+$$) and unstaking $$-$$) queues +- $$\mathcal{Q}_\pm$$ the Ether in the staking ($$+$$) and unstaking ($$-$$) queues The net change in time of a stock is written using a dot, such as $$\frac{dA}{dt}:=\dot{A}$$[^partial], the net change in accessible Ether @@ -139,7 +142,7 @@ cycles of the erratic base fee oscillations. So, our conceptual model: -$$\dislaystyle +$$\displaystyle \begn{array}{rcl} \dot{A} &=& I - B - J\\ \dot{V} &=& I + P - R - K @@ -148,7 +151,7 @@ $$\dislaystyle \end{array} $$ -| Flow Name | Symbol | Domain$$\to$$Codomain[^cats] | Constraint | +| Flow Name | Symbol | Domain $$\to$$ Codomain[^cats] | Constraint | | :-- | :-- | :-: | :-- | | Tx Fees | $$F$$ | $$U\to\cancel{O},V$$ | $$B+P=F Date: Mon, 13 Jan 2025 12:09:05 -0600 Subject: [PATCH 37/70] added final sections; typos and broken links remain --- _posts/2025-01-15-issuance-dynamics.md | 103 ++++++++++++++++++------- 1 file changed, 77 insertions(+), 26 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index aa2d698..61eb930 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -385,18 +385,20 @@ $$\displaystyle dA = \alpha dt &\leq& \left(ys-\beta(1-s)-\jmath s\right)dt\\ dA &\leq& ysdt = y_0(1)\sqrt{s/A}dt \leq y_0(1)/\sqrt{A}dt\\ \sqrt{A}dA &\leq& y_0(1)dt\\ -\int_{A(0)}^{A(t)}\sqrt{A}dA &\leq& \int_0^t y_0(1)dt\\ -\frac{2}{3}x^3\Big|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ +\int_{A(0)}^{A(t)}\sqrt{A}dA &\leq& \int_0^{\ t} y_0(1)dt\\ +\left.\frac{2}{3}x^3\right|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ A(t) &\leq& \left(A(0)+\frac{3}{2}y_0(1)t\right)^{2/3}\\ \therefore A(t) &\ll& e^{kt} ~\forall ~\mathrm{const.}~k>0 \end{array} $$ -The last line is Vinogradov asymptotic notation.[^asym] For two positive functions, $$g$$ -dominates $$f$$, written $$f(t)\ll g(t)$$ just when -$$\lim_{t\to\infty}[f(t)/g(t)]=0$$. Since supply $$A(t)$$ is *eventually* -less than a powerlaw of $$t$$, it is *subexponential*. Thus, no -positive inflation rate can maintain indefinitely. +The last line is Vinogradov asymptotic notation.[^asym] For two +positive functions, $$g$$ dominates $$f$$, written $$f(t)\ll g(t)$$ +just when $$\lim_{t\to\infty}[f(t)/g(t)]=0$$; when the limit is a +non-zero constant we say $f\sim g$. Since supply $$A(t)$$ is +*eventually* less than a powerlaw of $$t$$, it is *subexponential*. +Thus, no positive rate of Ether suply expansion can maintain +indefinitely. This does not mean we would find every intermediate inflation rate pleasant. Following surges in $$Q_+$$, inflation can accelerate quite @@ -605,27 +607,52 @@ wether $$s^\star$$ is (un)stable. We will be ignoring the partial derivatives ("sensitivities") by assuming they are small in comparison with their corresponding intensives.[^small-part] +The full no-churn stability condition, including variations in $\alpha$ is +$$\displaystyle +\alpha^\star/f^\star + r + \jmath^\star/f^\star + > +\left(\frac{\partial\log\ r}{\partial\log\ x}\right|^\star\left( +\alpha^\star/f^\star + \left(\frac{1}{r^\star}-\frac{1}{x^\star}\right) +\left.\frac{\partial(\alpha/f)}{\partial r}\right|^\star\right) +$$ -### Weak Deflation - -If $$\alpha_{const}\in(-rf-\jmath,0]$$ then an interior market equilibrium -$$s^\star$$ requires high slashing, as we argued before based on -Churn. All other things being equal, it is also less stable. - - - -### Strong Deflation - -Consider now $$\alpha<-rf-\jmath$$. This could happen for instance if -the issuance curve were reduced particularly bluntly, or changes in -fundamentals drove either MEV or the base fee (and thus $$f$$) to a -persistently higher amount, such that $$ys\ll bf(1-s)+\jmath s$$. - -These conditions cannot maintain of course. You don't need -differential equations to see that $$\alpha<0$$ shrinks $$A$$, which -eventually raises $$y(sA)$$. But as a temporary intervention to tame -runaway staking fraction how would this work? +If we assume that sensitivities are dominated by their respective +intensives, this reduces to a simple $$\alpha^\star+r^\star f^\star \gtsim +-\jmath^\star$. + +Weak Deflation. If $$\alpha_{const}\in(-rf-\jmath,0]$$ then an +interior market equilibrium $$s^\star$$ requires high slashing, as we +argued before based on Churn. All other things being equal, it is +also less stable; in the region where partial derivatiives might play +a role, but probably okay. Assuming low slashing continues, as +validators must pay the cost themselves so are incentivized to +minimize it, then $x^\star>1$ and runaway staking is inevitable if the +fixed point is stable: the numerator is larger than the denominator. +If unstable, $s$ is pushed toward zero, and becomes unpredictable. + +Strong deflation. Consider now $$\alpha<-rf-\jmath$$. This could +happen for instance if the issuance curve were reduced particularly +bluntly, or changes in fundamentals drove either MEV or the base fee +(and thus $$f$$) to a persistently higher amount, such that $$ys\ll +bf(1-s)+\jmath s$$. These conditions cannot maintain of course. You +don't need differential equations to see that $$\alpha<0$$ shrinks +$$A$$, which eventually raises $$y(sA)$$. But as a temporary +intervention to tame runaway staking fraction how would this work? +Our fixed point is negative and the simplistic "ignore the constants" +stability criteria is no longer met. So again, 100\% staking becomes +inevitable (the source $s^\star<0$ pushes $s$ instead of pulling it), +or the behavior becomes unpredictable as externalities intervene. + +There's a lot of potential complexity here, but none of it is +desirable! If you don't like inflation, wait until you try deflation! +In all seriousness, while we agree largely with the aversion to +inflation in the context of national economies, it is really important +to recognize that those economies have a level of demand diversity and +robustness that Ethereum at present can only dream of. One day +hopefully we will have the luxury of bemoaning inflation's effects on +hodlers, and reducing it, without fear of the infrastructural dynamics +of these market equilibria running amok. ### Low Inflation; Even Lower Fees @@ -844,6 +871,30 @@ research!" # Dynamical Systems References +We are developing the ethode guide in the hope that it can serve a +pedagogical role. For now we have assumed some basic familiarity with +nonlinear dynamics, asymptotic methods, etc. at the level of the first +few of Prof. S. Strogatz’ youtube lectures. + +- [Nonlinear Dynamics](https://youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V&si=iN5YCipB_CeIfrbB) + +- [Perturbation Theory](https://youtube.com/playlist?list=PL5EH0ZJ7V0jV7kMYvPcZ7F9oaf_YAlfbI&si=lFFFJIMAH6nE5BX1) + +Highly Recommended books in order of increasing difficulty and +sophistication if you want to understand this stuff: + +- Kun (2020) [A Programmer’s Introduction to Mathematics](https://www.amazon.com/Programmers-Introduction-Mathematics-Second/dp/B088N68LTJ/) + +- Strogatz (2024) [Nonlinear Dynamics and Chaos](https://www.amazon.com/Nonlinear-Dynamics-Chaos-Steven-Strogatz/dp/0367026503/) + +- Hirsch, Smale and Devaney (2003) [Differential Equations …](https://www.amazon.com/Differential-Equations-Dynamical-Introduction-Mathematics/dp/0123497035/) + +- Bender and Orszag (1997) [Advanced Mathematical Methods for Scientists and Engineers](https://www.amazon.com/Advanced-Mathematical-Methods-Scientists-Engineers/dp/0387989315/) + +- Arnol\’d (Ed.) the Dynamical Systems Series + - esp. V (1994) [Bifurcation and Catastrophe](https://www.amazon.com/Dynamical-Systems-Bifurcation-Encyclopaedia-Mathematical/dp/0387181733/) + + # Footnotes [^humor]: [Open Zeppelin]() is an early icon of smart contract best From f9731b608fa95f9c34edc0ccaa3929761c479103 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 12:27:13 -0600 Subject: [PATCH 38/70] fixed some latex and images --- _posts/2025-01-15-issuance-dynamics.md | 55 +++++++++++++------------- 1 file changed, 28 insertions(+), 27 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 61eb930..f11afc9 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -145,7 +145,7 @@ So, our conceptual model: $$\displaystyle \begn{array}{rcl} \dot{A} &=& I - B - J\\ -\dot{V} &=& I + P - R - K +\dot{V} &=& I + P - R - K\\ \dot{U} - \dot{V} &=& K + Q_- - Q_+ - F\\ \dot{S} &=& R + Q_+ - Q_- - J\\ \end{array} @@ -255,9 +255,9 @@ $$b(A,S,t)=B/F$$.[^time] | Flow Name | Symbol | Domain$$\to$$Codomain[^cats] | Constraint | Intensive | Range \[Units\] | | :-- | :-- | :-: | :-- | :-- | :-- | | Tx Fees | $$F$$ | $$U\to\cancel{O},V$$ | $$00$$, these terms draw $$s$$ toward respective points $$r,1,0$$. We emphasize that the action of yield $$y$$ is $$x\to r$$, which may not be the same as $$x\to1$$. @@ -462,9 +462,9 @@ derivative relations for variables $$(A,\alpha,s,t)$$[^partial] we have $$\displaystyle \begin{array}{rcl} -\dot{A} &=& \pm\alpha A\\ -\dot{s} &=& \pm\alpha(r-s) + (rf+q_+)(1-s) - (q_-+(1-r)j)s -\dot{\alpha} &=& \pm\xi\dot{s} - \gamma\alpha s \pm\chi +\dot{A} &=& \alpha A\\ +\dot{s} &=& \alpha(r-s) + (rf+q_+)(1-s) - (q_-+(1-r)j)s\\ +\dot{\alpha} &=& \xi\dot{s} - \gamma\alpha s +\chi \end{array} $$ @@ -533,8 +533,8 @@ point $$s^\star$$ during a period in which we may treat inflation as constant $$\alpha=\alpha_{const}$$. $$\displaystyle -s^\star = \frac{ - r^\star(\alpha_{const} + f^\star) + q_+^\star +s^\star = + \frac{r^\star(\alpha_{const} + f^\star) + q_+^\star }{ m^\star := (\alpha_{const} + r^\star f^\star + q_+^\star + q_-^\star + (1-r)\jmath^\star) } @@ -580,8 +580,8 @@ s^\star \approx r^\star \frac{ \alpha_const} + r^\star f^\star + (1-r)\jmath^\star} = 1 - \frac{ \jmath^\star}{ - \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} - - \frac{\alpha_{const}(1-2)}{\alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} + \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} - + \frac{\alpha_{const}(1-2)}{\alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} $$ We will explore the *stability* of this fixed point below, and based on @@ -599,7 +599,8 @@ stable *sink*. If the derivative is zero, the fixed point is a degenerate *center*, unrealistic outside of physics. If positive, the fixed point is an unstable *source*. -![1D Stability Conditions](../assetsPosts/1d-stab.png) +![1D Stability Conditions]( + ../assetsPosts/2025-01-15-issuance-dynamics/1d-stab.png) Specifically, we want the sign of $$\left.\frac{\partial\dot{s}}{\partial s}\right|^\star$$ to determine @@ -610,16 +611,15 @@ with their corresponding intensives.[^small-part] The full no-churn stability condition, including variations in $\alpha$ is $$\displaystyle -\alpha^\star/f^\star + r + \jmath^\star/f^\star - > +\alpha^\star/f^\star + r + \jmath^\star/f^\star > \left(\frac{\partial\log\ r}{\partial\log\ x}\right|^\star\left( \alpha^\star/f^\star + \left(\frac{1}{r^\star}-\frac{1}{x^\star}\right) \left.\frac{\partial(\alpha/f)}{\partial r}\right|^\star\right) $$ If we assume that sensitivities are dominated by their respective -intensives, this reduces to a simple $$\alpha^\star+r^\star f^\star \gtsim --\jmath^\star$. +intensives, this reduces to a simple +$$\alpha^\star+r^\star f^\star \gtrsim -\jmath^\star$$. Weak Deflation. If $$\alpha_{const}\in(-rf-\jmath,0]$$ then an interior market equilibrium $$s^\star$$ requires high slashing, as we @@ -662,7 +662,8 @@ be seen in the contours of the market equilibrium staking fraction $$s^\star$$ corresponding to $$\dot{s}=0$$, shown here with $$\alpha^\star=\alpha_{const}$$. -![alpha vs. s Figure](../assetsPosts/2025-01-15-issuance-fundamentals/staking-fixpoint.png). +![alpha vs. s Figure]( + ../assetsPosts/2025-01-15-issuance-dynamics/staking-fixpoint.png). To find the equilibrium values $$(\alpha^\star/f^\star,\,r^\star)$$ necessary to acheive a desired staking fraction $$x^\star$$, simply @@ -713,10 +714,10 @@ for inflation to push the market equilibrium $$s^\star$$ itself into runaway staking is (see below for explanation): $$\displaystyle -1 ~~ < ~~ -\left.\frac{\partial\log\ r}{\partial\log\ \alpha}\right|^\star -\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} ~~ + ~~ -\left.\frac{\partial\log\ f}{\partial\log\ \alpha}\right|^\star +1 \ \ < \ \ +\left(\frac{\partial\log\ r}{\partial\log\ \alpha}\right)^\star +\cdot \frac{1 + \alpha^\star/f^\star}{1 - r^\star} \ \ + \ \ +\left(\frac{\partial\log\ f}{\partial\log\ \alpha}\right)^\star $$ In practtice it all depends on what ETH users consider sufficiently @@ -766,7 +767,7 @@ moderate $$s$$, which brings us to the quotient rule. ## The Quotient Rule $$\displaystyle -\dot{s}=\frac{\dot{S}}{A}-s\frac{\dot{A}}{A}=\frac{\dot{S}}{S}-s\alpha +\dot{s}\ =\ \frac{\dot{S}}{A}\ -\ s\frac{\dot{A}}{A}\ =\ \frac{\dot{S}}{S}\ -\ s\alpha $$ What increases $$s$$ is any increase in staked Ether $$S$$, but also any @@ -845,7 +846,7 @@ question about user preferences. Austrian School devotees may be so inflation-averse that they are already staking all their previously-liquid ETH at $$\alpha\approx0.5$$\%/yr. In contrast users who were content to grow up with fiat currencies during periods of -$$\approx$$3\% inflation or even worse might not care, or would just +$$\approx$$ 3\% inflation or even worse might not care, or would just stake in Compound or buy stETH. For users seeking to passively preserve wealth, staking in Compound @@ -874,7 +875,7 @@ research!" We are developing the ethode guide in the hope that it can serve a pedagogical role. For now we have assumed some basic familiarity with nonlinear dynamics, asymptotic methods, etc. at the level of the first -few of Prof. S. Strogatz’ youtube lectures. +few of Prof. S. Strogatz youtube lectures. - [Nonlinear Dynamics](https://youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V&si=iN5YCipB_CeIfrbB) From 5a211effdcd050d842208af61b7e0d536c3136b6 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 12:31:43 -0600 Subject: [PATCH 39/70] trying to fix more latex --- _posts/2025-01-15-issuance-dynamics.md | 23 +++++++++++------------ 1 file changed, 11 insertions(+), 12 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index f11afc9..ffa6f8a 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -257,7 +257,7 @@ $$b(A,S,t)=B/F$$.[^time] | Tx Fees | $$F$$ | $$U\to\cancel{O},V$$ | $$0\jmath+q_-$$ staked ETH $$S$$ just continues growing and growing. In contrast it is harder for $$U$$ to get as big, limited by its own loss term -$$-\left(rf+(1-r)bf+q_+\right)\cdot U$$. At some point in the (far) +$$-\left(rf+(1-r)bf+q_+\right) U$$. At some point in the (far) future $$S$$ becomes big enough that $$ry(S)<\jmath+q_-$$ and the system becomes *capable* of oscillation, depending on parameters and a zoo of partial derivatives.[^SU] @@ -428,9 +428,9 @@ massage, we obtain for staking fraction $$\displaystyle \begin{array}{rcl} -\dot{s} &=& y(sA)\cdot(r-s) + \\ +\dot{s} &=& y(sA)\ (r-s) + \\ && \left[q_++f(1-s)\left(bs +(1-b)r\right)\right]\cdot(1-s) + \\ - && \left[j(1-s+r)+q_-\right]\cdot(0-s). + && \left[j(1-s+r)+q_-\right]\ (0-s). \end{array} $$ @@ -525,7 +525,7 @@ Recall our approximate equation for the fraction of staked ETH $$s$$, in which all coefficients are positive but inflation $$\alpha$$: $$\displaystyle -\dot{s} &=& \alpha\cdot(r-s) + (rf+q_+)\cdot(1-s) + (q_-+(1-r)j)\cdot(0-s) +\dot{s} &=& \alpha\ (r-s) + (rf+q_+)\ (1-s) + (q_-+(1-r)j)\ (0-s) $$ So assuming $$|\dot{\alpha}|\ll|\dot{s}|$$, let us examine the fixed @@ -575,13 +575,12 @@ $$q_+^\star\ll(rf)^\star$$. Thus, the fixed point $$s^\star$$ simplifies to $$\displaystyle -s^\star \approx r^\star \frac{ - \alpha_const} + f^\star}{ - \alpha_const} + r^\star f^\star + (1-r)\jmath^\star} -= 1 - \frac{ - \jmath^\star}{ - \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} - - \frac{\alpha_{const}(1-2)}{\alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} +s^\star \approx r^\star \frac{\alpha_{const} + f^\star}{ + \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} \ =\ 1\ -\ +\frac{\jmath^\star}{ + \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star}\ -\ +\frac{\alpha_{const}(1-2)}{ + \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} $$ We will explore the *stability* of this fixed point below, and based on From 26b098929dacc616bd50e4c8fbce9c553d940f56 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 12:33:41 -0600 Subject: [PATCH 40/70] trying to fix more latex --- _posts/2025-01-15-issuance-dynamics.