Issuance Issues — Tertiary Treatise

^ i know, i know. another alliteration, an arguably arduous articulation at that – as if anyone actually asked admonish away; any advice appreciated.

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^ p.s. we can only aspire to be both precise and pleasant while measuring pie slices. 🥧📐


by mike – friday, june 21, 2024.
^turns out 2024 is the only year in the entire range of 2021-2027 where the summer solstice is on june 20
th
instead of the 21
st
"missed it by that much."


many thanks to Nate for discussions around the framing of this article.

tl;dr;
Hello & welcome to the third installment of the Issuance Issues®. The first issue 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 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 The Great Issuance Debate© 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 increases 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."


Contents
(1) Nominal yield & inflation refresher
(2) A new dimension – network ownership
   (2.1) Unstaked ETH, no burn
   (2.2) Staked ETH, no burn
   (2.3) Ownership gap, no burn
(3) Bring on the burn
   (3.1) Unstaked ETH, burn
   (3.2) Staked ETH, burn
   (3.3) Ownership gap, burn
(4) Wrapping up

The plotting code is available here.[1]

Related work

Article Description
Minimum Viable Issuance Anders' first post
Properties of issuance level (part 1) Anders' second post
Endgame Staking Economics: A Case for Targeting Caspar & Ansgar's post
Electra: Issuance Curve Adjustment Proposal Caspar & Ansgar's Electra proposal
UCC2: Ethereum's Staking Endgame Jon, Hasu, Caspar, & Ansgar's discussion
Initial Analysis of Stake Distribution Julian's post
Minimum Viable Issuance Anders' third post
Reward curve with tempered issuance Anders' fourth post
Foundations of MVI Anders' fifth post
FAQ: Ethereum issuance reduction 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,

issuance=2.664stake.

Here,

stake is denominated ETH, and the
issuance
is the annual yield from staking. For example, let
stake=30mm
ETH, then
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 formula,
inflation=new supplyold supplyold supply=(old supply+issuancestake)old supplyold supply=issuancestakeold supply.

We use the supply of

120mm ETH, so we can rewrite the inflation as,
inflation=2.664stake120,000,000.

For example, let
stake=30mm
ETH, then
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.

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The main characteristic of the yield curve (left) is that it grows extremely quickly as

stake0; the curve highly incentivizes non-zero stake. On the other end, as
stake120mm
, the yield declines increasingly slowly, meaning the curve doesn't meaningfully differentiate between
60mm
and
80mm
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

90mm staked ETH (dashed lines) for reference.

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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. 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

πu denote the portion of the network you own as a non-staker (
subscript
u
for "unstaked"); we use
π
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,

πu=new network ownership - old network ownershipold network ownership.

Again, we use the relative change equation to see how much the network ownership decreases after a year. Note that, by definition,

πu must be negative because, without the burn, the total supply is monotone increasing, but the number of non-staked tokens stays fixed, implying
new network ownership<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
new network ownership
term, we see this playing out as
new network ownership=my tokensnew supply=my tokensold supply+inflation.

The

old network ownership is simply
old network ownership=my tokensold supply.

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
πu
above,

πu=u/(T(1+i(s)))u/Tu/T=11+i(s)1.

The figure below shows

πu for all possible stake values,
s
. We include two reference values:

  • stake=30mm
    ETH
    πu=0.75%
  • stake=90mm
    ETH
    πu=1.30%

These are interpreted as, e.g., "with

30mm staked ETH, the percent of the total supply a non-staker owns decreases by
0.75%
after a year."

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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

πs denote the annual change in the portion of the supply that a staker owns. Using the same relative change formula, we have

πs=new network ownership - old network ownershipold network ownership

Again focusing in on

new network ownership, we now have to incorporate the yield from staking into the numerator,
new network ownership=my tokens+my yieldnew supply=my tokens+my yieldold supply+inflation

Putting this into algebra, let

s be the staked tokens and
y(s)
denote the yield for staking (as defined in Section 1). Then we have
πs=s(1+y(s))/(T(1+i(s)))s/Ts/T=1+y(s)1+i(s)1.

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
πs=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

πs along with
πu
(from Section 2.1) for various staking levels. We include two new reference values (in addition to the
πu
values from above):

  • stake=30mm
    ETH
    πs=2.26%
  • stake=90mm
    ETH
    πs=0.43%

These are interpreted as, e.g., "with

30mm staked ETH, the percent of the total supply a staker owns increases by
2.26%
after a year."

