There was a write up a few years ago by Anders at the Ethereum Foundation looking to determine the circulating supply equilibrium for ETH. It leaves out some critical pieces but, assuming there’s some truth to it, there are some interesting extensions of the model, as alluded to here. We will be using the model to determine the equilibrium base fee in ETH and the ETHUSD rate. This piece, as with most things I write on here, are attempts to think out loud and are, thus, probably incomplete. What follows may ultimately be tautological, but an interesting thought experiment nonetheless.

The model starts with annual issuance $I$ equal to:

\[I=cF\sqrt{D} \tag{1}\]

where $F$ is the base reward factor, $D$ is the amount deposited in the staking contract, and $c$ is a constant calculated as:

\[c=\frac{82181.25}{\sqrt{10^9}} \approx 2.6 \tag{2}\]

where $82181.25$ is the number of epochs per year and $10^9$ is the amount of gwei per ether. As a function of yield $y$ demanded by stakers:

\[yD=cF\sqrt{D} \tag{3}\]

Solving for $D$ we get:

\[D=\frac{c^2F^2}{y^2} \tag{4}\]

Plugging $D$ into equation $(1)$ we get:

\[I = cF\sqrt{\frac{c^2F^2}{y^2}} \tag{5}\]

Expanding on $y$ and simplifying:

\[I = \frac{c^2F^2}{r_f+r_p} \tag{6}\]

where $r_f$ is the risk-free rate and $r_p$ is a risk premium. We can also formulate the amount burned $B$ as a function of the burn rate $b$ and total supply $S$ as well as the annualized amount of gas used $G$ and the gas price in ETH $p$:

\[B = bS = Gp \tag{7}\]

Therefore, net issuance $N$ is equal to:

\[N = \frac{c^2F^2}{r_f+r_p}-bS = \frac{c^2F^2}{r_f+r_p}-Gp \tag{8}\]

The original piece uses $bS$ but we will use $Gp$. The equilibrium when $I = B$ is:

\[\frac{c^2F^2}{r_f+r_p}=Gp \tag{9}\]

Solving for $p$ we get:

\[p = \frac{c^2F^2}{(r_f+r_p)G} \tag{10}\]

We can also express $G$ as a function of block utilization $u$, a block gas cap $C$, and an annualization factor $\lambda=2628000$ blocks per year:

\[G = uC\lambda \tag{11}\]

Equation $(10)$ then becomes:

\[p=\frac{c^2F^2}{(r_f+r_p)uC\lambda} \tag{12}\]

For example, the current gas target is $15\times10^6$ which corresponds to $u=.5$ and $C=30\times10^6$. These parameters imply a $G=3.942 \times 10^{13}$. Assuming $r_f=.05$ and $r_p=.02$, the base fee in equilibrium would be:

\[\frac{2.6^2 64^2}{(.05+.02)3.942 \times 10^{13}}\approx \frac{1.003 \times 10^{-8}}{10^{-9}} \approx 10 \text{ gwei} \tag{13}\]

However, $p$ is the base fee in ETH. Users take into consideration the ETHUSD rate when setting their gas prices in ETH. We can express $p$ as a function of the ETHUSD rate $\theta$:

\[p=\alpha+\beta\theta \tag{14}\]

Therefore:

\[\alpha+\beta\theta=\frac{c^2F^2}{(r_f+r_p)uC\lambda} \tag{15}\]

Solving for $\theta$:

\[\theta=\frac{\frac{c^2F^2}{(r_f+r_p)uC\lambda}-\alpha}{\beta} \tag{16}\]

Using the data from my previous post, and using the same assumptions we made above, we get an implied ETHUSD rate of:

\[\ln(\theta) =\frac{\ln\left(\frac{2.6^2 64^2}{(.05+.02)3.942 \times 10^{13}}\right)+8.4}{-1.12}\approx\exp(8.94)\approx\$7668 \tag{17}\]

The interval for this value ranges from $\$3781$ to $\$16612$.

We can glean some intuition for equations $(12)$ and $(16)$ when $I=B$, keeping the assumptions we just made in mind. First, we can identify which variables are endogenous and exogenous to the current Ethereum protocol. The exogenous variables $x$ are $c$, $F$, $C$, $\lambda$, and, arguably, $u$, which leaves $r_f$, $r_p$, $\alpha$, and $\beta$ as endogenous $n$. When $r_f$ and $r_p$ decrease, issuance $I$ increases and $B$ increases due to an increase in ETH denominated gas prices. Also, the equilibrium ETHUSD rate decreases. In this case, there are a few levers, not considering practicality, that the protocol can adjust if, say, targeting $p$. The protocol could lower the base reward factor $F$, increase the gas cap $C$, and increase the number of blocks per year. These adjustments would obviously have effects elsewhere.

I need to think more about why the case of $I=B$ is a particularly important state of the world and, as mentioned by Barnabe at the Ethereum Foundation, why they would come into equilibrium at all. If it is, then I would like to think more about incorporating additional variables, such as convenience yields, into the model and coming up with better causality assumptions. For example, the implied ETHUSD rate just falls out of the equation and regression coefficients.