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index ffa6f8a..16f9457 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -293,7 +293,7 @@ class ConstParams(Params): With the above, you should be able to construct the following $$(S,U)$$ system: $$\displaystyle -\begn{array}{ccrlcrl} +\begin{array}{ccrlcrl} \dot{S} &=& (ry-\jmath-q_-) & S & + & \left(q_++r(1-b)f\right) & U\\ \dot{U} &=& \left((1-r)y+q_-\right) & S & - & \left(rf+(1-r)bf+q_+\right) & U\\ \end{array} @@ -525,7 +525,7 @@ Recall our approximate equation for the fraction of staked ETH $$s$$, in which all coefficients are positive but inflation $$\alpha$$: $$\displaystyle -\dot{s} &=& \alpha\ (r-s) + (rf+q_+)\ (1-s) + (q_-+(1-r)j)\ (0-s) +\dot{s} = \alpha\ (r-s) + (rf+q_+)\ (1-s) + (q_-+(1-r)j)\ (0-s) $$ So assuming $$|\dot{\alpha}|\ll|\dot{s}|$$, let us examine the fixed From 9a65e13bbcbfb926f2825d5af7ae011aa8baf2f5 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 12:36:15 -0600 Subject: [PATCH 41/70] trying to fix more latex --- _posts/2025-01-15-issuance-dynamics.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 16f9457..792c2fd 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -94,7 +94,7 @@ inflation is held 1. low enough, that concerns over inflation do not dominate the reinvestment of profits by staking businesses at equilibrium, -$$\left\frac{dr}{d\alpha}\right|^\star<0$$ (no news here) but simultaneously +$$\left(\frac{dr}{d\alpha}\right)^\star<0$$ (no news here) but simultaneously 1. high enough to numerically dominate priority fees and MEV, as a fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. @@ -137,13 +137,13 @@ units \[ETH/yr\]. By averaging over "long" timescales (at least quarterly)[^aves] we approximate the staking and unstaking queues as equilibrated; -$$\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$$, and average over many +$$\mathcal{Q}_+ \approx 0 \approx \mathcal{Q}_-$$, and average over many cycles of the erratic base fee oscillations. So, our conceptual model: $$\displaystyle -\begn{array}{rcl} +\begin{array}{rcl} \dot{A} &=& I - B - J\\ \dot{V} &=& I + P - R - K\\ \dot{U} - \dot{V} &=& K + Q_- - Q_+ - F\\ From 14dc3e670a8f27f4de6767f0559b567f50b42a93 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 12:57:05 -0600 Subject: [PATCH 42/70] blog doesnt parse left and right latex parens --- _posts/2025-01-15-issuance-dynamics.md | 28 +++++++++++++------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 792c2fd..9bac422 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -257,7 +257,7 @@ $$b(A,S,t)=B/F$$.[^time] | Tx Fees | $$F$$ | $$U\to\cancel{O},V$$ | $$00 +dA = \alpha dt &\leq& \left(ys-\beta(1-s)-\jmath s\right)dt\\ +dA &\leq& ysdt = y_0(1)\sqrt{s/A}dt \leq y_0(1)/\sqrt{A}dt\\ +\sqrt{A}dA &\leq& y_0(1)dt\\ +\int_{A(0)}^{A(t)}\sqrt{A}dA &\leq& \int_0^{\ t} y_0(1)dt\\ +\left.\frac{2}{3}x^3\right|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ +A(t) &\leq& \left(A(0)+\frac{3}{2}y_0(1)t\right)^{2/3}\\ +\therefore A(t) &\ll& e^{kt} ~\forall ~\mathrm{const.}~k>0 \end{array} $$ @@ -610,10 +610,10 @@ with their corresponding intensives.[^small-part] The full no-churn stability condition, including variations in $\alpha$ is $$\displaystyle -\alpha^\star/f^\star + r + \jmath^\star/f^\star > -\left(\frac{\partial\log\ r}{\partial\log\ x}\right|^\star\left( -\alpha^\star/f^\star + \left(\frac{1}{r^\star}-\frac{1}{x^\star}\right) -\left.\frac{\partial(\alpha/f)}{\partial r}\right|^\star\right) +(\alpha^\star/f^\star)\ +\ r\ +\ (\jmath^\star/f^\star)\ \ >\ \ +\log{r}_{\log x}^\star\left( +(\alpha^\star/f^\star)\ +\ (r^\star)^{-1}-(x^\star)^{-1} +(\alpha/f)_r^\star\right) $$ If we assume that sensitivities are dominated by their respective From 72a88249a9f4185e3d6705622d756fd5a90a2643 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 13:05:02 -0600 Subject: [PATCH 43/70] adjusting to blogs poor kerning --- _posts/2025-01-15-issuance-dynamics.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 9bac422..77d6283 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -265,7 +265,7 @@ $$b(A,S,t)=B/F$$.[^time] | Costs & Profits | $$K$$ | $$V\to U$$ | .. | $$1-r=K/(I+P)$$ | $$1-r\in(0,1)$$ \[1\] | The use of intensive variable parameters and the approximation -$$\dot{V}\approx0$$[^flowfrac] allows us to reshape our conceptual model +$$\dot{V}\approx0$$ allows us to reshape our conceptual model into one that is defined in its own dynamical and intensive variables, a *dynamical system*. We'll build this up one step at a time. As you follow along you may find it useful to look at models in python. You @@ -397,7 +397,7 @@ positive functions, $$g$$ dominates $$f$$, written $$f(t)\ll g(t)$$ just when $$\lim_{t\to\infty}[f(t)/g(t)]=0$$; when the limit is a non-zero constant we say $f\sim g$. Since supply $$A(t)$$ is *eventually* less than a powerlaw of $$t$$, it is *subexponential*. -Thus, no positive rate of Ether suply expansion can maintain +Thus, no positive rate of Ether supply expansion can maintain indefinitely. This does not mean we would find every intermediate inflation rate @@ -611,9 +611,9 @@ The full no-churn stability condition, including variations in $\alpha$ is $$\displaystyle (\alpha^\star/f^\star)\ +\ r\ +\ (\jmath^\star/f^\star)\ \ >\ \ -\log{r}_{\log x}^\star\left( -(\alpha^\star/f^\star)\ +\ (r^\star)^{-1}-(x^\star)^{-1} -(\alpha/f)_r^\star\right) +\log{r}_{\log x}^\star\left[ +(\alpha^\star/f^\star)\ +\ (1/r^\star-1/x^\star) +(\alpha/f)_r^\star\right] $$ If we assume that sensitivities are dominated by their respective From 84019119cc36326c665eec8a46865e7fd6b7c208 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 13:06:18 -0600 Subject: [PATCH 44/70] adjusting to blogs poor kerning --- _posts/2025-01-15-issuance-dynamics.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 77d6283..51cd19c 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -607,7 +607,8 @@ wether $$s^\star$$ is (un)stable. We will be ignoring the partial derivatives ("sensitivities") by assuming they are small in comparison with their corresponding intensives.[^small-part] -The full no-churn stability condition, including variations in $\alpha$ is +The full no-churn stability condition, including variations in +$\alpha$ is (with subscripts denoting partials) $$\displaystyle (\alpha^\star/f^\star)\ +\ r\ +\ (\jmath^\star/f^\star)\ \ >\ \ From a22b036c0aaf073f8207affffc549a717890201f Mon Sep 17 00:00:00 2001 From: Eric Siu <94845277+randomishwalk@users.noreply.github.com> Date: Mon, 13 Jan 2025 13:26:25 -0600 Subject: [PATCH 45/70] Update 2025-01-15-issuance-dynamics.md low value add typo fixes :) --- _posts/2025-01-15-issuance-dynamics.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 77d6283..fa1a3b6 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -3,7 +3,7 @@ layout: post title: "Ethereum Macroeconomics via Dynamics 1/2" author: Eric Downes categories: ["ethereum", "macroeconomics", "dynamics", "issuance"] -excerpt: Work on ETH Macro supported by the Ethereumm Foundation +excerpt: Work on ETH Macro supported by the Ethereum Foundation usemathjax: true thanks: We are deeply grateful to the Ethereum Foundation for their support of this work. I, the author, feel indebted to the profound patience and forebearance of Eric Siu and my colleagues at 20 Squares, as this is *months* overdo. This work has benefited from conversations with Danieli and Philipp, as well as Eric Siu, Andrew Sudbury, Ansgar Dietrichs, and Mac Budkowski. --- @@ -46,19 +46,19 @@ podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). In this blog post we address the first of these concerns "runaway (near 100\%) staking" $$s\to1$$ and how it relates to the second, using a "stock and flow" macroecnomics model built with guidance from -dynamical system theory. In contrast with other research we find +dynamical system theory. In contrast with other research, we find inflation playing a positive role in moderating runaway staking, but eventually inflation must subside, along with the moderation it provides. In the second post, we look more closely at governance centralization -and discuss a means for evaluating macroeconomic interventions inspred +and discuss a means for evaluating macroeconomic interventions inspired by bifurcation theory. Briefly, we are not optimstic that reducing issuance will prevent governance centralization, either. In both posts, we provide a few code examples using -[ethode](https://github.com/20squares/ethode/) a thin units-aware -wrapper we built around `scipy.integrate.solve_ivp` to streamline +[ethode](https://github.com/20squares/ethode/), a thin units-aware +wrapper we built around `scipy.integrate.solve_ivp`, to streamline model evaluation. Readers desiring to follow our derivations, dive into technical mathematical points not covered here, run their own simulations, or learn some dynamical systems are recommended to look @@ -70,7 +70,7 @@ enough to get you going. ## For The Impatient! -Issuance does not all get dumped into liquid unstaked Ether. Some +Issuance does not all get dumped into native Ether. Some portion of it is reinvested by staking businesses; indeed this process is coded into LST smart contracts. @@ -81,7 +81,7 @@ is dominated by reinvestment instead of speculation, inflation persistently decreases. We use our macroeconmics model to identify a "low inflation; even -lower fees" regime (LI;ELF). Outside of LI;ELF convergence to a +lower fees" regime (LI;ELF). Outside of LI;ELF, convergence to a desirable staking future without runaway staking is unlikely. Strong deflation, in which the magnitude of deflation exceeds the reinvestment of transaction fees, probably corresponds to unstable @@ -406,7 +406,7 @@ alarmingly. A good example will be the 2032 Ethereum staking-mania. This can be modeled in future retrospect by the following graph showing a simulation with `qs=.05, qu=.01`; a constant $$q_+=5$$\% of unstaked ETH is added to the staking queue each year. We aren't -excited to hodl through multiple decades of 10\% inflation, and we +excited to HODL through multiple decades of 10\% inflation, and we expect you aren't either. ![US President-Elect Swift Endorses Ethereum!]( From 9b3286e2e801c17314ebb5145c3769490438d6e1 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Mon, 13 Jan 2025 13:37:16 -0600 Subject: [PATCH 46/70] fixed power mistake in subexp proof --- _posts/2025-01-15-issuance-dynamics.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 51cd19c..43ac947 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -387,7 +387,7 @@ dA &\leq& ysdt = y_0(1)\sqrt{s/A}dt \leq y_0(1)/\sqrt{A}dt\\ \sqrt{A}dA &\leq& y_0(1)dt\\ \int_{A(0)}^{A(t)}\sqrt{A}dA &\leq& \int_0^{\ t} y_0(1)dt\\ \left.\frac{2}{3}x^3\right|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ -A(t) &\leq& \left(A(0)+\frac{3}{2}y_0(1)t\right)^{2/3}\\ +A(t) &\leq& \left(A(0)^{3/2}+\frac{3}{2}y_0(1)t\right)^{2/3}\\ \therefore A(t) &\ll& e^{kt} ~\forall ~\mathrm{const.}~k>0 \end{array} $$ @@ -819,11 +819,11 @@ prove this is impossible? So, finally... should the Ethereum community reduce issuance? -* Glib answer: +### Glib answer Nope. -* Short answer: +### Short answer If you are very inflation averse or you want to slow down the transition to high staking, please consider and simulate downward @@ -834,7 +834,7 @@ solve, and does not claim to solve, the long term problems, but we have been forced to conclude that reducing issuance doesn't solve them either! -* Long answer: +### Long answer Ask the users, especially the validators, especially the LSTs. Model user preferences so that the demand curve becomes semi-empirical From 4f4eca5d11b27857748b6c92ee412425c4526176 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 18:02:58 -0600 Subject: [PATCH 47/70] final draft --- _posts/2025-01-15-issuance-dynamics.md | 633 +++++++++++++------------ 1 file changed, 338 insertions(+), 295 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 43ac947..59f44a5 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -38,8 +38,8 @@ confluence of three interrelated challenges: The view of inflation emphasized in this work in particular feels quite different to us, than the views expressed, for instance, in this -[very helpful review -podcast](https://www.youtube.com/watch?v=ivynR3RI3_Y). +[very helpful review podcast]( + https://www.youtube.com/watch?v=ivynR3RI3_Y). ## Lookahead @@ -71,8 +71,8 @@ enough to get you going. ## For The Impatient! Issuance does not all get dumped into liquid unstaked Ether. Some -portion of it is reinvested by staking businesses; indeed this process is -coded into LST smart contracts. +portion of it is reinvested by staking businesses at a ratio $r$; +indeed this process is coded into LST smart contracts. It is important to distinguish between transient behavior, such as speculation in staking, and medium/long-term behavior, such as the @@ -80,26 +80,28 @@ reinvestment of staking rewards by staking businesses. When staking is dominated by reinvestment instead of speculation, inflation persistently decreases. -We use our macroeconmics model to identify a "low inflation; even -lower fees" regime (LI;ELF). Outside of LI;ELF convergence to a -desirable staking future without runaway staking is unlikely. Strong +We use our macroeconmics model to identify a friend "Mr.LI;ELF": +"Medium $r$. Low Inflation; Even Lower Fees". Without Mr.LI;ELF +convergence to a desirable staking future is unlikely. Strong deflation, in which the magnitude of deflation exceeds the reinvestment of transaction fees, probably corresponds to unstable dynamics. Under zero or weak deflation, the tendency toward runaway staking can be moderated only by high churn and/or slashing. -In contrast within LI;ELF staked ETH fraction approaches near the -reinvestment ratio. Thus, runaway staking can be avoided only while -inflation is held +In contrast with Mr.LI;ELF staked ETH fraction approaches to a value +above moderate reinvestment ratio $r$. How far above depends on the +"ELF" part. Thus, runaway staking can be avoided only while inflation +is held 1. low enough, that concerns over inflation do not dominate the reinvestment of profits by staking businesses at equilibrium, $$\left(\frac{dr}{d\alpha}\right)^\star<0$$ (no news here) but simultaneously -1. high enough to numerically dominate priority fees and MEV, as a -fraction of unstaked Ether; $$\alpha^\star\gg f^\star$$. +1. high enough to not be numerically dominated by the reinvestment of +priority fees and MEV, as a fraction of unstaked Ether; +$$\alpha^\star\gtrsim r^\star f^\star$$. -Inflation should *eventually* decay, driving $$s\to1$$ if it is not +Inflation will *eventually* decay though, driving $$s\to1$$ if it is not already there, though it may take considerable time. This "L2 future" has been recognized by many others: most Ether is staked, with the majority used for settlement of L2 rollups. We'll discuss it more @@ -174,12 +176,12 @@ upgrade](https://github.com/ethereum/consensus-specs/blob/dev/specs/deneb/beacon implemented [EIP 7514](https://eips.ethereum.org/EIPS/eip-7514), an upper limit on $$R+Q_+$$ chosen so as to not limit any present flows. We also ignore the pre-existing symmmetric limits on (un)staking -$$Q_\pm$$. The purpose of our models is to show, in the absence of such -limits, where the dynamics push the system. If you wish to study the -dynamics post-Deneb, you could make $$R+Q_+$$ constant; we will revisit -EIP 7514 in our next post, on staking *composition*. So the -constraints ($$Q_- 1/Yr: @@ -299,13 +304,14 @@ $$\displaystyle \end{array} $$ -These coefficients of $$S,U$$ are miserably complicated-looking, but as -written they are all (but one) positive, and so we can reason about -this model's evolution. Specifically, so long as $$ry(S)>\jmath+q_-$$ -staked ETH $$S$$ just continues growing and growing. In contrast it is -harder for $$U$$ to get as big, limited by its own loss term -$$-\left(rf+(1-r)bf+q_+\right) U$$. At some point in the (far) -future $$S$$ becomes big enough that $$ry(S)<\jmath+q_-$$ and the system +These coefficients of staked $$S$$ and unstaked $$,U$$ ETH are +miserably complicated-looking, but as written they are all (but one) +positive, and so we can reason about this model's evolution. +Specifically, so long as $$ry(S)>\jmath+q_-$$ staked ETH $$S$$ just +continues growing and growing. In contrast it is harder for $$U$$ to +get as big, limited by its own loss term +$$-\left(rf+(1-r)bf+q_+\right) U$$. At some point in the (far) future +$$S$$ becomes big enough that $$ry(S)<\jmath+q_-$$ and the system becomes *capable* of oscillation, depending on parameters and a zoo of partial derivatives.