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We see that

πs0 as
stake120mm
. 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

πsπu. Using the expressions derived in the previous sections, we have

πsπu=1+y(s)1+i(s)1(11+i(s)1)=y(s)1+i(s).

The figure below helps visualize this. The left plot shows

πs,πu 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
stake=20mm
ETH to help with the scaling.

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We can interpret this as, e.g., "at

stake=40mm 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.

No burn, non-staker

(πu)
Ownership share before
=30/1,000,000(0.3 bips).

Ownership share after
=30/1,010,000(0.297 bips).

Ownership change
πu=(0.2970.3)/0.3=0.011%
decrease.

No burn, staker

(πs)
Ownership share before
=30/1,000,000(0.3 bips).

Ownership share after
=31/1,010,000(0.3069 bips).

Ownership change
πs=(0.30690.3)/0.3=0.0232.3%
increase.

No burn, ownership gap

(πsπu)
Gap
πsπu=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

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 increases 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

new network ownership. Let
πuβ
denote "the change in percentage of network ownership for a non-staker while including the burn" (
superscriptβ
for 'burn'). We continue using our relative change formula,

πuβ=new network ownership - old network ownershipold network ownership.

Now the

new network ownership term includes the burn as a factor in calculating the new supply,
new network ownership=my tokensnew supply=my tokensold supply+inflationburn.

The

old network ownership remains as
old network ownership=my tokensold supply.

In algebraic terms, let

b denote the number of tokens burned annually. Then,

πuβ=u/(T(1+i(s)b))u/Tu/T=11+i(s)b1

Notice that this is very similar to our

πu calculation, just with the burn subtracted from the denominator. The intuition here is that the burn decreases the denominator, making the overall value of
πuβ
less negative (e.g., more burn
"smaller decrease"[4] in network ownership). In fact, if
i(s)=b
, then
πuβ=0
, meaning the non-staker continues owning a constant portion of the supply. Further, if
i(s)<b
(there is *more burn* than inflation[5]),
πuβ>0
, which means the portion of the network owned by the non-staker goes up!

The figure below shows the values of

πuβ 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.

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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

πsβ denote the change in network ownership of a staker while incorporating the burn. Again, we use the familiar relative change formula.

πsβ=new network ownership - old network ownershipold network ownership

Just as in Section 3.1, the burn shows up when we calculate the

new network ownership.

new network ownership=my tokens+my yieldnew supply=my tokens+my yieldold supply+inflationburn.

Putting this into algebraic terms, we have

πsβ=s(1+y(s))/(T(1+i(s)b))s/Ts/T=1+y(s)1+i(s)b1.

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

πsβ and
πuβ
values respectively. The black curves represent the no-burn situation – the tan and blue lines in Section 2.2.

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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

πsβπuβ.

πsβπuβ=1+y(s)1+i(s)b1(11+i(s)b1)=y(s)1+i(s)b.

In Section 2.3, 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
πsβ,πuβ
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[6] zooming in on the
3040mm
staked ETH range.

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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
0.07%
(
7
bips) – quite small.

Let's include the burn in our calculations by extending our numerical example from Section 2.3.

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).

Burn, non-staker

(πuβ)
Ownership share before
=30/1,000,000(0.3 bips).

Ownership share after
=30/995,000(0.3015 bips).

Ownership change
πuβ=(0.30150.3)/0.3=0.0050.5%
increase.

Burn, staker

(πsβ)
Ownership share before
=30/1,000,000(0.3 bips).

Ownership share after
=31/995,000(0.3116 bips).

Ownership change
πsβ=(0.31160.3)/0.3=0.0393.9%
increase.

Burn, ownership gap

(πsβπuβ)
Gap
πsβπuβ=0.0390.005=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 always 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
    higher disparity between stakers and non-stakers.
  6. But this
    ()
    disparity isn't that big.


Thanks for reading!
— made with ♥ by mike.


  1. Buyer beware; we get saucy in the matplotlib (YMMV).[2] ↩︎

  2. Me 'n matplotlib go waaaaaay back – see, e.g., Figure 4 in this article from six(!) years ago.[3] ↩︎

  3. Nested footnotes; learning from the best. ↩︎

  4. I don't dislike double negatives. :) ↩︎

  5. This is what is sometimes referred to as "ultra sound". ↩︎

  6. Matplotlib makes these inset axes soooo easy – check it out. ↩︎