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# Issuance Issues — Tertiary Treatise<br><p class="small"><small>*^ i know, i know. another alliteration, an arguably arduous articulation at that – as if anyone actually asked... admonish away; any advice appreciated.*</small></p>

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_1a62d211dbf0a319810db8b50fcdbbd2.png width=87%>

_{***^ p.s. we can only aspire to be both precise and pleasant while measuring pie slices.*** 🥧📐}
$\cdot$
*by [mike](https://twitter.com/mikeneuder) – friday, june 21, 2024.*
_{*^turns out 2024 is the only year in the entire range of 2021-2027 where the [summer solstice](https://en.wikipedia.org/wiki/Summer_solstice) is on june 20$^{th}$ instead of the 21$^{st}$ -- "[missed it by that much](https://www.youtube.com/watch?v=8WbXc6zOjMQ)."*}
$\cdot$
*many thanks to [Nate](https://x.com/nan0gloss) for discussions around the framing of this article.*
$\cdot$
**tl;dr;**
Hello & welcome to the third installment of the <u>Issuance Issues</u>®. The [first issue](https://notes.ethereum.org/@mikeneuder/iiii) highlighted the broad-stroke arguments for why we may end up in a high-stake rate paradigm and the associated negative externalities. The [second issue](https://notes.ethereum.org/@mikeneuder/subsol) focused on real & nominal yield, supply curves, and how a lower issuance may result in higher real yield in equilibrium.

This article deep dives 🔥the burn🔥 and its role in the issuance discourse. One of the most common themes in <span style="color:red">The Great Issuance Debate(c)</span> is confusion around the burn. "***How is issuance too high if the net supply of `ETH` is barely inflationary (or even deflationary)?***" This confused me for a long time, so in this article, we start slowly and use a lot of plots and numerical examples to build intuition. We make our way to the key insight that **higher burn <u>increases</u> the gap between staking and not staking**. Hopefully, this article demonstrates that *regardless of the burn, the issuance remains a redistribution mechanism that shifts network ownership from non-stakers to stakers.* Once we convince ourselves of that, it follows that the burn is mostly orthogonal to the issuance discussion (and marginally *increases* the ownership gap). Let's get it -- "[and here ... we ... go.](https://youtu.be/24cQfQJ79Rs?t=71)"
$\cdot$
**Contents**
(1) [Nominal yield \& inflation refresher](#1-Nominal-yield-amp-inflation-refresher)
(2) [A new dimension – network ownership](#2-A-new-dimension-–-network-ownership)
&nbsp;&nbsp;&nbsp;(2.1) [Unstaked `ETH`, no burn](#21-Unstaked-ETH-no-burn)
&nbsp;&nbsp;&nbsp;(2.2) [Staked `ETH`, no burn](#22-Staked-ETH-no-burn)
&nbsp;&nbsp;&nbsp;(2.3) [Ownership gap, no burn](#23-Ownership-gap-no-burn)
(3) [Bring on the burn](#3-Bring-on-the-burn)
&nbsp;&nbsp;&nbsp;(3.1) [Unstaked `ETH`, burn](#31-Unstaked-ETH-burn)
&nbsp;&nbsp;&nbsp;(3.2) [Staked `ETH`, burn](#32-Staked-ETH-burn)
&nbsp;&nbsp;&nbsp;(3.3) [Ownership gap, burn](#33-Ownership-gap-burn)
(4) [Wrapping up](#4-Wrapping-up)
$\cdot$
The plotting code is [available here](https://github.com/michaelneuder/issuance/blob/main/tertiary-treatise.ipynb).[^1]

[^1]: Buyer beware; we get saucy in the matplotlib (YMMV).[^3]

[^3]: Me 'n matplotlib go waaaaaay back – see, e.g., Figure 4 in this [article](https://arxiv.org/pdf/1806.10936) from six(!) years ago.[^5]

[^5]: Nested footnotes; learning from [the best](https://www.reddit.com/r/davidfosterwallace/comments/t8ijuy/footnotes_for_footnotes/).