[^SU] @@ -346,7 +352,7 @@ Inflation is used to refer to many things, but here we mean specifically the quarterly fractional change in accessible Ether. Consider $$\alpha:=\dot{A}/A\approx(I-B-J)/A$$ in light of the [above table](#table-of-flows). In general and under -the existing yield curve we have (where $$\beta=bf=B/U$$): +the existing yield curve we have (where $$\beta=bf=B/U$$):[^noburn] $$\displaystyle \alpha\ \approx\ y(sA)s-\beta(1-s)-\jmath s \ =\ y_0(1)\sqrt{s/A}-\beta(1-s)-\jmath s @@ -356,7 +362,7 @@ You can explore this by noting `alpha(), sfrac()` as `@output` methods ```python @dataclass -class SUaSim(SUSim): +class SUaSim(SUSimConst): @output def sfrac(self, S:ETH, U:ETH) -> One: return S / (S + U) @@ -364,7 +370,7 @@ class SUaSim(SUSim): def alpha(self, S:ETH, U:ETH) -> 1/Yr: s = self.sfrac(S,U) return p.yld(S) * s - p.b * p.f * (1 - s) - p.j * s -su_a = SUSim_alpha() +su_a = SUaSim() su_a.sim() ``` @@ -376,48 +382,66 @@ is sublinear issuance $$I\leq yS\lesssim S$$, chosen to avoid attacks](https://raw.githubusercontent.com/ethereum/research/master/papers/discouragement/discouragement.pdf). Because of this, positive inflation cannot maintain indeinitely. We will demonstrate with the existing yield curve, but the argument is -general. Unusually for this blog post, we will show most of the steps +general. Unusually for this blog post, we show most of the steps so the argument is hopefully understood. Elsewhere we use $I\approx -yS$, but here we use $I\leq yS$ as this rigor is required.[^ycov] +yS$, but here we use $I\leq yS$ as greater rigor is appropriate.[^ycov] $$\displaystyle \begin{array}{rcl} -dA = \alpha dt &\leq& \left(ys-\beta(1-s)-\jmath s\right)dt\\ -dA &\leq& ysdt = y_0(1)\sqrt{s/A}dt \leq y_0(1)/\sqrt{A}dt\\ -\sqrt{A}dA &\leq& y_0(1)dt\\ -\int_{A(0)}^{A(t)}\sqrt{A}dA &\leq& \int_0^{\ t} y_0(1)dt\\ -\left.\frac{2}{3}x^3\right|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ -A(t) &\leq& \left(A(0)^{3/2}+\frac{3}{2}y_0(1)t\right)^{2/3}\\ -\therefore A(t) &\ll& e^{kt} ~\forall ~\mathrm{const.}~k>0 +dA = \alpha Adt &\leq& \left(ys-\beta(1-s)-\jmath s\right)Adt\\ +dA &\leq& ysAdt = y_0(1)\sqrt{sA}dt \leq y_0(1)\sqrt{A}dt\\ +\int_{A(0)}^{A(t)}A^{-1/2}dA &\leq& \int_0^{\ t} y_0(1)dt\\ +\left.\frac{1}{2}\sqrt{A}\right|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ +A(t) &\leq& \left(\sqrt{A(0)}+2y_0(1)t\right)^2\\ +\therefore A(t) \sim t^2 &\ll& e^{kt} ~\forall ~\mathrm{const.}~k>0 \end{array} $$ -The last line is Vinogradov asymptotic notation.[^asym] For two -positive functions, $$g$$ dominates $$f$$, written $$f(t)\ll g(t)$$ -just when $$\lim_{t\to\infty}[f(t)/g(t)]=0$$; when the limit is a -non-zero constant we say $f\sim g$. Since supply $$A(t)$$ is -*eventually* less than a powerlaw of $$t$$, it is *subexponential*. -Thus, no positive rate of Ether supply expansion can maintain -indefinitely. +The last line is Vinogradov asymptotic notation, used in the rest of +this post[^asym] For two positive functions, $$g$$ dominates $$f$$, +written $$f(t)\ll g(t)$$ just when +$$\lim_{t\to\infty}[f(t)/g(t)]=0$$. When the limit is a non-zero +constant we say $f\sim g$. We use $\gg/\ll/\sim$ for numbers as well, +by which we mean the order of magnitude is much larger / much smaller +/ similar. + +The point. Since supply $$A(t)$$ is *eventually* less than a powerlaw of $$t$$, +it is *subexponential*. Thus, no positive rate of Ether supply +expansion can maintain indefinitely. This does not mean we would find every intermediate inflation rate -pleasant. Following surges in $$Q_+$$, inflation can accelerate quite -alarmingly. A good example will be the 2032 Ethereum staking-mania. -This can be modeled in future retrospect by the following graph -showing a simulation with `qs=.05, qu=.01`; a constant $$q_+=5$$\% of -unstaked ETH is added to the staking queue each year. We aren't -excited to hodl through multiple decades of 10\% inflation, and we -expect you aren't either. - -![US President-Elect Swift Endorses Ethereum!]( - ../assetsPosts/swifties.jpg) - -Instead of downplaying fears of inflation, rather we mean to separate -concerns. Unpleasantly high inflation in the medium term, even if -that "medium term" lasts decades, is a *dynamics* problem, not an -equilibrium problem, and so dynamical solutions (like EIP 7514) seem -better suited. Unfortunately we will see that given the above, -$$s\to1$$ is an *equilibrium* problem. +pleasant. Following surges in $$Q_+$$ and/or drops in supply, +inflation can accelerate quite alarmingly. A good example will be the +Ethereum staking-mania following the 2132 Atlantia-v-Eurasia market +crash, in which 99\% of present-day Ether will have been burned. We +aren't excited to hodl through multiple decades of 10\% inflation, and +we expect you aren't either! + +```python +@dataclass +class MegaBurnParams(ConstParams): + b: One = 1e-3 + qs: 1/Yr = 2e-1 +@dataclass +class MegaBurnSim(SUaSim): + ic: tuple[ETH, ETH] = (1.20e6 * .4, 1.20e6 * .6) + tspan: tuple[Yr, Yr] = (0, 200) + params: Params = MegaBurnParams() +zomg = MegaBurnSim() +zomg.sim() +``` + +![Disaster Scenario]( + ../assetsPosts/2025-issuance-dynamics/megaburn.png) + +Silliness aside, we do not mean to downplay fears of inflation, and we +encourage you to find more realistic scenarios in which such sustained +inflation occurs. Rather, we mean to separate concerns. Unpleasantly +high inflation in the medium term, even if that "medium term" lasts +decades, is a *dynamics* problem, not an equilibrium problem, and so +dynamical solutions (like EIP 7514) seem better suited. Unfortunately +we will see that given the above, $$s\to1$$ *is* an equilibrium +problem. ### Staking Fraction @@ -494,6 +518,9 @@ $$y':=\frac{dy}{dS}$$. neglect externalities $$\chi\approx0$$ because we have nothing intelligent to say about them, but you might not want to. +We don't recommend modeling this numerically; these equations *are* +stiff. We use it for mathematical analysis only. + #### Separation of Timescales and $$\alpha>0$$ Observe that every term in $$(\xi,\gamma)$$ are small fractions, @@ -508,16 +535,18 @@ of its own dynamic variable, with staking fraction equilibrating to $$s^\star$$ more quickly than $$\alpha^\star$$ equilibrates. Does this hold presently? For a sanity-check, a quick look at YCharts -since The Merge shows that $$s,\dot{s}$$ do indeed seem to vary over a -much greater range than $$(\log{A},\alpha)$$. +since Sept 2022 (the Merge) shows that $$s,\dot{s}$$ do indeed seem to vary over a +much greater range than $$(\log{A},\alpha)$$ under Proof-of-Stake. -![The staking fraction from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg) -![The inflation rate from YCharts](../assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg) +![The staking fraction from YCharts]( + ../assetsPosts/2025-01-15-issuance-dynamics/YCharts-x.jpg) +![The inflation rate from YCharts]( + ../assetsPosts/2025-01-15-issuance-dynamics/YCharts-alpha.jpg) For the remainder of this post, we will assume this obtains. Anecdotally, even when it does not, the revealed interplay between inflation and staking fraction shows up, and is a very important -concept for Ethereum Macroeconomics. +concept for Ethereum macroeconomics. # Staking Fraction in 1D @@ -588,27 +617,27 @@ the range of $$\alpha$$ categorize three basic behaviors. ## Stability in One Dimension -Fixed-points are market equilibria just when they are stable. Every -stable fixed point in a one-dimensional flow is locally stable. In -general a fixed point is locally stable when small changes -(*perturbations*) shrink over time. For a continuous map like ours, -this concerns the derivative of the RHS at the fixed point: if it is -negative, then small perturbations shrink and the fixed point is a -stable *sink*. If the derivative is zero, the fixed point is a -degenerate *center*, unrealistic outside of physics. If positive, the -fixed point is an unstable *source*. +Fixed-points are market equilibria just when they are +stable.[^stability] Local stability is easy to asses for one +dimensional maps. In general a fixed point is locally stable when +small changes (*perturbations*) shrink over time. For a continuous +map $$\dot{x}(x)$$ like ours, this concerns the derivative of the RHS +at the fixed point. If it is negative, then small perturbations +shrink and the fixed point is a stable *sink*, and $x$ +"flows"[^flowflux] toward it. If the derivative is zero, the fixed +point is a degenerate *center*, unrealistic outside of physics. If +positive, the fixed point is an unstable *source* and repels $x$. ![1D Stability Conditions]( ../assetsPosts/2025-01-15-issuance-dynamics/1d-stab.png) -Specifically, we want the sign of +Specifically for staking fraction, we want the sign of $$\left.\frac{\partial\dot{s}}{\partial s}\right|^\star$$ to determine wether $$s^\star$$ is (un)stable. We will be ignoring the partial derivatives ("sensitivities") by assuming they are small in comparison -with their corresponding intensives.[^small-part] - -The full no-churn stability condition, including variations in -$\alpha$ is (with subscripts denoting partials) +with their corresponding intensives.[^small-part] The full no-churn +stability condition, including variations in $\alpha$ is (with +subscripts denoting partials) $$\displaystyle (\alpha^\star/f^\star)\ +\ r\ +\ (\jmath^\star/f^\star)\ \ >\ \ @@ -624,12 +653,12 @@ $$\alpha^\star+r^\star f^\star \gtrsim -\jmath^\star$$. Weak Deflation. If $$\alpha_{const}\in(-rf-\jmath,0]$$ then an interior market equilibrium $$s^\star$$ requires high slashing, as we argued before based on Churn. All other things being equal, it is -also less stable; in the region where partial derivatiives might play -a role, but probably okay. Assuming low slashing continues, as -validators must pay the cost themselves so are incentivized to -minimize it, then $x^\star>1$ and runaway staking is inevitable if the -fixed point is stable: the numerator is larger than the denominator. -If unstable, $s$ is pushed toward zero, and becomes unpredictable. +also less stable. Assuming low slashing continues, as validators must +pay the cost themselves so are incentivized to minimize it, then +$x^\star>1$ and runaway staking is inevitable if the fixed point is +stable: the numerator is larger than the denominator. If unstable, +$s$ is pushed toward zero, and in practice becomes unpredictable by +our model: externalities intervene to do... something. Strong deflation. Consider now $$\alpha<-rf-\jmath$$. This could happen for instance if the issuance curve were reduced particularly @@ -654,7 +683,7 @@ hopefully we will have the luxury of bemoaning inflation's effects on hodlers, and reducing it, without fear of the infrastructural dynamics of these market equilibria running amok. -### Low Inflation; Even Lower Fees +### Medium $r$; Low Inflation; Even Lower Fees Well, that was deflating! Let's cheer ourselves up by considering the behaviors under $$\alpha_{const}>0$$. A positive role for inflation can @@ -675,15 +704,20 @@ the equilibrium reinvestment ratio $$r^\star$$ is the y-value. A breakdown of limitng behaviors is illustrative under positive inflation. For any value of non-negative inflation, $$r^\star$$ is a lower bound for the equilibrium staking fraction we should expect. If -inflation dominates fees, $$\alpha_{const}\gg f^\star$$ then $$s^\star$$ -is larger by a small amount than $$r^\star$$, while if fees dominate -inflation $$0<\alpha_{const}\ll f^\star$$ then $$s^\star$$ becomes -insensitive to non-zero reinvestment ratio and $$s^\star\to1$$. For a -numerical comparison, eyeballing charts (so *extremely* rough -approximations here, possibly off by an order of magnitude) $$f -\approx .002<.005\approx\alpha$$ so to within 10\% error above, +inflation dominates fees, $$\alpha_{const}\gg f^\star$$ then +$$s^\star$$ is larger by a very small amount than $$r^\star$$, while +if fees dominate inflation $$0<\alpha_{const}\ll f^\star$$ then +$$s^\star$$ becomes insensitive to non-zero reinvestment ratio and +$$s^\star\to1$$. In the intermediate range $r^\star +f^\star\sim\alpha^\star$, $r^\star < s^\star$ still but the gap is +bigger. + +For a numerical comparison, eyeballing charts (so *extremely* rough +approximations here, possibly off by an order of magnitude) $$rf +\approx .004\sim.005\approx\alpha$$ so to within 15-20\% error above, $$s^\star\approx r^\star$$ over the range of $$r\in(.5,.75)$$ inferred -from the Lido yield rate. +from the Lido yield rate. Clearly we are not yet at equilibrium, or +$r$ is much lower than our extremely rough estimates. How the transient values $$(\alpha_{now}/f,r)$$ relate to the true equilibrium values $$(\alpha^\star/f^\star,r^\star)$$ depends on @@ -706,7 +740,7 @@ intermediate timescales we see $$s\to1$$? This is certainly possible; per the arguments of Ethereum researchers, high inflation could still lead to runaway staking if $$r$$ is sensitive enough. -In our model the net effect of inflation on LI;ELF staking fraction +In our model the net effect of inflation on staking fraction equilibrium is reflected by taking the derivative $$0<\left.\frac{ds^\star}{d\alpha}\right|^\star$$ assuming $$r,f$$ are implicit functions of $$\alpha$$. That is, the necessary condition @@ -728,6 +762,30 @@ conditions satisfy the above condition. No mean feat, but it would help focus inflationary pressure arguments into empirically measurable assertions that can be tracked for Ethereum health. +### Whence "Mr"; Dangers of Low $r$ + +Before you start advocating for "EIP 7514 On Steroids" on crypto +twitter (that is, throttling the staking queue to prevent $r$ from +rising, such as due to inflation-pressure), consider the danger of low +staking, that is low $r$ in the absence of significant churn. In the +simplified staking fraction equation, the term $\alpha(r-s)$ is +positive only when $s0$... what if +$r\to\epsilon\ll{s_{now}}$ for some reason? Then the only positive +term $(rf+q_+)(1-s)$ gets very small in the absence of new staking. +But, given sufficient inflation and fees and very marginal +sensitivities, the stability requirement $\alpha+rf+\jmath>0$ could +still obtain, so the new fixed point, presuming inflation maintains in +LI;ELF territory, $s^\star\sim\epsilon$ remains stable. + +That's quite a lot of ifs, but consider exponential growth of supply +under present issuance: $y_1\sqrt{sA}$... $A\sim e^{\alpha t}$ can +maintain so long as $s\sim\epsilon + s_{now}e^{-\alpha t}$. For +roughly $\alpha^{-1}\log\epsilon^{-1}$ Ethereum could in principle +experience simultaneously lowered security, and consistent inflation. +For small $\epsilon$ and $\alpha\sim.1$ that's more than a decade in +principle. Whether this has any bearing on reality, we don't know, +but its worth being aware of. + # Concluding Discussion @@ -821,68 +879,74 @@ So, finally... should the Ethereum community reduce issuance? ### Glib answer -Nope. +No. ### Short answer -If you are very inflation averse or you want to slow down the -transition to high staking, please consider and simulate downward -adjusting the constants in EIP 7514, adopted during the Deneb upgrade. -This already directly limits $$R+Q_+$$, but was very nicely designed -to not interfere with existing staking flows. EIP 7514 does not -solve, and does not claim to solve, the long term problems, but we -have been forced to conclude that reducing issuance doesn't solve them -either! +Let's adopt a dynaical solution to a dynamical problem. If you are +very inflation averse or you want to slow down the transition to high +staking, please study and simulate downward adjusting the constants in +EIP 7514, adopted during the Deneb upgrade. This already directly +limits $$R+Q_+$$, but was very nicely designed to not interfere with +existing staking flows. It would be particularly interesting to see +if $R$ can be safely decoupled from limits on $Q_+$, to avoid the +danger of low-$r$ scenarios. EIP 7514 does not solve, and does not +claim to solve, the long term problems, but we have been forced to +conclude that reducing issuance doesn't solve them either! ### Long answer Ask the users, especially the validators, especially the LSTs. Model user preferences so that the demand curve becomes semi-empirical instead of theorized. Near-100\% staking seems to be baked-in -eventually, and high inflation cannot sustain under an issuance yield -curve designed to avoid discouragement. So the question becomes -essentially "How bad will it get in the meantime?" This is actually a +eventually, as persistent inflation cannot sustain under an issuance yield +curve designed to avoid discouragement. + +So the question becomes essentially "How bad will it get in the +meantime?" We recommend of course that you use these tools to run +simulations. But that is only half the answer... this is really a question about user preferences. Austrian School devotees may be so inflation-averse that they are already staking all their previously-liquid ETH at $$\alpha\approx0.5$$\%/yr. In contrast users who were content to grow up with fiat currencies during periods of -$$\approx$$ 3\% inflation or even worse might not care, or would just +$$\approx3$$\% inflation or even worse might not care, or would just stake in Compound or buy stETH. For users seeking to passively preserve wealth, staking in Compound (or Aave, or whatever) is ideal for generating raw Ether demand, of course. Staking in LSTs, though concerning from a governance angle, may present less of an issue than some have feared. LSTs must share -some yield with users to be in business, but if they want to profit -themselves must maintain $$r<1$$. +some yield with users in order to have users, but if they want to stay +in business they must maintain $$r_{LST}<1$$, and both profit-taking +and covering fiat-denominated costs go into unstaked Ether $U$. So how high can $$r_{LST}$$ go before LSTs find the loss of gross profits unacceptable? Great question. We will see next time that as -many have recognized, the entity/group that can maintain the highest -$$r$$ wins the race eventually. It's worth noting however that if you +many have recognized, the entity/group $i$ that can maintain the highest +$$r_i$$ wins the race eventually. It's worth noting however that if you apply price uncertainty via a risk-discounting rate of, well anything -really, you will see that the "maximize $$r$$" strategy is far from the -most profitable, risk-adjusted. Of course, a speculative investor -seeking to profit from our analysis would find our study woefully -short of details. Put anoher way, since you have insisted on reading -the "long answer", we will end with the classic and cowardly refrain -of academics and academic-adjacents everywhere "it requires more -research!" +really, you will see that the "maximize $$r_{mine}$$" strategy is far +from the most profitable, risk-adjusted. Of course, a speculative +investor seeking to profit from our analysis would find our study +woefully short of details. Put anoher way, since you have insisted on +reading the "long answer", we will end with the classic and cowardly +refrain of academics and academic-adjacents everywhere "it requires +more research!" # Dynamical Systems References -We are developing the ethode guide in the hope that it can serve a -pedagogical role. For now we have assumed some basic familiarity with -nonlinear dynamics, asymptotic methods, etc. at the level of the first -few of Prof. S. Strogatz youtube lectures. +We hope to develop the ethode guide so it can serve a pedagogical +role. For now we have assumed some basic familiarity with nonlinear +dynamics, asymptotic methods, etc. at the level of the first few of +Prof. S. Strogatz youtube lectures. - [Nonlinear Dynamics](https://youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V&si=iN5YCipB_CeIfrbB) - [Perturbation Theory](https://youtube.com/playlist?list=PL5EH0ZJ7V0jV7kMYvPcZ7F9oaf_YAlfbI&si=lFFFJIMAH6nE5BX1) Highly Recommended books in order of increasing difficulty and -sophistication if you want to understand this stuff: +sophistication if you decide you want to understand this stuff: - Kun (2020) [A Programmer’s Introduction to Mathematics](https://www.amazon.com/Programmers-Introduction-Mathematics-Second/dp/B088N68LTJ/) @@ -904,193 +968,174 @@ high demand. They have absolutely no connection to this post, our models, etc. and hopefully they will not sue us for using their name in a bad dynamical systems joke. -[^aves]: We use moving quarterly averages, though any timescale -$$\tau$$ sufficiently long that the [fast dynammics]() of [the base]() -fee are [integrated out](), and the lags from (un)staking queues are -not appreciable; see [guide](https://github.com/20squares/ethode/blob/master/guide/guide.md) for details. As we are averaging -quarterly, we set the staking $$\dot{\mathcal{Q}}_+$$, unstaking -$$\dot{\mathcal{Q}}_-$$, and reward $$\dot{W}$$ queues to zero, including -their respective flows ($$R+Q_+,Q_-,I+P$$) in their target stocks -($$S,U,V$$); even if ethereum produces empty blocks, so long as the -reward queue is not empty $$U>0$$. - -[^IyS]: The approximation is due to the use of quarterly averages. A -more precise approximation is $$\displaystyle -I=\frac{1}{\tau}\int_{t-\tau}^ty^\bullet S^\bullet dt'=yS-|\kappa| $$ -where $$S^\bullet,y^\bullet$$ are spot values and the quarterly -covariance between staking and yield is -$$\kappa=\int_{t-\tau}^t(y^\bullet-y)(S^\bullet-S)dt'<0$$. See the -[ethode guide](https://github.com/20squares/ethode/blob/master/guide/guide.md) for more discussion. - -[^regex]: This regex script is provided to translate the $$\LaTeX$$ -within the markdown source to (our esimatimate of) terminology more -common at [issuance.wtf](https://issuance.wtf) - -[^partial]: Sometimes $$\dot{x}=dx/dt$$ is used e.g. for the partial -derivative $$\frac{\partial x}{\partial t}$$, *but not here*. For a model -in which you assume dynamical variables $$(A,s,\alpha)$$, these derivatives -are related for some parameter $$x(A,s,\alpha,t)$$ -$$\displaystyle -\frac{dx}{dt} := \dot{x} = -\frac{\partial x}{\partial t} + -\frac{\partial x}{\partial A}\dot{A} + -\frac{\partial x}{\partial s}\dot{s} + -\frac{\partial x}{\partial\alpha}\dot{\alpha} =: -x_t + x_A\dot{A} + x_s\dot{s} + x_\alpha\dot{\alpha} -$$ -in which each partial is taken holding all the other variables constant, -and $$\frac{\partial x}{\partial t}$$ is used in practice to capture any -variability from non-dynamical variables. In principle -$$\partial_A{x}$$ and $$\partial_\alpha{x}$$ are distinct; a quantity can -depend on supply (how big ETH market cap is compared to BTC, say) -and inflation independently. +[^reasons]: For derivations involving differential equations, "D" +(used for staking Deposit) and its corresponding intensive "d" are +cursed variables. "s" was already in use in some places for staking +fraction, and we are resolute on keeping the intensive and its +corresponding extensive the same letter. "C" is a more natural choice +for circulating supply, but then the three variables of most interest +are somethiing like (C,s,ς) which is masochistic in its silibance, +even for squares. We prefer "accessible" to circulating because the +former implies you *could* access it, at some cost, while the latter +sometimes implies a velocity of money. A velocity which S and much of +U may lack depending on dynamics: backed up unstaking queue, leveraged +or looped CDPs, etc. But even if our terminology were actually +superior, we're not going to change economic jargon any time soon. + +[^aves]: We use moving quarterly averages, though any timescale τ +sufficiently long that the +[erratic](https://decentralizedthoughts.github.io/2022-03-10-eip1559/) +and [fast dynammics](https://arxiv.org/pdf/2102.10567) of [the base +fee](https://ethereum.github.io/abm1559/notebooks/eip1559.html) are +integrated out, and the lags from (un)staking queues are not +appreciable. As we are averaging quarterly, we set the staking, +unstaking, and reward queues to zero, including their respective flows +(R+Q+,Q-,I+P) in their codomain stocks (S,U,V); even if ethereum +produces empty blocks, so long as the reward queue is not empty U > 0. +Se also our next footmnote on I<=yS.[^ycov] + +[^ycov]: (Here the lack of latex in footnotes really sucks.) The +inequality I <= yS is due to the use of quarterly averages. Using +"int" for integral, and "S.", "y." for spot values +``` +I = int[t-τ to t][y.(S.) S. dt'/τ] + = yS - int[t-τ to t][(y - y.)(S - S.)dt/τ + = yS - |COV(y., S.)| +``` +This approximation works for any yield curve, erring in a +conservative direction without explicit depenedence on the present-day +yield curve `y. = y0(1)/sqrt(S)` where y0(1)~166.3/yr. + +[^partial]: Sometimes "dot x" = dx/dt is used for the partial +derivative of x with time t, which we denote x_t. The full relation +is ```dx = x_t + x_A dA + x_s ds + x_α dα``` in which each partial is +taken holding all the other variables constant, and x_t is used in +practice to smuggle in any variability from non-dynamical variables. +In principle x_A and x_α are distinct; a quantity can depend on supply +(how big ETH market cap is compared to BTC, say) and inflation +independently. [^yield]: For anyone from finance, this is *not* the same as a [bond yield curve](https://www.investopedia.com/terms/y/yieldcurve.asp); there is essentially no necessary lag for vaidator rewards, accounting quarterly. - -[^reasons]: The entire human project of using symbols to refer to -things is deeply cursed and probably hopeless. Nonetheless, we all -presist in providing excuses for - [^cats]: We use domain/codomain in imprecise analogy with category theory mainly because we want to reserve "source" for an attractor, as -per dynamical systems. The analogy, whie imprecise is not +per dynamical systems. The analogy, whie inexact is not inappropriate. It is routine to implicitly use associativity to account for fibers of flows through multiple steps; "electricity from wind/nuclear/gas" even though the electrons are indistinguishable. -Flows such as tx fees $$U\overset{F}{\longrightarrow}V,\cancel{O}$$ -involve a categorical product $$V\times\cancel{O}$$ in that the -fractional flows $$U\overset{B}{\longrightarrow}\cancel{O}$$ and must -factor through $$V\times\cancel{O}$$. For any flow with contributions -to stocks $$V,\cancel{O}$$ it is not hard imagining a uu2i universal -object $$V\times\cancel{O}$$ mediating. Similarly the staking queue -$$V+(U-V)\overset{R+Q_+}{\longrightarrow}S$$ should involve a -categorical coproduct. Wether there is content here beyond "flows are -functions in $$\sf Set$$" is unclear. None of this matters in the least +Flows such as tx fees U--F-->V,Ø involve a categorical product VxØ in +that the smaller fractional flows U--B-->Ø must factor through it. +Similarly the staking queue V+(U-V)---R+Q+-->S inolves a coproduct in +he domain. Whether there is content here beyond "flows are injective +morphisms in Set" is unclear. None of this matters in the least for Ethereum dynamics, of course. If you're reading it consider this an easter egg / attempt to detect a living and alert audience. -[^elowex]: Why not simply choose \"$$B=bA$$\" as was done more simply (modulo -syntax) in [Elowsson 2020]()?[^syntax] Obviously if there is no -unstaked Ether no one can afford tx fees, but this might not matter. -More importantly $$S$$ is a *dynamical variable*, so $$b=B(A-S)$$ is more -appropriate here. The function $$B$$ might do all kinds of complicated -nonsense, but it can never go negative and it can never exceed $$U$$. - -[^whyr]: Intensives expressed as fractions of flows such as $$R/(I+P)$$), instead -of fractional rates of sources (like $$J/S$$ or $$Q_-/S$$) occur when the -source dynamical variable, here $$V$$, is assumed to equilibrate -$$\dot{V}\approx0$$. Then the outging flows $$R+K$$ must equal the -incoming flows $$I+P$$, so we choose $$R=r(I+P)$$. If onchain data -indicates %\approx$$70\% reinvestment of staking rewards into $$S$$ takes -a lot longer than a quarter we would revisit this assumption, though +[^elowex]: Why not simply choose "B = bA" as was done in a 2020 post +by Elowsson? Obviously if there is no unstaked Ether no one can +afford tx fees. Here s is a *dynamical variable*, so b = B(A-S) = +bf(1-s)A is more appropriate for our model. The function B might do +all kinds of complicated nonsense, but it can never go negative and it +can never exceed U. + +[^whyr]: Intensives expressed as fractions of flows such as R/(I+P), instead +of fractional rates of sources (like J/S or Q-/S) occur when the +source dynamical variable, here V, is assumed to equilibrate +dV/dt=0. Then the outging flows R+K must equal the +incoming flows I+P, so we choose R=r(I+P). If onchain data +indicates, say, ~70\% reinvestment of staking rewards into S takes +a lot longer than three months, we would revisit this assumption, though we do not expect our qualitative results to change re inflation and staking fraction. -[^rlst]: A non-zero $$r=R/(I+P)$$ is built into the smart contract of every +[^rlst]: A non-zero r=R/(I+P) is built into the smart contract of every Liquid Staking Provider (LSP). Here, token-holders provide Ether and receive a redeemable token (LST) that shares some staking rewards -with them. This fraction of rewards $$r_{LST}$$ is a lower bound on -the long term behavior $$r_{LST}\leq r_{LSP}\leq -R^\star(I^\star+P^\star)$$; more on this next time. +with them. This fraction of rewards r_LST is a lower bound on +the long term behavior r_LST <= equilibrium R(I+P). -[^rdyn]: Splitting the staking queue into $$R+Q_+$$ allows us to +[^rdyn]: Splitting the staking queue into R + Q+ allows us to somewhat separate short-term *transient* behavior from long-term dynamics. Speculative investment in staking by venture capitalists -and novice stakers is expected to die down eventually -$$\lim_{t\to\infty}Q_+\approx0$$; either they give up or they run +and novice stakers is expected to die down eventually; either they give up or they run staking like a business where making a profit matters. Every business that wants to stay in business reinvests some portion of its profits, -so $$r,R>0$$ and is what matters in the long run, once most everyone who -wants to stake is indeed staking. +so r,R > 0 is what matters in the long run, once most everyone who +wants to stake is staking. -[^metastable]: Metastability involves a region of space where -trajectories are bounded surrounding a fixed point with at least one -eigenvalue with positive real part. We won't rule it out, but look -first for stable fixed points. +[^flowflux]: Confusingly the movement of a dynamic variable +toward/away-from a fixed point is often also called a "flow". Terms +in equations lke R,J,... could then be called "fluxes". But you're not +confused, right? -[^time]: We can often use the dependence on $$t$$ to smuggle in +[^time]: We can often use the dependence on t to smuggle in any forces, like market panics, etc. that we neglected to include as dynamical variables. If not, we must add a dynamical variable. -[^asym]: For computer scientists $f\ll g$ is equivalent to -$f=\mathcal{O}(g)$, which you may be more familiar with. Specifically -$$I\ll S$$ means that $$\lim_{S\to\infty}(I/S)=0$$. We say this -instead of $I\leq S$ to avoid confusion with inequality due merely to -coefficients. Asymptotic Notation is well-explained on +[^asym]: For computer scientists f << g is equivalent to F=O(g) if +you're more familiar with big-O notation. Specifically I << S means +that the limit of I/S as t gets very large is 0. Contrast to I <= yS, +which could just be a matter of coefficients. Asymptotic Notation is +well-explained on [wikipedia](https://en.wikipedia.org/wiki/Big_O_notation); see the bottom for Vinogradov. The art of using it to your advantage in calculations is demonstrated by [Prof. Carl Bender](https://www.youtube.com/watch?v=LYNOGk3ZjFM). -[^ycov]: The relation between quarterly averaged issuance and -the yield curve $$y^\bullet$$ is: -$$\displaystyle -I:=\int_{t-\tau}^t y^\bullet(S^\bullet)S^\bullet(t')dt'\approx