**Related work**
| Article | Description|
|---|---|
|[*Minimum Viable Issuance*](https://notes.ethereum.org/@anderselowsson/MinimumViableIssuance) | Anders' first post |
|[*Properties of issuance level (part 1)*](https://notes.ethereum.org/@anderselowsson/HyUIqjo_6) | Anders' second post|
|[*Endgame Staking Economics: A Case for Targeting*](https://ethresear.ch/t/endgame-staking-economics-a-case-for-targeting/18751) | Caspar & Ansgar's post |
|[*Electra: Issuance Curve Adjustment Proposal*](https://ethereum-magicians.org/t/electra-issuance-curve-adjustment-proposal/18825) | Caspar & Ansgar's Electra proposal |
| [*UCC2: Ethereum's Staking Endgame*](https://www.ucc2.xyz/podcast/episode/28ec02cd/ethereums-staking-endgame) | Jon, Hasu, Caspar, & Ansgar's discussion |
| [*Initial Analysis of Stake Distribution*](https://ethresear.ch/t/initial-analysis-of-stake-distribution/19014) | Julian's post |
| [*Minimum Viable Issuance*](https://notes.ethereum.org/@anderselowsson/MinimumViableIssuance) | Anders' third post |
| [*Reward curve with tempered issuance*](https://ethresear.ch/t/reward-curve-with-tempered-issuance-eip-research-post/19171) | Anders' fourth post |
| [*Foundations of MVI*](https://notes.ethereum.org/@anderselowsson/Foundations-of-MV) | Anders' fifth post |
| [*FAQ: Ethereum issuance reduction*](https://ethresear.ch/t/faq-ethereum-issuance-reduction/19675) | Anders' sixth post |

---

## (1) Nominal yield \& inflation refresher

As a quick refresher, recall that the current Ethereum issuance rate is what we referred to as the "inverse-root curve" in the [previous post](https://notes.ethereum.org/@mikeneuder/subsol#1-Inverse-Root-Curve-current-issuance),
$$
\text{issuance} = 2.6 \cdot \frac{64}{\sqrt{\text{stake}}}.
$$

Here, $\text{stake}$ is denominated `ETH`, and the $\text{issuance}$ is the annual yield from staking. For example, let $\text{stake} = 30\text{mm}$ `ETH`, then $\text{issuance} = 0.0304$ meaning the staking yield is $3.04\%$ annually. Using the issuance, we can calculate the annual inflation (in proportional terms) using the simple [relative change](https://en.wikipedia.org/wiki/Relative_change) formula,
$$
\begin{alignat}{2}
\text{inflation} &= \frac{\text{new supply} - \text{old supply}}{\text{old supply}}\\
&= \frac{(\text{old supply} + \text{issuance} \cdot \text{stake}) - \text{old supply}}{\text{old supply}}\\
&= \frac{\text{issuance} \cdot \text{stake}}{ \text{old supply}}.
\end{alignat}
$$

We use the supply of $120\text{mm}$ `ETH`, so we can rewrite the inflation as,
$$
\begin{alignat}{2}
\text{inflation} &= \frac{2.6 \cdot 64 \cdot \sqrt{\text{stake}}}{120,000,000}.
\end{alignat}
$$
For example, let $\text{stake}=30\text{mm}$ `ETH`, then $\text{inflation}=0.0076$ meaning the annual inflation rate is $0.7\%$. The figure below plots the yield (left) and the inflation (right) for various staked `ETH` values. The dotted line shows the example values we calculated on both plots.

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_c3b88c0c9d00299a5ce3d15ec243089e.png width=100%>

The main characteristic of the yield curve (left) is that it grows extremely quickly as $\text{stake} \to 0$; the curve highly incentivizes non-zero stake. On the other end, as $\text{stake} \to 120\text{mm}$, the yield declines increasingly slowly, meaning the curve doesn't meaningfully differentiate between $60\text{mm}$ and $80\text{mm}$ staked `ETH` (because the yields are relatively similar). The inflation curve (right) is monotone increasing, so each marginal new staker increases the overall inflation rate of the system.

The figure below includes example values for $90\text{mm}$ staked `ETH` (dashed lines) for reference.

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_97a9869a9ca0b5c9b22bd99ba7393081.png width=100%>

OK, great. Hopefully, this was mainly a review and pretty easy to follow. Now, to the new stuff.

## (2) A new dimension – network ownership

Before building up to the burn, we first need to examine changes in "network ownership," which we define here as "the portion of the `ETH` supply you own." We start with the non-burn case as it is simpler to understand and helps set the table for incorporating the burn in [Section 3](#3-Bring-on-the-burn). To begin, `ETH` is either staked or unstaked. Using this partition, we can examine the annual *change* in network ownership for each case.