yS-|COV(y^\bullet,S^\bullet)| -$$ -where we approximate $$y,S$$ as constant on the interval $$(t-\tau,t)$$ -and the covariance is negative -$$0>COV(y^\bullet,S^\bullet)=\tau^{-2}\int_{t-\tau}^t(y^\bullet - -y)(S^\bullet-S)dt'$$. The use of covariance here may seem a bit silly; -$$y^\bullet(S^\bullet)$$ is a deterministic relationship! -We do this because the community is discussing changing the yield curve, and we want to discuss -generic features of Ethereum. -So we have used approximations that we felt err in a conservative direction without explicit -depenedence on the present-day yield curve $$y^\bullet_0(S^\bullet)=y_0(1)\sqrt{S_1/S}$$ where -$$y_0(1)\approx166.3$$/yr and $$S_1=1$$ETH. - [^params]: Variable parameters that are positive fractions cannot contribute fixed-points themselves, but they can strongly influence -*where* a fixed point is. Example: as $$s\to1$$, if the leading terms -were $$\beta\sim(1-s),j\sim(1-s)^2$$ this gives increasingly larger -$$A^\star$$ as $$s\to1$$. - -[^vitalikp]: Our $$\frac{\partial{d\log~y}}{\partial{d\log~S}}$$ is -$$p-1$$ in the [discouragement paper](https://github.com/20squares/ethode/blob/master/guide/guide.md) - -[^noburn]: While slashing could believably go to zero on quarterly -timescales, no burn $$B=0$$ implies blocks are empty, at least. -Obviously this isn't really a functioning state for Ethereum, but -think of it as a limiting behavior. A more nuanced treatment of -$$s\to1$$ can be obtained assuming $$s^\star=1-\epsilon$$ where -$$\epsilon\ll1$$, making $$A^\star\sim (y(1)/b^\star -f^\star)^2\epsilon^{-2}$$; as $$\epsilon$$ is very small $$\epsilon^{-2}$$ -is very large. Detailed treatment of the forces using expansions in -$$\epsilon$$ are useful, and we would need to model churn, slashing, and -burn in light of stochasticity/quantization. One can use difference -equations, but a useful generic behavior of such systems (a "weak -coupling" limit) is that such perturbations move the dynamics away from the -fixed point, apparently randomly, yet the average rate of precession -about $$s^\star=1-\epsilon$$ is often still given by the imaginary -component of the largest eigenvalue. - -[^mu]: If anything the fractional rates - of slashing and burn are positive with small changes in inflation, - due to either a single ETH potentially being of less real value, or - stimulation of economic activity attracting more validators and - higher average burn. - -[^ics]: $$\gamma$$ expresses the sensitvity of inflation to supply - initial conditions; the partial $$x_A$$ always holds $$(\alpha,s)$$ - consant, but $$\dot{A}=\alpha A$$, and the only partials in $$\gamma$$ - are $$\jmath_A,\beta_A$$. +*where* a fixed point is. Example: as s -> 1, if the leading terms +were bf~(1-s) and j~(1-s)**2 this gives increasingly larger +equilibrium A as s goes to 1. + +[^vitalikp]: Our (dlog y)/(dlog S) = 1 - p in the [discouragement +paper](https://github.com/20squares/ethode/blob/master/guide/guide.md) + +[^noburn]: Regarding B = bf(1-s)A the burn. While slashing could +believably go to zero on quarterly timescales, no burn B=0 implies +blocks are empty. Obviously s=1, B=0 isn't really a +functioning state for Ethereum. A better asymptotic limit would be s += 1-ε making A ~ (1/ε)**2 (that's squared... so very big as ε is very +small). Detailed treatment of the burn, staking queues using +expansions in ε would be useful, and we would need to model churn, +slashing, and burn in light of stochasticity/quantization. One can +use difference equations, but a useful generic behavior of such +systems (a "weak coupling" limit) is that such perturbations move the +dynamics away from the fixed point, apparently randomly. Yet! +Somehow, the average rate of precession about s=1-ε is often given by +the imaginary component of the largest eigenvalue of the simpler +model. + +[^mu]: If anything the fractional rates of slashing and burn are + positive with small changes in inflation, due to either a single ETH + potentially being of less real value, or stimulation of economic + activity attracting more validators and higher average burn. + +[^ics]: gamma γ expresses the sensitvity of inflation to supply + initial conditions; the partial x_A always holds (alpha,s) consant, + but dA = alpha dt and the partials in gamma are j_A and (bf)_A. [^small-part]: Small sensitivities wrt intensives is not gauranteed. -Certainly a large magnitude, say $$\beta_s>1$$ cannot maintain for too -long: $$\beta\in(0,1)$$ afterall, but locally, a large derivative -$$\beta_s>\beta$$ is possible. - -[^stability]: Assesing the stability of very long-term -$$\alpha^\star=0$$ in the $$(A,s,\alpha)$$ model directly is interesting -but academic. One of the eigenvalues at any fixed point with -$$\alpha^\star=0$$ must be zero, so higher-order terms matter (the fixed -point is degenerate), and linear-stability analysis is insufficient. -A reader imbued with mathematical athleticism and free time is -encouraged to think of a Lyapunovv function $$л(\alpha=0)\geq0$$ for issuance yield, -and obtain a contraction mapping $$\dot{л}(y)<0$$. +Certainly a large magnitude, say (bf)_s>1 cannot maintain for too +long; 0 +bf is still possible. + +[^stability]: Global stability involves either trajectories infinitely +returning to a region of the fixed point (think comets) or a +contraction map showing the system shrinking to a limit set. We won't +rule out global stability, but recommend you look first for locally +stable fixed points. Assesing the stability of equilibrium zero +inflation in the (A,s,alpha) model is interesting but probably +academic. One of the eigenvalues at any fixed point with alpha=0 is +zero, so higher-order terms matter (the fixed point is degenerate), +and linear-stability analysis is insufficient: we need to care about +global stability not just local. A reader imbued with mathematical +athleticism and free time is encouraged to think of a Lyapunovv +function л(alpha=0) >= 0, and obtain a contraction +mapping dл(alpha)/dt <= 0. [^SU]: Readers wishing for more detail are encouraged to use the two dimensional local stability criterion (see @@ -1098,6 +1143,4 @@ dimensional local stability criterion (see the condition of eigenvalues with an imaginary part. But simulate it too! -[^asym]: - From fcecb4704523a9a4f6f8d5b117fc2c24dfb16c31 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 18:39:27 -0600 Subject: [PATCH 48/70] small fixes --- _posts/2025-01-15-issuance-dynamics.md | 16 +++++++--------- 1 file changed, 7 insertions(+), 9 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 95cba4e..21aa8a3 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -393,7 +393,7 @@ dA &\leq& ysAdt = y_0(1)\sqrt{sA}dt \leq y_0(1)\sqrt{A}dt\\ \int_{A(0)}^{A(t)}A^{-1/2}dA &\leq& \int_0^{\ t} y_0(1)dt\\ \left.\frac{1}{2}\sqrt{A}\right|^{\sqrt{A(t)}}_{\sqrt{A(0)}} &\leq& y_0(1)t\\ A(t) &\leq& \left(\sqrt{A(0)}+2y_0(1)t\right)^2\\ -\therefore A(t) \sim t^2 &\ll& e^{kt} ~\forall ~\mathrm{const.}~k>0 +\therefore A(t) &\lesssim& t^2 \ll e^{kt} ~\forall ~\mathrm{const.}~k>0 \end{array} $$ @@ -432,7 +432,7 @@ zomg.sim() ``` ![Disaster Scenario]( - ../assetsPosts/2025-issuance-dynamics/megaburn.png) + ../assetsPosts/2025-issuance-dynamics/mega-burn.png) Silliness aside, we do not mean to downplay fears of inflation, and we encourage you to find more realistic scenarios in which such sustained @@ -629,7 +629,7 @@ point is a degenerate *center*, unrealistic outside of physics. If positive, the fixed point is an unstable *source* and repels $x$. ![1D Stability Conditions]( - ../assetsPosts/2025-01-15-issuance-dynamics/1d-stab.png) + ../assetsPosts/2025-01-15-issuance-dynamics/1d-stab.jpg) Specifically for staking fraction, we want the sign of $$\left.\frac{\partial\dot{s}}{\partial s}\right|^\star$$ to determine @@ -996,12 +996,10 @@ Se also our next footnote on I<=yS.[^ycov] [^ycov]: (Here the lack of latex in footnotes really sucks.) The inequality I <= yS is due to the use of quarterly averages. Using -"int" for integral, and "S.", "y." for spot values -``` -I = int[t-τ to t][y.(S.) S. dt'/τ] - = yS - int[t-τ to t][(y - y.)(S - S.)dt/τ - = yS - |COV(y., S.)| -``` +"int" for integral, and "S.", "y." for spot values. + * I = int[t-τ to t][y.(S.) S. dt'/τ] + * I = yS - int[t-τ to t][(y - y.)(S - S.)dt/τ + * I = yS - |COV(y., S.)| <= yS This approximation works for any yield curve, erring in a conservative direction without explicit dependence on the present-day yield curve `y. = y0(1)/sqrt(S)` where y0(1)~166.3/yr. 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zAi;VH#W*iI+Ayh1V+Miyv_y1@((q75Wpy6isd>1o1|5Y`1mk;7UC)#iPT}vV7Xq73 z2zFd-oygQD;YYObzLr~6Tujl<(8>pzff}k7c$4okZG@486Q&aZs{^?=SiQLOhLRBgBIcu#-k_&9 z4Fx#|j_5WFtMxQrRP}ylY2{c^$m6C7NYSLBewosv&-xJ$3VV?cQvV!V8@aY9Is%nW zJbAiv$CJlTby1}nuP>g0Znh#rHViO0jJbd*ORt2|L71#`ZkZ$KAW6ArSt>|k41pS=n%-tGPSXcD)c{r9RkYJ&f7{ZCn5CMF%Y&yCrtt;bH) HvN-cUEE}dn literal 0 HcmV?d00001 From 4ab449681452821abb2852689e73d0212ac408a6 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 19:06:05 -0600 Subject: [PATCH 50/70] pulled broken footnoe into text, fixed images and a code typo --- _posts/2025-01-15-issuance-dynamics.md | 40 +++++++++++++++----------- 1 file changed, 24 insertions(+), 16 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 21aa8a3..8264e85 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -189,12 +189,9 @@ A few flows deserve specific comment: $$I$$ and $$(R,Q_+,K)$$. All of these stocks and flows, $$(I,S,\ldots)$$, are moving time-averages over *spot values* $$(I^\bullet,S^\bullet,\ldots)$$ -defined at a given block; $$S:=\tau^{-1}\int_{t-\tau}^t -S^\bullet(t')dt'$$, etc. In the case of issuance, we -can express spot issuance as a known function of the yield curve -$$I^\bullet = y^\bullet S^\bullet$$. We'll assume that +defined at a given block. For issuance, we'll assume that -1. Issuance is sublinear $$\boxed{1\ll I\approx yS\ll S}$$[^asym] to +1. Issuance is sublinear $$1\ll I\approx yS\ll S$$[^asym] to avoid [discouragement attacks]( https://github.com/20squares/ethode/blob/master/guide/guide.md), and that @@ -205,8 +202,25 @@ $$\frac{\partial{d\log{y}}}{\partial{d\log{S}}} \approx\frac{\partial{d\log{y}^\bullet}}{\partial{d\log{S^\bullet}}}$$.[^vitalikp] The first is common and almost certainly an overestimate with -$I\leq{yS}$ more precise.[^ycov] We deem this a good direction in -which to err in light of our results concerning (the lack of) runaway +$I\leq{yS}$ more precise. We reason as follows. We can express spot +issuance as a known function of the yield curve $$I^\bullet = +y^\bullet S^\bullet$$. From this we obtain an inequality for the +quarterly-averaged issuance $I\leq yS$ using time-covariance. + +$$\displaystyle +\begin{array}{rcl} +I &=& \frac{1}{\tau}\int_{t-\tau}^ty^\bullet S^\bullet dt'\\ + &\approx& yS + \frac{1}{\tau}\int_{t-\tau}^t(y^\bullet-y)(S^\bullet-S)dt'\\ + &=& yS - |COV(y^\bullet,S^\bullet)|\\ +I &\leq& yS +\end{array} +$$ + +This approximation should work for any positive definite yield curve +with finite slope, erring in a conservative direction without explicit +dependence on the present curve $y^\bullet = y_0(1)/sqrt(S^\bullet)$ +with $y_0(1)\approx166.3$/yr. We deem this a good direction in which +to err in light of our results concerning (the lack of) runaway inflation. ### Bounding Reinvestment $$R$$ @@ -278,7 +292,9 @@ with $A_{now}\approx120e6,\ s_{now}\approx.3$ as a start for the models that follow. ```python +from numpy import sqrt from ethode import * + @dataclass class ConstParams(Params): y1: 1/Yr = 166.3 @@ -358,7 +374,7 @@ $$\displaystyle \alpha\ \approx\ y(sA)s-\beta(1-s)-\jmath s \ =\ y_0(1)\sqrt{s/A}-\beta(1-s)-\jmath s $$ -You can explore this by noting `alpha(), sfrac()` as `@output` methods +You can explore this by adding `alpha(), sfrac()` as `@output` methods ```python @dataclass @@ -995,14 +1011,6 @@ produces empty blocks, so long as the reward queue is not empty U > 0. Se also our next footnote on I<=yS.[^ycov] [^ycov]: (Here the lack of latex in footnotes really sucks.) The -inequality I <= yS is due to the use of quarterly averages. Using -"int" for integral, and "S.", "y." for spot values. - * I = int[t-τ to t][y.(S.) S. dt'/τ] - * I = yS - int[t-τ to t][(y - y.)(S - S.)dt/τ - * I = yS - |COV(y., S.)| <= yS -This approximation works for any yield curve, erring in a -conservative direction without explicit dependence on the present-day -yield curve `y. = y0(1)/sqrt(S)` where y0(1)~166.3/yr. [^partial]: Sometimes "dot x" = dx/dt is used for the partial derivative of x with time t, which we denote x_t. The full relation From 6284a0e9a92040d15c62e89accacd080fb32e9a8 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 19:14:15 -0600 Subject: [PATCH 51/70] fixed image link --- _posts/2025-01-15-issuance-dynamics.md | 27 +++++++++++++------------- 1 file changed, 14 insertions(+), 13 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 8264e85..a91eaa5 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -429,9 +429,7 @@ This does not mean we would find every intermediate inflation rate pleasant. Following surges in $$Q_+$$ and/or drops in supply, inflation can accelerate quite alarmingly. A good example will be the Ethereum staking-mania following the 2132 Atlantia-v-Eurasia market -crash, in which 99\% of present-day Ether will have been burned. We -aren't excited to hodl through multiple decades of 10\% inflation, and -we expect you aren't either! +crash, in which 99\% of present-day Ether will have been burned. ```python @dataclass @@ -448,16 +446,19 @@ zomg.sim() ``` ![Disaster Scenario]( - ../assetsPosts/2025-issuance-dynamics/mega-burn.png) - -Silliness aside, we do not mean to downplay fears of inflation, and we -encourage you to find more realistic scenarios in which such sustained -inflation occurs. Rather, we mean to separate concerns. Unpleasantly -high inflation in the medium term, even if that "medium term" lasts -decades, is a *dynamics* problem, not an equilibrium problem, and so -dynamical solutions (like EIP 7514) seem better suited. Unfortunately -we will see that given the above, $$s\to1$$ *is* an equilibrium -problem. + ../assetsPosts/2025-01-15-issuance-dynamics/1d-stab.jpg) + +We aren't excited to hodl through multiple decades of 10\% inflation, +and we expect you aren't either! Silliness aside, we do not mean to +downplay fears of inflation, and we encourage you to find more +realistic scenarios in which such sustained inflation occurs. Rather, +we mean to separate concerns. + +Unpleasantly high inflation in the medium term, even if that "medium +term" lasts decades, is a *dynamics* problem, not an equilibrium +problem, and so dynamical solutions (like EIP 7514) seem better +suited. Unfortunately we will see that given the above, $$s\to1$$ +*is* an equilibrium problem. ### Staking Fraction From ea6ca43e153e4725dc597aecd989ad020463bbdd Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 19:16:12 -0600 Subject: [PATCH 52/70] fixed typo --- _posts/2025-01-15-issuance-dynamics.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index a91eaa5..dcdc56c 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -302,7 +302,7 @@ class ConstParams(Params): f: 1/Yr = 8e-3 j: 1/Yr = 1e-5 r: One = .65 - W's: 1/Yr = 1e-4 + qs: 1/Yr = 1e-4 qu: 1/Yr = 1e-4 s1: ETH = 1 def yld(self, S:ETH, **kwargs) -> 1/Yr: From 7c74d7e5c7f6ce7c6164b821fea97e852658a085 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 19:17:55 -0600 Subject: [PATCH 53/70] fixed right link wrong place --- _posts/2025-01-15-issuance-dynamics.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index dcdc56c..a49c308 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -446,7 +446,7 @@ zomg.sim() ``` ![Disaster Scenario]( - ../assetsPosts/2025-01-15-issuance-dynamics/1d-stab.jpg) + ../assetsPosts/2025-01-15-issuance-dynamics/mega-burn.png) We aren't excited to hodl through multiple decades of 10\% inflation, and we expect you aren't either! Silliness aside, we do not mean to From 3dd56651cde3dd89ce14e9d1d09c7458ab7d9fd9 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 19:21:13 -0600 Subject: [PATCH 54/70] fixed latex err --- _posts/2025-01-15-issuance-dynamics.md | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index a49c308..43fb5c2 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -584,7 +584,7 @@ s^\star = }{ m^\star := (\alpha_{const} + r^\star f^\star + q_+^\star + q_-^\star + (1-r)\jmath^\star) } -= 1 - \frac{(1-r)\jmath+q_-}{m^star} - \frac{\alpha_0(1-r^\star)}{m^\star} += 1 - \frac{(1-r)\jmath+q_-}{m^\star} - \frac{\alpha_0(1-r^\star)}{m^\star} $$ We will see that without $$\alpha>0$$ an interior market equilibrium is @@ -622,10 +622,6 @@ simplifies to $$\displaystyle s^\star \approx r^\star \frac{\alpha_{const} + f^\star}{ - \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} \ =\ 1\ -\ -\frac{\jmath^\star}{ - \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star}\ -\ -\frac{\alpha_{const}(1-2)}{ \alpha_{const} + r^\star f^\star + (1-r)\jmath^\star} $$ From 31bd19209fd9824353621ed35331bfa76a8c9b74 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Tue, 14 Jan 2025 19:25:05 -0600 Subject: [PATCH 55/70] fixed latex err --- _posts/2025-01-15-issuance-dynamics.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 43fb5c2..85e3c3e 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -270,16 +270,16 @@ $$b(A,S,t)=B/F$$.[^time] ### Table of Flows -| Flow Name | Symbol | Domain$$\to$$Codomain[^cats] | Constraint | Intensive | Range \[Units\] | +| Flow Name | Symbol | Domain $$\to$$ Codomain[^cats] | Constraint | Intensive | Range \[Units\] | | :-- | :-- | :-: | :-- | :-- | :-- | | Tx Fees | $$F$$ | $$U\to\cancel{O},V$$ | $$0 Date: Tue, 14 Jan 2025 19:34:59 -0600 Subject: [PATCH 56/70] ok im done, somebody merge the damn PR --- _posts/2025-01-15-issuance-dynamics.md | 12 +++++------- 1 file changed, 5 insertions(+), 7 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 85e3c3e..3d9273e 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -185,7 +185,7 @@ but more accurate upper limits would play little role in our analysis. A few flows deserve specific comment: $$I$$ and $$(R,Q_+,K)$$. -### Bounding Issuance $$I$$ +### Bounding Issuance All of these stocks and flows, $$(I,S,\ldots)$$, are moving time-averages over *spot values* $$(I^\bullet,S^\bullet,\ldots)$$ @@ -223,7 +223,7 @@ with $y_0(1)\approx166.3$/yr. We deem this a good direction in which to err in light of our results concerning (the lack of) runaway inflation. -### Bounding Reinvestment $$R$$ +### Bounding Reinvestment Reinvestment of staking rewards by validators $$R$$ is achieved by staking a new validator from existing rewards. While clearly a @@ -339,7 +339,7 @@ class SUSimConst(ODESim): params: Params = ConstParams() @staticmethod def func(t:Yr, v:tuple[ETH, ETH], p:Params) -> tuple[ETH/Yr, ETH/Yr]: - x = {'S': (S := v[0]), 'U': (U := v[1])} + S, U = v dS = (p.r * (y := p.yld(S)) - p.j - p.qu) * S + \ ((rf := p.r * p.f) * (1 - p.b) + p.qs) * U dU = ((1 - p.r) * y + p.qu) * S - \ @@ -400,7 +400,7 @@ Because of this, positive inflation cannot maintain indefinitely. We will demonstrate with the existing yield curve, but the argument is general. Unusually for this blog post, we show most of the steps so the argument is hopefully understood. Elsewhere we use $I\approx -yS$, but here we use $I\leq yS$ as greater rigor is appropriate.[^ycov] +yS$, but here we use $I\leq yS$ as greater [rigor](#bounding-issuance) is appropriate. $$\displaystyle \begin{array}{rcl} @@ -1005,9 +1005,7 @@ appreciable. As we are averaging quarterly, we set the staking, unstaking, and reward queues to zero, including their respective flows (R+Q+,Q-,I+P) in their codomain stocks (S,U,V); even if ethereum produces empty blocks, so long as the reward queue is not empty U > 0. -Se also our next footnote on I<=yS.[^ycov] - -[^ycov]: (Here the lack of latex in footnotes really sucks.) The +Se also our [section](bounding-issuance) on I<=yS. [^partial]: Sometimes "dot x" = dx/dt is used for the partial derivative of x with time t, which we denote x_t. The full relation From 17a1f19bcb77ef951b3efd67dcd7e07082459ccb Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Wed, 15 Jan 2025 01:47:15 -0600 Subject: [PATCH 57/70] integrating feedback --- _posts/2025-01-15-issuance-dynamics.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 3d9273e..211973a 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -101,11 +101,11 @@ $$\left(\frac{dr}{d\alpha}\right)^\star<0$$ (no news here) but simultaneously priority fees and MEV, as a fraction of unstaked Ether; $$\alpha^\star\gtrsim r^\star f^\star$$. -Inflation will *eventually* decay though, driving $$s\to1$$ if it is not -already there, though it may take considerable time. This "L2 future" -has been recognized by many others: most Ether is staked, with the -majority used for settlement of L2 rollups. We'll discuss it more -next time. +Inflation will *eventually* decay though, driving the equilibrium +point itself $$s^\star\to1$$, though it may +take considerable time. This "L2 future" has been recognized by many +others: most Ether is staked, with the majority used for settlement of +L2 rollups. We'll discuss it more next time. Given all the above, we advise caution. Intervening to reduce the issuance yield curve seems quite capable of exacerbating the very @@ -457,7 +457,7 @@ we mean to separate concerns. Unpleasantly high inflation in the medium term, even if that "medium term" lasts decades, is a *dynamics* problem, not an equilibrium problem, and so dynamical solutions (like EIP 7514) seem better -suited. Unfortunately we will see that given the above, $$s\to1$$ +suited. Unfortunately we will see that given the above, $$s^\star\to1$$ *is* an equilibrium problem. ### Staking Fraction @@ -749,7 +749,7 @@ some considerations: ### Runaway $$r$$ from Inflation pressure Could the sensitivity $$r_\alpha$$ be sufficient such that even at -intermediate timescales we see $$s\to1$$? This is certainly possible; +intermediate timescales we see $$s^\star\to1$$? This is certainly possible; per the arguments of Ethereum researchers, high inflation could still lead to runaway staking if $$r$$ is sensitive enough. @@ -807,7 +807,7 @@ We saw above a few things: 1. Reinvestment $$r$$ is a lower-bound for $$s^\star$$. 1. Low (but positive) inflation moderates staking fraction closer to this lower bound at intermediate timescales -1. Positive inflation cannot maintain indefinitely, so eventually $$s\to1$$. +1. Positive inflation cannot maintain indefinitely, so eventually $$s^]star\to1$$. Conceptually, how can inflation *moderate* staking fraction, though? Shouldn't more staking lead to more issuance, which leads to more From feab7630ec1c6ccfd72dabe3b722af24648c2b6d Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Wed, 15 Jan 2025 21:46:43 -0600 Subject: [PATCH 58/70] mod code --- _posts/2025-01-15-issuance-dynamics.md | 18 ++++++++---------- 1 file changed, 8 insertions(+), 10 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 3d9273e..190992c 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -292,11 +292,11 @@ with $A_{now}\approx120e6,\ s_{now}\approx.3$ as a start for the models that follow. ```python -from numpy import sqrt from ethode import * - @dataclass class ConstParams(Params): + init_conds: ETH_Data = (('S', 120e6 * .3), ('U', 120e6 * .7)) + tspan: tuple[Yr, Yr] = (0, 100) y1: 1/Yr = 166.3 b: One = 5e-1 f: 1/Yr = 8e-3 @@ -306,7 +306,7 @@ class ConstParams(Params): qu: 1/Yr = 1e-4 s1: ETH = 1 def yld(self, S:ETH, **kwargs) -> 1/Yr: - return self.y1 * sqrt(self.s1 / S) + return self.y1 * np.sqrt(self.s1 / S) ``` ### $$(S,U)$$; not just a UNIX Command! @@ -334,18 +334,16 @@ partial derivatives.[^SU] ```python @dataclass class SUSimConst(ODESim): - ic: tuple[ETH, ETH] = (120e6 * .3, 120e6 * .7) - tspan: tuple[Yr, Yr] = (0, 100) - params: Params = ConstParams() + params: Params = field(default_factory = ConstParams) @staticmethod def func(t:Yr, v:tuple[ETH, ETH], p:Params) -> tuple[ETH/Yr, ETH/Yr]: - S, U = v - dS = (p.r * (y := p.yld(S)) - p.j - p.qu) * S + \ + S, U = v + dS = (p.r * (y := p.yld(S)) - p.j - p.qu) * S + \ ((rf := p.r * p.f) * (1 - p.b) + p.qs) * U dU = ((1 - p.r) * y + p.qu) * S - \ (rf + (1 - p.r) * p.b * p.f + p.qs) * U - return dS, dU -su = SUSim() + return dS, dU +su = SUSimConst() su.sim() ``` From fc6ffc61d6701157d070fa24a8db3ab6a516d135 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 03:30:37 -0600 Subject: [PATCH 59/70] final commit --- _posts/2025-01-15-issuance-dynamics.md | 35 +++++++++++++------------- 1 file changed, 18 insertions(+), 17 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index c58661d..0dc1962 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -295,8 +295,6 @@ models that follow. from ethode import * @dataclass class ConstParams(Params): - init_conds: ETH_Data = (('S', 120e6 * .3), ('U', 120e6 * .7)) - tspan: tuple[Yr, Yr] = (0, 100) y1: 1/Yr = 166.3 b: One = 5e-1 f: 1/Yr = 8e-3 @@ -333,8 +331,12 @@ partial derivatives.[^SU] ```python @dataclass -class SUSimConst(ODESim): - params: Params = field(default_factory = ConstParams) +class SUConstParams(ConstParams): + init_conds: ETH_Data = (('S', 120e6 * .3), ('U', 120e6 * .7)) + tspan: tuple[Yr, Yr] = (0, 100) +@dataclass +class SUConstSim(ODESim): + params: Params = field(default_factory = SUConstParams) @staticmethod def func(t:Yr, v:tuple[ETH, ETH], p:Params) -> tuple[ETH/Yr, ETH/Yr]: S, U = v @@ -343,7 +345,7 @@ class SUSimConst(ODESim): dU = ((1 - p.r) * y + p.qu) * S - \ (rf + (1 - p.r) * p.b * p.f + p.qs) * U return dS, dU -su = SUSimConst() +su = SUConstSim() su.sim() ``` @@ -376,7 +378,7 @@ You can explore this by adding `alpha(), sfrac()` as `@output` methods ```python @dataclass -class SUaSim(SUSimConst): +class SUaConstSim(SUConstSim): @output def sfrac(self, S:ETH, U:ETH) -> One: return S / (S + U) @@ -384,7 +386,7 @@ class SUaSim(SUSimConst): def alpha(self, S:ETH, U:ETH) -> 1/Yr: s = self.sfrac(S,U) return p.yld(S) * s - p.b * p.f * (1 - s) - p.j * s -su_a = SUaSim() +su_a = SUaConstSim() su_a.sim() ``` @@ -431,13 +433,13 @@ crash, in which 99\% of present-day Ether will have been burned. ```python @dataclass -class MegaBurnParams(ConstParams): +class MegaBurnParams(SUConstParams): + init_conds: ETH_Data = (('S', 120e6 * .4), ('U', 120e6 * .6)) + tspan: tuple[Yr, Yr] = (0, 200) b: One = 1e-3 qs: 1/Yr = 2e-1 @dataclass -class MegaBurnSim(SUaSim): - ic: tuple[ETH, ETH] = (1.20e6 * .4, 1.20e6 * .6) - tspan: tuple[Yr, Yr] = (0, 200) +class MegaBurnSim(SUaConstSim): params: Params = MegaBurnParams() zomg = MegaBurnSim() zomg.sim() @@ -447,16 +449,15 @@ zomg.sim() ../assetsPosts/2025-01-15-issuance-dynamics/mega-burn.png) We aren't excited to hodl through multiple decades of 10\% inflation, -and we expect you aren't either! Silliness aside, we do not mean to -downplay fears of inflation, and we encourage you to find more -realistic scenarios in which such sustained inflation occurs. Rather, -we mean to separate concerns. +and we expect you aren't either! Silliness aside, we encourage you to find more +realistic scenarios in which such sustained inflation occurs. +We mean to separate concerns, not dismiss inflation as a problem. Unpleasantly high inflation in the medium term, even if that "medium term" lasts decades, is a *dynamics* problem, not an equilibrium problem, and so dynamical solutions (like EIP 7514) seem better -suited. Unfortunately we will see that given the above, $$s^\star\to1$$ -*is* an equilibrium problem. +suited. Unfortunately we will see that given the above, +$$s^\star\to1$$ *is* an equilibrium problem. ### Staking Fraction From d1f3b388c6d2aafdd52b43b2da672c1363d982cc Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 03:33:57 -0600 Subject: [PATCH 60/70] Daniele --- _posts/2025-01-15-issuance-dynamics.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 0dc1962..1bccf43 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -5,7 +5,7 @@ author: Eric Downes categories: ["ethereum", "macroeconomics", "dynamics", "issuance"] excerpt: Work on ETH Macro supported by the Ethereum Foundation usemathjax: true -thanks: We are deeply grateful to the Ethereum Foundation for their support of this work. I, the author, feel indebted to the profound patience and forbearance of Eric Siu and my colleagues at 20 Squares, as this is *months* overdo. This work has benefited from conversations with Danieli and Philipp, as well as Eric Siu, Andrew Sudbury, Ansgar Dietrichs, and Mac Budkowski. +thanks: We are deeply grateful to the Ethereum Foundation for their support of this work. I, the author, feel indebted to the profound patience and forbearance of Eric Siu and my colleagues at 20 Squares, as this is *months* overdo. This work has benefited from conversations with Daniele and Philipp, as well as Eric Siu, Andrew Sudbury, Ansgar Dietrichs, and Mac Budkowski. --- # Ethereum Macroeconomics From 561a517ec924ba1eda3fc389e5d57bcff00c281c Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 04:17:24 -0600 Subject: [PATCH 61/70] hopefully final commit for reals --- _posts/2025-01-15-issuance-dynamics.md | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 1bccf43..6f32772 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -378,14 +378,17 @@ You can explore this by adding `alpha(), sfrac()` as `@output` methods ```python @dataclass -class SUaConstSim(SUConstSim): +class SUaConstParams(SUConstParams): @output def sfrac(self, S:ETH, U:ETH) -> One: return S / (S + U) @output def alpha(self, S:ETH, U:ETH) -> 1/Yr: s = self.sfrac(S,U) - return p.yld(S) * s - p.b * p.f * (1 - s) - p.j * s + return self.yld(S) * s - self.b * self.f * (1 - s) - self.j * s +@dataclass +class SUaConstSim(SUConstSim): + params: Params = field(default_factory = SUaConstParams) su_a = SUaConstSim() su_a.sim() ``` @@ -433,14 +436,14 @@ crash, in which 99\% of present-day Ether will have been burned. ```python @dataclass -class MegaBurnParams(SUConstParams): +class MegaBurnParams(SUaConstParams): init_conds: ETH_Data = (('S', 120e6 * .4), ('U', 120e6 * .6)) tspan: tuple[Yr, Yr] = (0, 200) b: One = 1e-3 qs: 1/Yr = 2e-1 @dataclass class MegaBurnSim(SUaConstSim): - params: Params = MegaBurnParams() + params: Params = field(default_factory = MegaBurnParams) zomg = MegaBurnSim() zomg.sim() ``` From cf542fceb309e4c9c9c8a01eb58130c64dcf3805 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 04:20:20 -0600 Subject: [PATCH 62/70] valid latex not compiling whyyyyy --- _posts/2025-01-15-issuance-dynamics.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 6f32772..e9bffd2 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -139,7 +139,7 @@ units \[ETH/yr\]. By averaging over "long" timescales (at least quarterly)[^aves] we approximate the staking and unstaking queues as equilibrated; -$$\mathcal{Q}_+ \approx 0 \approx \mathcal{Q}_-$$, and average over many +$\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$, and average over many cycles of the erratic base fee oscillations. So, our conceptual model: From 617c3ba6531073647773f6a494414c3263c1e922 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 04:31:15 -0600 Subject: [PATCH 63/70] j -> jmath in text --- _posts/2025-01-15-issuance-dynamics.md | 23 ++++++++++++----------- 1 file changed, 12 insertions(+), 11 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index e9bffd2..3b9064b 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -138,9 +138,10 @@ from their derivatives. Here all flows are positive real numbers with units \[ETH/yr\]. By averaging over "long" timescales (at least quarterly)[^aves] we -approximate the staking and unstaking queues as equilibrated; -$\mathcal{Q}_+\approx0\approx\mathcal{Q}_-$, and average over many -cycles of the erratic base fee oscillations. +approximate the staking and unstaking queues as equilibrated, and +average over many cycles of the +[erratic](https://decentralizedthoughts.github.io/2022-03-10-eip1559/) +base fee oscillations. So, our conceptual model: @@ -259,7 +260,7 @@ considered here, but please, prove us wrong! Flows obey inequalities, usually as a fraction of the source, except for $r,b$. We convert these inequalities; for each uppercase *extensive* flow $$(J,F,B,\ldots)$$ we define a lowercase *intensive -variable* $$(j,f,b,\ldots)$$: the fractions \[1\] and fractional rates +variable* $$(\jmath,f,b,\ldots)$$: the fractions \[1\] and fractional rates \[1/yr\]. In forming these, the ideal is to apply the tightest available bounds that still capture the asymptotic behavior[^asym] in the limit of interest $$S\to A$$. We do not assume the intensive @@ -276,7 +277,7 @@ $$b(A,S,t)=B/F$$.