### (2.1) Unstaked `ETH`, no burn
Let's start with unstaked `ETH`. We want to answer, **"If I don't stake my `ETH`, what is the annual change in the percentage of the total supply that I own?"** You could also think of this as "dilution," but the "change in network ownership" framing may be more intuitive (albeit more verbose). Let $\pi_u$ denote the *portion* of the network you own as a non-staker ($_{\text{subscript}}$ $u$ for "unstaked"); we use $\pi$ both for the letter "p" and because slices of pie are a nice mental model for these "proportional ownership" ideas (hence the cover photo of the document :). Then,

$$
\begin{align}
\pi_u &= \frac{\text{new network ownership - old network ownership}}{\text{old network ownership}}.
\end{align}
$$

Again, we use the relative change equation to see how much the network ownership decreases after a year. Note that, by definition, $\pi_u$ must be negative because, without the burn, the total supply is monotone increasing, but the number of non-staked tokens stays fixed, implying $\text{new network ownership} < \text{old network ownership}$. This is the clearest demonstration of **staking as a redistribution mechanism by which unstaked `ETH` pays for security through a reduction in network ownership.** Examining the $\text{new network ownership}$ term, we see this playing out as
$$
\begin{align}
\text{new network ownership} &= \frac{\text{my tokens}}{\text{new supply}} \\
&= \frac{\text{my tokens}}{\text{old supply} + \text{inflation}}.
\end{align}
$$

The $\text{old network ownership}$ is simply
$$
\begin{align}
\text{old network ownership} &= \frac{\text{my tokens}}{\text{old supply}}.
\end{align}
$$

Putting this into algebra, let $u$ be the number of unstaked tokens you own and $T$ be the total supply of `ETH`. Also, let $i(s)$ denote the inflation as a function of the amount of stake, $s$. Using our "word equation" for $\pi_u$ above,

$$
\begin{align}
\pi_u &= \frac{u / (T\cdot(1+ i(s))) - u/T}{u/T} \\
&= \frac{1}{1+i(s)} - 1.
\end{align}
$$

The figure below shows $\pi_u$ for all possible stake values, $s$. We include two reference values:
- $\text{stake}=30\text{mm}$ `ETH` $\implies \pi_u = -0.75\%$
- $\text{stake}=90\text{mm}$ `ETH` $\implies \pi_u = -1.30\%$

These are interpreted as, e.g., "with $30\text{mm}$ staked `ETH`, the percent of the total supply a non-staker owns **decreases** by $0.75\%$ after a year."

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_2451f45cede9ab3fc6119bb3d29b39ea.png width=80%>

Notice that these values are negative because the non-stakers *always* decrease the portion of the supply they own (because the stakers are creating new `ETH` while the non-stakers retain the same number of tokens). Also, the values are monotone decreasing in the amount of stake because the inflation is always increasing, meaning the the total supply is growing faster with more `ETH` staked.

### (2.2) Staked `ETH`, no burn

Moving to staked `ETH`, the situation is much different. Here, the change in my network ownership is, by definition *increasing*. The rate at which it increases depends on both the yield and the overall inflation rate. Let $\pi_s$ denote the annual change in the *portion* of the supply that a staker owns. Using the same relative change formula, we have

$$
\begin{align}
\pi_s &= \frac{\text{new network ownership - old network ownership}}{\text{old network ownership}} \\
\end{align}
$$

Again focusing in on $\text{new network ownership}$, we now have to incorporate the yield from staking into the numerator,
$$
\begin{align}
\text{new network ownership} &= \frac{\text{my tokens} + \text{my yield}}{\text{new supply}} \\
&= \frac{\text{my tokens} + \text{my yield}}{\text{old supply} + \text{inflation}}
\end{align}
$$

Putting this into algebra, let $s$ be the staked tokens and $y(s)$ denote the yield for staking (as defined in [Section 1](#1-Nominal-yield-amp-inflation-refresher)). Then we have
$$
\begin{align}
\pi_s &= \frac{s(1+y(s)) / (T\cdot(1+i(s))) - s/T}{s/T} \\
&= \frac{1+y(s)}{1+i(s)} - 1.
\end{align}
$$

Notice that this value is positive if and only if $y(s) >i(s)$ (yield is greater than inflation), which is the case unless the total supply is staked. If the entire supply is staked, then $y(s) = i(s)$ (because all tokens are earning the exact yield, meaning the overall inflation is the same rate) and $\pi_s=0$. This makes intuitive sense because if everyone is staking, the change in the percentage of the supply ownership is constant (no redistribution is happening).