[^time] | Base Fees[^aves] | $$B$$ | $$U\to\cancel{O}$$ | .. | $$b:=B/F$$ | $$b\in(0,1)$$ \[1\] | | Priority Fees | $$P$$ | $$U\to V$$ | .. | $$1-b=P/F$$ | $$1-b\in(0,1)$$ \[1\] | | Issuance[^aves] | $$I$$ | $$\cdot \to V$$ | $$0 Date: Thu, 16 Jan 2025 04:39:41 -0600 Subject: [PATCH 64/70] minor fixed, should be it --- _posts/2025-01-15-issuance-dynamics.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 3b9064b..7332d2a 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -533,7 +533,7 @@ $$y':=\frac{dy}{dS}$$. issuance implies $$y'<0$$; under the current yield curve the term $$sA|y'|=\frac{1}{2}y_0(sA)$$. -* $$\chi:=-j\math_ts-\beta_t(1-s)$$ represents externalities affecting the +* $$\chi:=-\jmath_ts-\beta_t(1-s)$$ represents externalities affecting the inflation loss term encoded as explicit time-dependencies. We neglect externalities $$\chi\approx0$$ because we have nothing intelligent to say about them, but you might not want to. @@ -972,7 +972,7 @@ sophistication if you decide you want to understand this stuff: - Bender and Orszag (1997) [Advanced Mathematical Methods for Scientists and Engineers](https://www.amazon.com/Advanced-Mathematical-Methods-Scientists-Engineers/dp/0387989315/) -- Arnol\’d (Ed.) the Dynamical Systems Series +- Arnol’d (Ed.) the Dynamical Systems Series - esp. V (1994) [Bifurcation and Catastrophe](https://www.amazon.com/Dynamical-Systems-Bifurcation-Encyclopaedia-Mathematical/dp/0387181733/) From b12dc1874258a2ef263ebb81a1c4b387ae32aee8 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 05:18:32 -0600 Subject: [PATCH 65/70] added table at end based on feedback --- _posts/2025-01-15-issuance-dynamics.md | 62 ++++++++++++++++++-------- 1 file changed, 44 insertions(+), 18 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 7332d2a..b3e040f 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -807,10 +807,11 @@ but its worth being aware of. We saw above a few things: -1. Reinvestment $$r$$ is a lower-bound for $$s^\star$$. +1. Reinvestment $$r$$ is a lower-bound for the staking fraction fixed point $$s^\star$$ + unless Ether is deflating, when likely $s\to1$ or the dynamics are unstable. 1. Low (but positive) inflation moderates staking fraction closer to this lower bound at intermediate timescales -1. Positive inflation cannot maintain indefinitely, so eventually $$s^]star\to1$$. +1. Positive inflation cannot maintain indefinitely, so eventually $$s^\star\to1$$. Conceptually, how can inflation *moderate* staking fraction, though? Shouldn't more staking lead to more issuance, which leads to more @@ -857,15 +858,15 @@ So, while we could certainly model reinvestment differently, and there are lags we are blithely integrating over, we think that these market forces will still act as described above in a different model. It is possible that even during sustained inflation, these effects will be -unable to prevent the upward creep in $$s$$, because $$r$$ is too large, -or the sensitivity of $$r$$ to inflation at equilibrium is too great, a -condition which we mathematized above. In fact we expect that every -argument about inflation effects driving increased staking, overpaying -for security, etc. could (perhaps should) be rephrased in terms of -reinvestment of staking rewards. All these critically depend on the -preferences of ETH users for, and thus their behavior in reaction to, -inflation rate etc., which thusfar are not measured. We encourage the -community to rectify this! +unable to prevent the upward creep in $$s$$, because $$r$$ is too +large, or the sensitivity of $$r$$ to inflation at equilibrium is too +great, a condition which we mathematized above. In fact we expect +that every argument about inflation effects driving increased staking, +overpaying for security, etc. could (perhaps should) be rephrased in +terms of reinvestment of staking rewards. All these critically depend +on the preferences of ETH users for, and thus their behavior in +reaction to, inflation rate etc., which thusfar are not measured, as +far as we know. We encourage the community to rectify this! ## Can reflexivity prevent $$s\to1$$? @@ -875,7 +876,8 @@ capable of such behavior under different parameters or when coupled to price. Why such negligence? If cycles *do* arise, we expect market participants, anticipating such cycles, would act to profit off of these cycles in a way that should reduce them. Buy late in the -inflation cycle, sell, late in the deflation cycle, etc. +inflation cycle, sell, late in the deflation cycle, etc. This would show +up in our model via the partial derivatives including externalities. Notably though, we only expect this to happen because it does not require the coordination of market participants: each @@ -976,13 +978,37 @@ sophistication if you decide you want to understand this stuff: - esp. V (1994) [Bifurcation and Catastrophe](https://www.amazon.com/Dynamical-Systems-Bifurcation-Encyclopaedia-Mathematical/dp/0387181733/) +## Glossary of Things What Have Dots or Dot-Adjacent Shapes + +Unfortunately the fonts used in markdown on the blog are not the +greatest at rendering nicely for some of the chosen syntax, especially +on certain monitors/browsers. Due to feedback from people with +bleeding eyes, we can at least offefr this table. We also included, +or tried to include, the common variables not present in other tables. + +| :-- | :-- | +| Symbol | Description | +| $:=$ | Equality by definition, as opposed to a result which is $=$ | +| $S$ | Staked Ether, | +| $U$ | Unstaked Ether | +| $A:=S+U$ | Circulating/Accessible Supply of Ether | +| $\cancel{O}$ | Burned Ether | +| $\alpha:=\dot{A}/A$ | Inflation | +| $y$ | $\tau$-averaged Issuance yield | +| $y':=dy/dS$ | "y prime" the derivative of the issuance yield curve | +| $\dot{X}:=dX/dt$ | Change in time of $X$, meant as a generic | +| $X^\star$ | "X star" a fixed point where $\dot{X}=0$ | +| $X^\bullet$ | "X spot" the value of $X$ at a given block; anything without a spot is averaged over $\tau$, at least quarterly | +| $(\ldots)^\star$ | anything including derivatives etc. evaluated at the fixed point | + + # Footnotes -[^humor]: [Open Zeppelin]() is an early icon of smart contract best -practices, and continues to provide templates and auditing services in -high demand. They have absolutely no connection to this post, our -models, etc. and hopefully they will not sue us for using their name -in a bad dynamical systems joke. +[^humor]: [Open Zeppelin](https://www.openzeppelin.com/) is an early +icon of smart contract best practices, and continues to provide +templates and auditing services in high demand. They have absolutely +no connection to this post, our models, etc. and hopefully they will +not sue us for using their name in a bad dynamical systems joke. [^reasons]: For derivations involving differential equations, "D" (used for staking Deposit) and its corresponding intensive "d" are @@ -1012,7 +1038,7 @@ Se also our [section](bounding-issuance) on I<=yS. [^partial]: Sometimes "dot x" = dx/dt is used for the partial derivative of x with time t, which we denote x_t. The full relation -is ```dx = x_t + x_A dA + x_s ds + x_α dα``` in which each partial is +is `dx = x_t + x_A dA + x_s ds + x_α dα` in which each partial is taken holding all the other variables constant, and x_t is used in practice to smuggle in any variability from non-dynamical variables. In principle x_A and x_α are distinct; a quantity can depend on supply From a606a41d041a2c2d922f8302dc8e1c22c2cbe6d3 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 05:21:31 -0600 Subject: [PATCH 66/70] added table at end based on feedback --- _posts/2025-01-15-issuance-dynamics.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index b3e040f..8cfbc46 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -989,7 +989,7 @@ or tried to include, the common variables not present in other tables. | :-- | :-- | | Symbol | Description | | $:=$ | Equality by definition, as opposed to a result which is $=$ | -| $S$ | Staked Ether, | +| $S$ | Staked Ether | | $U$ | Unstaked Ether | | $A:=S+U$ | Circulating/Accessible Supply of Ether | | $\cancel{O}$ | Burned Ether | @@ -998,8 +998,8 @@ or tried to include, the common variables not present in other tables. | $y':=dy/dS$ | "y prime" the derivative of the issuance yield curve | | $\dot{X}:=dX/dt$ | Change in time of $X$, meant as a generic | | $X^\star$ | "X star" a fixed point where $\dot{X}=0$ | -| $X^\bullet$ | "X spot" the value of $X$ at a given block; anything without a spot is averaged over $\tau$, at least quarterly | -| $(\ldots)^\star$ | anything including derivatives etc. evaluated at the fixed point | +| $X^\bullet$ | "X spot" the value of $X$ at a given block; *not* a $\tau$-average | +| $(\ldots)^\star$ | any expression evaluated at the fixed point | # Footnotes From 0c05879ea9285a9f155060db9fe6724554ccedb2 Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 05:22:28 -0600 Subject: [PATCH 67/70] and again with the table --- _posts/2025-01-15-issuance-dynamics.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 8cfbc46..d40d3d5 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -986,8 +986,8 @@ on certain monitors/browsers. Due to feedback from people with bleeding eyes, we can at least offefr this table. We also included, or tried to include, the common variables not present in other tables. -| :-- | :-- | | Symbol | Description | +| :-- | :-- | | $:=$ | Equality by definition, as opposed to a result which is $=$ | | $S$ | Staked Ether | | $U$ | Unstaked Ether | From 733d4c4d02c0c945e6b9145fa0391f02b8ad5b6e Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 05:30:03 -0600 Subject: [PATCH 68/70] final spell check --- _posts/2025-01-15-issuance-dynamics.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index d40d3d5..5afcd9a 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -983,7 +983,7 @@ sophistication if you decide you want to understand this stuff: Unfortunately the fonts used in markdown on the blog are not the greatest at rendering nicely for some of the chosen syntax, especially on certain monitors/browsers. Due to feedback from people with -bleeding eyes, we can at least offefr this table. We also included, +bleeding eyes, we can at least offer this table. We also included, or tried to include, the common variables not present in other tables. | Symbol | Description | @@ -1148,7 +1148,7 @@ model. initial conditions; the partial x_A always holds (alpha,s) constant, but dA = alpha dt and the partials in gamma are j_A and (bf)_A. -[^small-part]: Smallness of sensitivities wrt intensives is not gauranteed. +[^small-part]: Smallness of sensitivities wrt intensives is not guaranteed. Certainly a large magnitude, say (bf)_s>1 cannot maintain for too long; 0 bf is still possible. @@ -1157,13 +1157,13 @@ bf is still possible. returning to a region of the fixed point (think comets) or a contraction map showing the system shrinking to a limit set. We won't rule out global stability, but recommend you look first for locally -stable fixed points. Assesing the stability of equilibrium zero +stable fixed points. Assessing the stability of equilibrium zero inflation in the (A,s,alpha) model is interesting but probably academic. One of the eigenvalues at any fixed point with alpha=0 is zero, so higher-order terms matter (the fixed point is degenerate), and linear-stability analysis is insufficient: we need to care about global stability not just local. A reader imbued with mathematical -athleticism and free time is encouraged to think of a Lyapunovv +athleticism and free time is encouraged to think of a Lyapunov function л(alpha=0) >= 0, and obtain a contraction mapping dл(alpha)/dt <= 0. From 926aa0016e4b71bcfdfab2f35ad21546b4e6985b Mon Sep 17 00:00:00 2001 From: Eric Downes Date: Thu, 16 Jan 2025 06:01:29 -0600 Subject: [PATCH 69/70] fixed i.c. for megaburn --- _posts/2025-01-15-issuance-dynamics.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index 5afcd9a..a9229f9 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -438,7 +438,7 @@ crash, in which 99\% of present-day Ether will have been burned. ```python @dataclass class MegaBurnParams(SUaConstParams): - init_conds: ETH_Data = (('S', 120e6 * .4), ('U', 120e6 * .6)) + init_conds: ETH_Data = (('S', 1.2e6 * .4), ('U', 1.2e6 * .6)) tspan: tuple[Yr, Yr] = (0, 200) b: One = 1e-3 qs: 1/Yr = 2e-1 From 16163b720c1fa43e232c9b10a2e2d068f89c4293 Mon Sep 17 00:00:00 2001 From: Daniele Date: Fri, 17 Jan 2025 11:59:13 +0100 Subject: [PATCH 70/70] fixed typographical mishaps --- _posts/2025-01-15-issuance-dynamics.md | 213 +++++-------------------- 1 file changed, 43 insertions(+), 170 deletions(-) diff --git a/_posts/2025-01-15-issuance-dynamics.md b/_posts/2025-01-15-issuance-dynamics.md index a9229f9..33ba951 100644 --- a/_posts/2025-01-15-issuance-dynamics.md +++ b/_posts/2025-01-15-issuance-dynamics.md @@ -127,7 +127,7 @@ size of each is measured by *stocks* - ($$S$$)taked Ether (participating in consensus) is a compartment, as is - ($$U$$)nstaked unburnt Ether, -- containing the ($$V$$)alidator reward queue. -- ($$\mathrm{\{O}}$$) is all irrecoverable (burned, lost, etc.) Ether, and +- ($$\cancel{O}$$) is all irrecoverable (burned, lost, etc.) Ether, and - ($$A$$)ccessible/Circul($$A$$)ting Ether supply, $$A=S+U\approx120.4\times10^6$$ in Dec 2024. - $$\mathcal{Q}_\pm$$ the Ether in the staking ($$+$$) and unstaking ($$-$$) queues @@ -978,7 +978,7 @@ sophistication if you decide you want to understand this stuff: - esp. V (1994) [Bifurcation and Catastrophe](https://www.amazon.com/Dynamical-Systems-Bifurcation-Encyclopaedia-Mathematical/dp/0387181733/) -## Glossary of Things What Have Dots or Dot-Adjacent Shapes +# Glossary of Things That Have Dots or Dot-Adjacent Shapes Unfortunately the fonts used in markdown on the blog are not the greatest at rendering nicely for some of the chosen syntax, especially @@ -1004,173 +1004,46 @@ or tried to include, the common variables not present in other tables. # Footnotes -[^humor]: [Open Zeppelin](https://www.openzeppelin.com/) is an early -icon of smart contract best practices, and continues to provide -templates and auditing services in high demand. They have absolutely -no connection to this post, our models, etc. and hopefully they will -not sue us for using their name in a bad dynamical systems joke. - -[^reasons]: For derivations involving differential equations, "D" -(used for staking Deposit) and its corresponding intensive "d" are -cursed variables. "s" was already in use in some places for staking -fraction, and we are resolute on keeping the intensive and its -corresponding extensive the same letter. "C" is a more natural choice -for circulating supply, but then the three variables of most interest -are something like (C,s,ς) which is masochistic in its sibilance, -even for squares. We prefer "accessible" to circulating because the -former implies you *could* access it, at some cost, while the latter -sometimes implies a velocity of money. A velocity which S and much of -U may lack depending on dynamics: backed up unstaking queue, leveraged -or looped CDPs, etc. But even if our terminology were actually -superior, we're not going to change economic jargon any time soon. - -[^aves]: We use moving quarterly averages, though any timescale τ -sufficiently long that the -[erratic](https://decentralizedthoughts.github.io/2022-03-10-eip1559/) -and [fast dynamics](https://arxiv.org/pdf/2102.10567) of [the base -fee](https://ethereum.github.io/abm1559/notebooks/eip1559.html) are -integrated out, and the lags from (un)staking queues are not -appreciable. As we are averaging quarterly, we set the staking, -unstaking, and reward queues to zero, including their respective flows -(R+Q+,Q-,I+P) in their codomain stocks (S,U,V); even if ethereum -produces empty blocks, so long as the reward queue is not empty U > 0. -Se also our [section](bounding-issuance) on I<=yS. - -[^partial]: Sometimes "dot x" = dx/dt is used for the partial -derivative of x with time t, which we denote x_t. The full relation -is `dx = x_t + x_A dA + x_s ds + x_α dα` in which each partial is -taken holding all the