The figure below shows $\pi_s$ along with $\pi_u$ (from [Section 2.1](#21-Unstaked-ETH-no-burn)) for various staking levels. We include two new reference values (in addition to the $\pi_u$ values from above):
- $\text{stake}=30\text{mm}$ `ETH` $\implies \pi_s = 2.26\%$
- $\text{stake}=90\text{mm}$ `ETH` $\implies \pi_s = 0.43\%$

These are interpreted as, e.g., "with $30\text{mm}$ staked `ETH`, the percent of the total supply a staker owns **increases** by $2.26\%$ after a year."

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_fc74c0e18713f1c4ddeb7257cb9ad9e2.png width=80%>

We see that $\pi_s \to 0$ as $\text{stake} \to 120\text{mm}$. OK, maybe you can see where we are going by now; *we care about the **difference** between the staker and non-staker change in network ownership.*

### (2.3) Ownership gap, no burn
Nifty! The natural next question is, **"What is the difference in the change of supply ownership for staked versus unstaked `ETH`?"** We define this quantity as the "ownership gap"; we just take the difference $\pi_s - \pi_u$. Using the expressions derived in the previous sections, we have

$$
\begin{align}
\pi_s - \pi_u &= \frac{1+y(s)}{1+i(s)} - 1 - \left(\frac{1}{1+i(s)} - 1\right) \\
&= \frac{y(s)}{1+i(s)}.
\end{align}
$$

The figure below helps visualize this. The left plot shows $\pi_s,\pi_u$ with the gap we are measuring filled in with gray; the right plot shows the ownership gap values. Notice that the x-axis now starts at $\text{stake}=20\text{mm}$ `ETH` to help with the scaling.

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_eadf9d1feeda658cfada7936eabe773b.png width=100%>

We can interpret this as, e.g., "at $\text{stake}=40\text{mm}$ `ETH`, the change in the stakers' ownership is about $2.6\%$ higher than the change in the non-stakers' ownership." To accentuate this point, consider the following numerical example.

> **Numerical example (no burn)** – Let the total supply be $1,000,000$ tokens. Consider a non-staker and a staker, each starting with $30$ tokens. The staker earns $1$ token in yield over a year, while the total inflation is $10,000$ tokens.
>
> <u>No burn, non-staker</u> $(\pi_u)$
> $\cdot$ Ownership share before $=30/1,000,000 \; (0.3 \text{ bips}).$
> $\cdot$ Ownership share after $=30/1,010,000 \; (0.297 \text{ bips}).$
> $\cdot$ Ownership change $\pi_u = (0.297 - 0.3)/0.3 = -0.01 \implies 1\%$ decrease.
>
> <u>No burn, staker</u> $(\pi_s)$
> $\cdot$ Ownership share before $=30/1,000,000 \; (0.3 \text{ bips}).$
> $\cdot$ Ownership share after $=31/1,010,000 \; (0.3069 \text{ bips}).$
> $\cdot$ Ownership change $\pi_s = (0.3069 - 0.3)/0.3 = 0.023 \implies 2.3\%$ increase.
>
> <u>No burn, ownership gap</u> $(\pi_s-\pi_u)$
> $\cdot$ Gap $\pi_s - \pi_u = 0.023 - (-0.01) = 0.033 = 3.3\%$ difference.

Right – hope you are hanging in there. We can now incorporate the burn to see how this all fits together!

## (3) Bring on the burn

The burn decreases the total supply of `ETH`, which affects the network ownership calculations because it changes the value of $\text{new supply}$. The critical intuition around the burn is that the supply reduction applies to both stakers and non-stakers and in fact, **a higher burn <u>increases</u> the ownership gap between staking and non-staking.** This is most intuitive when considered in absolute terms. Consider a staker who earns yield for a year. In a no-burn world, their change in network ownership is measured against the original supply plus the total inflation; in a burn world, their change in network ownership is measured against the original supply plus total inflation less the burn. Same for the non-staker; let's start there.

### (3.1) Unstaked `ETH`, burn

The quantity of interest remains the same but incorporates the burn into the derivation of $\text{new network ownership}$. Let $\pi_u^\beta$ denote "the change in percentage of network ownership for a non-staker while including the burn" ($^{\text{superscript}} \beta$ for 'burn'). We continue using our relative change formula,

$$
\begin{align}
\pi_u^{\beta} &= \frac{\text{new network ownership - old network ownership}}{\text{old network ownership}}.
\end{align}
$$

Now the $\text{new network ownership}$ term includes the burn as a factor in calculating the new supply,
$$
\begin{align}
\text{new network ownership} &= \frac{\text{my tokens}}{\text{new supply}} \\
&= \frac{\text{my tokens}}{\text{old supply} + \text{inflation}-\text{burn}}.
\end{align}
$$

The $\text{old network ownership}$ remains as
$$
\begin{align}
\text{old network ownership} &= \frac{\text{my tokens}}{\text{old supply}}.
\end{align}
$$

In algebraic terms, let $b$ denote the number of tokens burned annually. Then,

$$
\begin{align}
\pi^{\beta}_u &= \frac{u / (T\cdot(1+ i(s) -b))- u/T}{u/T} \\
&= \frac{1}{1+i(s)-b} - 1
\end{align}
$$

Notice that this is very similar to our $\pi_u$ calculation, just with the burn subtracted from the denominator. The intuition here is that the burn decreases the denominator, making the overall value of $\pi_u^{\beta}$ *less negative* (e.g., more burn $\implies$ "smaller decrease"[^6] in network ownership). In fact, if $i(s)=b$, then $\pi_u^{\beta}=0$, meaning the non-staker continues owning a constant portion of the supply. Further, if $i(s)<b$ (there is \*more burn\* than inflation[^2]), $\pi_u^{\beta} > 0$, which means the portion of the network owned by the non-staker goes up!

[^2]: This is what is sometimes referred to as "[ultra sound](https://ultrasound.money/)".

[^6]: I don't dislike double negatives. :)

The figure below shows the values of $\pi_u^{\beta}$ for various levels of burn (as indicated by the color bar) and multiple levels of stake. Again, at each burn level, the change in network ownership decreases as the stake increases because the inflation rises. However, with the burn, the values are no longer always negative; anything above $0$ implies an *increase in the network ownership* of a non-staker at that burn and stake level.

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_e7781bf6ea35e1ec58814ae80e64e3dd.png width=80%>

This seems counterintuitive; how can the network ownership of a non-staker go up? The key insight here is that because the burn adjusts the supply for both non-stakers and stakers, **the change in network ownership of the stakers must be going up faster.** There is still a redistribution going on! We must factor the burn into our staker network ownership change calculation to compare apples-to-apples.

### (3.2) Staked `ETH`, burn
Let $\pi^\beta_s$ denote the change in network ownership of a staker while incorporating the burn. Again, we use the familiar relative change formula.

$$
\begin{align}
\pi^\beta_s &= \frac{\text{new network ownership - old network ownership}}{\text{old network ownership}} \\
\end{align}
$$

Just as in [Section 3.1](#31-Unstaked-ETH-burn), the burn shows up when we calculate the $\text{new network ownership}$.

$$
\begin{align}
\text{new network ownership} &= \frac{\text{my tokens} + \text{my yield}}{\text{new supply}} \\
&= \frac{\text{my tokens} + \text{my yield}}{\text{old supply} + \text{inflation} - \text{burn}}.
\end{align}
$$

Putting this into algebraic terms, we have

$$
\begin{align}
\pi^{\beta}_s &= \frac{s(1+y(s)) / (T\cdot(1+ i(s) -b))-s/T}{s/T}\\
&= \frac{1+y(s)}{1+i(s)-b} - 1.
\end{align}
$$

This equation should look familiar by now, as the burn is included in the denominator. The figure below shows these values for various amounts of burn. The dashed and solid lines are the $\pi^{\beta}_s$ and $\pi^{\beta}_u$ values respectively. The black curves represent the no-burn situation – the tan and blue lines in [Section 2.2](#22-Staked-ETH-no-burn).

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_e839278c7a26db7fda4b278aadbf8246.png width=80%>

Again, we turn to the ownership gap, or the difference between the change in network ownership for staked versus unstaked `ETH`. In terms of the figure above, that means taking the differences between the solid and dashed lines that have the same color (same burn level).