other variables constant, and x_t is used in -practice to smuggle in any variability from non-dynamical variables. -In principle x_A and x_α are distinct; a quantity can depend on supply -(how big ETH market cap is compared to BTC, say) and inflation -independently. - -[^yield]: For anyone from finance, this is *not* the same as a [bond -yield curve](https://www.investopedia.com/terms/y/yieldcurve.asp); -there is essentially no necessary lag for validator rewards, accounting -quarterly. - -[^cats]: We use domain/codomain in imprecise analogy with category -theory mainly because we want to reserve "source" for an attractor, as -per dynamical systems. The analogy, while inexact is not -inappropriate. It is routine to implicitly use associativity to -account for fibers of flows through multiple steps; "electricity from -wind/nuclear/gas" even though the electrons are indistinguishable. -Flows such as tx fees U--F-->V,Ø involve a categorical product VxØ in -that the smaller fractional flows U--B-->Ø must factor through it. -Similarly the staking queue V+(U-V)---R+Q+-->S involves a coproduct in -he domain. Whether there is content here beyond "flows are injective -morphisms in Set" is unclear. None of this matters in the least -for Ethereum dynamics, of course. If you're reading it consider this -an easter egg / attempt to detect a living and alert audience. - -[^elowex]: Why not simply choose "B = bA" as was done in a 2020 post -by Elowsson? Obviously if there is no unstaked Ether no one can -afford tx fees. Here s is a *dynamical variable*, so b = B(A-S) = -bf(1-s)A is more appropriate for our model. The function B might do -all kinds of complicated nonsense, but it can never go negative and it -can never exceed U. - -[^whyr]: Intensives expressed as fractions of flows such as R/(I+P), instead -of fractional rates of sources (like J/S or Q-/S) occur when the -source dynamical variable, here V, is assumed to equilibrate -dV/dt=0. Then the outgoing flows R+K must equal the -incoming flows I+P, so we choose R=r(I+P). If onchain data -indicates, say, ~70\% reinvestment of staking rewards into S takes -a lot longer than three months, we would revisit this assumption, though -we do not expect our qualitative results to change re inflation and -staking fraction. - -[^rlst]: A non-zero r=R/(I+P) is built into the smart contract of every -Liquid Staking Provider (LSP). Here, token-holders provide Ether -and receive a redeemable token (LST) that shares some staking rewards -with them. This fraction of rewards r_LST is a lower bound on -the long term behavior r_LST <= equilibrium R(I+P). - -[^rdyn]: Splitting the staking queue into R + Q+ allows us to -somewhat separate short-term *transient* behavior from long-term -dynamics. Speculative investment in staking by venture capitalists -and novice stakers is expected to die down eventually; either they give up or they run -staking like a business where making a profit matters. Every business -that wants to stay in business reinvests some portion of its profits, -so r,R > 0 is what matters in the long run, once most everyone who -wants to stake is staking. - -[^flowflux]: Confusingly the movement of a dynamic variable -toward/away-from a fixed point is often also called a "flow". Terms -in equations like R,J,... could then be called "fluxes". But you're not -confused, right? - -[^time]: We can often use the dependence on t to smuggle in -any forces, like market panics, etc. that we neglected to include as -dynamical variables. If not, we must add a dynamical variable. - -[^asym]: For computer scientists f << g is equivalent to F=O(g) if -you're more familiar with big-O notation. Specifically I << S means -that the limit of I/S as t gets very large is 0. Contrast to I <= yS, -which could just be a matter of coefficients. Asymptotic Notation is -well-explained on -[wikipedia](https://en.wikipedia.org/wiki/Big_O_notation); see the -bottom for Vinogradov. The art of using it to your advantage in -calculations is demonstrated by [Prof. Carl -Bender](https://www.youtube.com/watch?v=LYNOGk3ZjFM). - -[^params]: Variable parameters that are positive fractions cannot -contribute fixed-points themselves, but they can strongly influence -*where* a fixed point is. Example: as s -> 1, if the leading terms -were bf~(1-s) and j~(1-s)**2 this gives increasingly larger -equilibrium A as s goes to 1. - -[^vitalikp]: Our (dlog y)/(dlog S) = 1 - p in the [discouragement -paper](https://github.com/20squares/ethode/blob/master/guide/guide.md) - -[^noburn]: Regarding B = bf(1-s)A the burn. While slashing could -believably go to zero on quarterly timescales, no burn B=0 implies -blocks are empty. Obviously s=1, B=0 isn't really a -functioning state for Ethereum. A better asymptotic limit would be s -= 1-ε making A ~ (1/ε)**2 (that's squared... so very big as ε is very -small). Detailed treatment of the burn, staking queues using -expansions in ε would be useful, and we would need to model churn, -slashing, and burn in light of stochasticity/quantization. One can -use difference equations, but a useful generic behavior of such -systems (a "weak coupling" limit) is that such perturbations move the -dynamics away from the fixed point, apparently randomly. Yet! -Somehow, the average rate of precession about s=1-ε is often given by -the imaginary component of the largest eigenvalue of the simpler -model. - -[^mu]: If anything the fractional rates of slashing and burn are - positive with small changes in inflation, due to either a single ETH - potentially being of less real value, or stimulation of economic - activity attracting more validators and higher average burn. - -[^ics]: gamma γ expresses the sensitivity of inflation to supply - initial conditions; the partial x_A always holds (alpha,s) constant, - but dA = alpha dt and the partials in gamma are j_A and (bf)_A. - -[^small-part]: Smallness of sensitivities wrt intensives is not guaranteed. -Certainly a large magnitude, say (bf)_s>1 cannot maintain for too -long; 0 -bf is still possible. - -[^stability]: Global stability involves either trajectories infinitely -returning to a region of the fixed point (think comets) or a -contraction map showing the system shrinking to a limit set. We won't -rule out global stability, but recommend you look first for locally -stable fixed points. Assessing the stability of equilibrium zero -inflation in the (A,s,alpha) model is interesting but probably -academic. One of the eigenvalues at any fixed point with alpha=0 is -zero, so higher-order terms matter (the fixed point is degenerate), -and linear-stability analysis is insufficient: we need to care about -global stability not just local. A reader imbued with mathematical -athleticism and free time is encouraged to think of a Lyapunov -function л(alpha=0) >= 0, and obtain a contraction -mapping dл(alpha)/dt <= 0. - -[^SU]: Readers wishing for more detail are encouraged to use the two -dimensional local stability criterion (see -[Strogatz](https://www.youtube.com/watch?v=9yh9DmNqdk4)) to solve for -the condition of eigenvalues with an imaginary part. But simulate it -too! +[^humor]: [Open Zeppelin](https://www.openzeppelin.com/) is an early icon of smart contract best practices, and continues to provide templates and auditing services in high demand. They have absolutely no connection to this post, our models, etc. and hopefully they will not sue us for using their name in a bad dynamical systems joke. + +[^reasons]: For derivations involving differential equations, "D" (used for staking Deposit) and its corresponding intensive "d" are cursed variables. "s" was already in use in some places for staking fraction, and we are resolute on keeping the intensive and its corresponding extensive the same letter. "C" is a more natural choice for circulating supply, but then the three variables of most interest are something like (C,s,ς) which is masochistic in its sibilance, even for squares. We prefer "accessible" to circulating because the former implies you *could* access it, at some cost, while the latter sometimes implies a velocity of money. A velocity which S and much of U may lack depending on dynamics: backed up unstaking queue, leveraged or looped CDPs, etc. But even if our terminology were actually superior, we're not going to change economic jargon any time soon. + +[^aves]: We use moving quarterly averages, though any timescale τ sufficiently long that the [erratic](https://decentralizedthoughts.github.io/2022-03-10-eip1559/) and [fast dynamics](https://arxiv.org/pdf/2102.10567) of [the base fee](https://ethereum.github.io/abm1559/notebooks/eip1559.html) are integrated out, and the lags from (un)staking queues are not appreciable. As we are averaging quarterly, we set the staking, unstaking, and reward queues to zero, including their respective flows (R+Q+,Q-,I+P) in their codomain stocks (S,U,V); even if ethereum produces empty blocks, so long as the reward queue is not empty U > 0. See also our [section](bounding-issuance) on I<=yS. + +[^partial]: Sometimes "dot x" = dx/dt is used for the partial derivative of x with time t, which we denote x_t. The full relation is `dx = x_t + x_A dA + x_s ds + x_α dα` in which each partial is taken holding all the other variables constant, and x_t is used in practice to smuggle in any variability from non-dynamical variables. In principle x_A and x_α are distinct; a quantity can depend on supply (how big ETH market cap is compared to BTC, say) and inflation independently. + +[^yield]: For anyone from finance, this is *not* the same as a [bond yield curve](https://www.investopedia.com/terms/y/yieldcurve.asp); there is essentially no necessary lag for validator rewards, accounting quarterly. + +[^cats]: We use domain/codomain in imprecise analogy with category theory mainly because we want to reserve "source" for an attractor, as per dynamical systems. The analogy, while inexact is not inappropriate. It is routine to implicitly use associativity to account for fibers of flows through multiple steps; "electricity from wind/nuclear/gas" even though the electrons are indistinguishable. Flows such as tx fees U--F-->V,Ø involve a categorical product VxØ in that the smaller fractional flows U--B-->Ø must factor through it. Similarly the staking queue V+(U-V)---R+Q+-->S involves a coproduct in he domain. Whether there is content here beyond "flows are injective morphisms in Set" is unclear. None of this matters in the least for Ethereum dynamics, of course. If you're reading it consider this an easter egg / attempt to detect a living and alert audience. + +[^elowex]: Why not simply choose "B = bA" as was done in a 2020 post by Elowsson? Obviously if there is no unstaked Ether no one can afford tx fees. Here s is a *dynamical variable*, so b = B(A-S) = bf(1-s)A is more appropriate for our model. The function B might do all kinds of complicated nonsense, but it can never go negative and it can never exceed U. + +[^whyr]: Intensives expressed as fractions of flows such as R/(I+P), instead of fractional rates of sources (like J/S or Q-/S) occur when the source dynamical variable, here V, is assumed to equilibrate dV/dt=0. Then the outgoing flows R+K must equal the incoming flows I+P, so we choose R=r(I+P). If onchain data indicates, say, ~70\% reinvestment of staking rewards into S takes a lot longer than three months, we would revisit this assumption, though we do not expect our qualitative results to change re inflation and staking fraction. + +[^rlst]: A non-zero r=R/(I+P) is built into the smart contract of every Liquid Staking Provider (LSP). Here, token-holders provide Ether and receive a redeemable token (LST) that shares some staking rewards with them. This fraction of rewards r_LST is a lower bound on the long term behavior r_LST <= equilibrium R(I+P). + +[^rdyn]: Splitting the staking queue into R + Q+ allows us to somewhat separate short-term *transient* behavior from long-term dynamics. Speculative investment in staking by venture capitalists and novice stakers is expected to die down eventually; either they give up or they run staking like a business where making a profit matters. Every business that wants to stay in business reinvests some portion of its profits, so r,R > 0 is what matters in the long run, once most everyone who wants to stake is staking. + +[^flowflux]: Confusingly the movement of a dynamic variable toward/away-from a fixed point is often also called a "flow". Terms in equations like R,J,... could then be called "fluxes". But you're not confused, right? + +[^time]: We can often use the dependence on t to smuggle in any forces, like market panics, etc. that we neglected to include as dynamical variables. If not, we must add a dynamical variable. + +[^asym]: For computer scientists f << g is equivalent to F=O(g) if you're more familiar with big-O notation. Specifically I << S means that the limit of I/S as t gets very large is 0. Contrast to I <= yS, which could just be a matter of coefficients. Asymptotic Notation is well-explained on [wikipedia](https://en.wikipedia.org/wiki/Big_O_notation); see the bottom for Vinogradov. The art of using it to your advantage in calculations is demonstrated by [Prof. Carl Bender](https://www.youtube.com/watch?v=LYNOGk3ZjFM). + +[^params]: Variable parameters that are positive fractions cannot contribute fixed-points themselves, but they can strongly influence *where* a fixed point is. Example: as s -> 1, if the leading terms were bf~(1-s) and j~(1-s)**2 this gives increasingly larger equilibrium A as s goes to 1. + +[^vitalikp]: Our (dlog y)/(dlog S) = 1 - p in the [discouragement paper](https://github.com/20squares/ethode/blob/master/guide/guide.md) + +[^noburn]: Regarding B = bf(1-s)A the burn. While slashing could believably go to zero on quarterly timescales, no burn B=0 implies blocks are empty. Obviously s=1, B=0 isn't really a functioning state for Ethereum. A better asymptotic limit would be s = 1-ε making A ~ (1/ε)**2 (that's squared... so very big as ε is very small). Detailed treatment of the burn, staking queues using expansions in ε would be useful, and we would need to model churn, slashing, and burn in light of stochasticity/quantization. One can use difference equations, but a useful generic behavior of such systems (a "weak coupling" limit) is that such perturbations move the dynamics away from the fixed point, apparently randomly. Yet! Somehow, the average rate of precession about s=1-ε is often given by the imaginary component of the largest eigenvalue of the simpler model. + +[^mu]: If anything the fractional rates of slashing and burn are positive with small changes in inflation, due to either a single ETH potentially being of less real value, or stimulation of economic activity attracting more validators and higher average burn. + +[^ics]: gamma γ expresses the sensitivity of inflation to supply initial conditions; the partial x_A always holds (alpha,s) constant, but dA = alpha dt and the partials in gamma are j_A and (bf)_A. + +[^small-part]: Smallness of sensitivities wrt intensives is not guaranteed. Certainly a large magnitude, say (bf)_s>1 cannot maintain for too long; 0 bf is still possible. + +[^stability]: Global stability involves either trajectories infinitely returning to a region of the fixed point (think comets) or a contraction map showing the system shrinking to a limit set. We won't rule out global stability, but recommend you look first for locally stable fixed points. Assessing the stability of equilibrium zero inflation in the (A,s,alpha) model is interesting but probably academic. One of the eigenvalues at any fixed point with alpha=0 is zero, so higher-order terms matter (the fixed point is degenerate), and linear-stability analysis is insufficient: we need to care about global stability not just local. A reader imbued with mathematical athleticism and free time is encouraged to think of a Lyapunov function л(alpha=0) >= 0, and obtain a contraction mapping dл(alpha)/dt <= 0. + +[^SU]: Readers wishing for more detail are encouraged to use the two dimensional local stability criterion (see [Strogatz](https://www.youtube.com/watch?v=9yh9DmNqdk4)) to solve for the condition of eigenvalues with an imaginary part. But simulate it too!