### (3.3) Ownership gap, burn
(Sixth and final algebraic thing, soz.) Using the expressions derived in the previous sections, we have the ownership gap with the burn as $\pi^\beta_s - \pi^\beta_u$.

$$
\begin{align}
\pi_s^\beta - \pi_u^\beta &= \frac{1+y(s)}{1+i(s)-b} - 1 - \left(\frac{1}{1+i(s)-b} - 1\right) \\
&= \frac{y(s)}{1+i(s)-b}.
\end{align}
$$

In [Section 2.3](#23-Ownership-gap-no-burn), we calculated the value when the burn was $0\%$ (shown below in black); the figure also includes the $2\%$ burn scenario (in tan). The left plot shows $\pi_s^\beta,\pi_u^\beta$ with the gap we measure filled in with gray and light tan for $0\%$ and $2\%$ burn respectively; the right plot shows the ownership gap values, with the inset axes[^4] zooming in on the $30-40\text{mm}$ staked `ETH` range.

[^4]: Matplotlib makes these inset axes soooo easy – check it [out](https://matplotlib.org/stable/gallery/subplots_axes_and_figures/zoom_inset_axes.html).

<img src=https://storage.googleapis.com/ethereum-hackmd/upload_8e3068e778cc7b4da17de083c6419fc5.png width=100%>

**Key takeaway:** *the ownership gap is greater when you increase the burn because the staking issuance is a relatively larger portion of the total supply.* This is shown on the right with the tan line ($2\%$ burn) being *higher* than the black line ($0\%$ burn). N.b., the difference in the ownership gap between the two is only $\approx0.07\%$ ($7$ bips) – quite small.

Let's include the burn in our calculations by extending our numerical example from [Section 2.3](#23-Ownership-gap-no-burn).

> **Numerical example (with burn)** – Let the total supply be $1,000,000$ tokens. Consider a non-staker and a staker, each starting with $30$ tokens. The staker earns $1$ token in yield over a year. Let the total inflation be $10,000$ tokens, and the total burn be $15,000$ tokens (net decrease of $5,000$ tokens in the supply).
>
> <u>Burn, non-staker</u> $\big(\pi_u^\beta\big)$
> $\cdot$ Ownership share before $=30/1,000,000 \; (0.3 \text{ bips}).$
> $\cdot$ Ownership share after $=30/995,000 \; (0.3015 \text{ bips}).$
> $\cdot$ Ownership change $\pi_u^\beta=(0.3015 - 0.3)/0.3 = 0.005 \implies 0.5\%$ increase.
>
> <u>Burn, staker</u> $\big(\pi_s^\beta\big)$
> $\cdot$ Ownership share before $=30/1,000,000 \; (0.3 \text{ bips}).$
> $\cdot$ Ownership share after $=31/995,000 \; (0.3116 \text{ bips}).$
> $\cdot$ Ownership change $\pi_s^\beta=(0.3116 - 0.3)/0.3 = 0.039 \implies 3.9\%$ increase.
>
>
> <u>Burn, ownership gap</u> $\big(\pi_s^\beta-\pi_u^\beta\big)$
> $\cdot$ Gap $\pi_s^\beta-\pi_u^\beta = 0.039 - 0.005 = 0.034 = 3.4\%$ difference.

Recall that the previous ownership gap (with no burn) was $3.3\%$. With the burn, the ownership gap grows by $0.1\%$ to $3.4\%$. Again, this highlights the main point, which is that the burn actually *increases* the gap between stakers and non-stakers.

## (4) Wrapping up

That was a lot of [n]umbers & [a]lgebra & [p]lots (need a [nap]?), so we'll keep the summary short and sweet.

1. Staking rewards are <u>**always**</u> a redistribution of network ownership from non-stakers to stakers.
2. *With no burn*, the change in network ownership of non-stakers is negative, and the change in network ownership of the stakers is positive.
3. *With burn*, the change in network ownership of non-stakers can be positive or negative (or zero if the burn is exactly inflation); the change in network ownership of the stakers is positive and always larger.
4. The ownership gap, or the difference between the percentage change of network ownership between stakers and non-stakers, is the metric of interest.
5. The ownership gap is increasing in the burn; **more burn $\implies$ higher disparity between stakers and non-stakers.**
6. But this $(\uparrow)$ disparity isn't that big.

$\cdot \circ \star \diamond \bigodot \bigotimes \bigodot \diamond\star \circ \; \cdot$
Thanks for reading!
— made with ♥ by mike.