**THIS IS A WORK IN PROGRESS. EXPECT ERRORS AND LACK OF COMPLETENESS. THIS IS LARGELY A SKETCH OF A FUTURE HETEROGENEOUS MODEL WITH BETTER PROPERTIES.**

### Problems
We look to answer four questions: what is the equilibrium staking ratio? what is the distribution of stake between solo stakers, LSTs, and centralized exchanges? Are the first two answers acceptable for Ethereum's optimization? What issuance curve is optimal?
## Model
Agents have the choice to either stake, lend, or hold raw ETH. Let $S$ be the total supply of ETH. Let the portfolio shares be the staking ratio $\lambda$, the lending ratio $\ell$, and the holding ratio $h$. The portfolio shares must sum to one: $\lambda + \ell + h = 1$ and each share is greater than 0.
### Staking yield
The amount of staked ETH is:

$$
S^s=\lambda S
$$

Ethereum issuance in ETH is:

$$
I(\lambda)=k\sqrt{\lambda S}
$$

So the issuance yield to stakers is:

$$
\hat{i}(\lambda)=\frac{k}{\sqrt{\lambda S}}
$$

We take the priority fee yield $f$ as exogenous. The total staking yield is:

$$
r_s(\lambda)=\frac{k}{\sqrt{\lambda S}}+f
$$

We assume the existence of MEV burn. Also, supply $S$ evolves as:

$$
S_{t+1}=S_t+I(\lambda)-B_t
$$

where $B$ is amount of ETH burnt.

The real issuance yield is:

$$
\hat{i}_r(\lambda)= (1-\lambda)\hat{i}(\lambda)
$$

When determining minimum viable yield for solo stakers we should consider it.
### Lending yield
Lending yield is endogenous to the share of ETH supply to lending markets:

$$
r_L(\ell)=\frac{\beta}{\ell}
$$

where $\beta \gt 0$ is the lending demand intensity. If more ETH is lent, lending yield falls. If less ETH is lent, lending yield rises. Probably a utilization curve of sorts is more appropriate here.
### Convenience yield
Holding raw, unencumbered ETH produces a convenience yield:

$$
r_H=c
$$

Ideally we would make the convenience yield a function of inflation, the staking ratio itself, or a combination of the staking and lending ratio. Consider:

$$
r_H=c(\lambda, \ell)
$$

The intuition: if more ETH is locked in staking or lending, unencumbered ETH becomes more valuable. This creates a natural brake on the staking ratio.

## Risk

Let's assume the risk of staking is $\rho_s$, the risk of lending $\rho_L$, and the risk of holding raw ETH $\rho_H$. We assume that $p_H=0$. A simple way to specify risk costs is:
$$
\begin{align}
\rho_s=\theta \tau_s+\chi_s \\
\rho_L=\theta \tau_L+\chi_L
\end{align}
$$

where $\theta$ is the required compensation per unit of illiquidity, $\tau$ is the withdrawal time, and $\chi$ is specific risk.

Risk adjusted returns are:

$$
\begin{align}
R_s(\lambda)=\frac{k}{\sqrt{\lambda S}}+f-\rho_s \\
R_L(\ell)=\frac{\beta}{\ell}-\rho_L \\
R_H=c
\end{align}
$$

The agent optimizes:

$$
\max_{\lambda_i, \ell_i, h_i} \lambda_iR_s(\lambda)+ \ell_i R_L(\ell) + h_iR_H
$$

Therefore, in equilibrium ETH flows between staking lending and holding until all three offer the same risk-adjusted return:

$$
R_s(\lambda^*)=R_L(\ell^*)=R_H
$$

Solving the staking condition:

$$
\begin{align}
\frac{k}{\sqrt{\lambda^* S}}+f-\rho_s=c
\end{align}
$$

The equilibrium staking ratio is thus:

$$
\lambda^*=\frac{k^2}{S(c+\rho_s-f)^2}
$$

Higher issuance $k$ and higher priority fee yield $f$ raise the staking ratio $\lambda^*$. Higher supply $S$, higher convenience yield $c$, and higher staking risk $\rho_s$ lowers the staking ratio.

The lending share is:

$$
\ell^*=\frac{\beta}{c+\rho_L}
$$

Higher borrowing demand $\beta$ raises the lending share $\ell^*$. Higher raw ETH convenience yield and higher lending risk lowers the lending share.

Again, the raw holding is the residual:

$$
h^*=1-\lambda^*-\ell^*
$$

ETH flows into staking and lending until their risk-adjusted excess returns over raw ETH are competed down to zero.

We probably also want to consider the variance of returns which scales by $\frac{1}{S}$ where S is amount of stake that a staker has.
## Distribution of stake

Let total staking share be:

$$
\lambda=\lambda_{solo} + \lambda_{LST} + \lambda_{CEX}
$$

where $\lambda_{solo}$ is the solo staking share, $\lambda_{LST}$ is the liquid staking token share, and $\lambda_{CEX}$ is the centralized exchange staking share.

Each staking channel receives the same gross protocol yield but has different costs, risks, and convenience benefits.

For each staking channel $j \in \{solo, LST, CEX\}$, define:

$$
R_j(\lambda)=(1-m_j)r_s(\lambda)-\rho_j-\kappa_j+q_j
$$

where $m_j$ is the commission charged by channel $j$, $\kappa$ is the access cost, and $q$ is the channel specific convenience yield. We assume that $m_{solo}=0$ and $q_{solo}=0$. We also assume that $\kappa_{solo} \gt \kappa_{LST} \gt \kappa_{CEX}$. For LST staking, the user pays a fee but receives liquidity and DeFi composability $m_{LST} \gt 0$ and $q_{LST} \gt 0$. Also, for CEX staking, the user pays a fee but also receives liquidity via a token therefore, $m_{CEX} \gt 0$ and $q_{CEX} \gt 0$. The convenience yield for LSTs and centralized exchanges are driven by their total value locked $q_j = f(TVL_j)$. We also assume that $\rho_j=f(\gamma_j)$ where $\gamma_j$ is slashing risk. We assume that $\gamma_j = f(TVL_j)$.

We can assume the convenience yield for LSTs and centralized exchanges scales with total value locked. If all agents are identical then all stake goes to the option with the highest risk-adjusted return. We are going to need heterogeneous agents. Obviously, in practice, agents differ. Some have less than 32 ETH. Some cannot operate validators. Some value liquidity. Some distrust centralized exchanges. Some distrust smart contracts.

Another wrinkle is we assume the agent can only lend ETH, but if they hold an LST they can also lend that. Also, consider leveraged liquid staking so the demand for looping LSTs increases the yield on borrowing raw ETH to further loop LSTs.

### Ethereum optimization

Ethereum’s cost to corrupt can be modeled as the dollar cost required to control or compromise enough stake to execute a one-third attack multiplied by a decentralization coefficient.

Let $M$ = ETH market capitalization in dollars. Then the naive cost to corrupt is:

$$
\text{CTC} = \frac{1}{3} \lambda M
$$

This says that an attacker must economically control one-third of all staked ETH. If we do not want to specify the exact function for validator decentralization, we can use a single multiplier:

$$
\text{CTC} = \frac{1}{3} \lambda M \cdot D
$$

where $D$ = decentralization multiplier. This keeps the output in dollar terms while still recognizing that a more decentralized validator set is harder to corrupt than a concentrated one. Perhaps $D$ could be the Nakamoto Coefficient of stake or ETH holdings. Certainly, D of stake makes more sense for the cost-to-corrupt, but there is a broader political economy to consider.

We need to account for the contribution to economic security from each type of staker. The problem with centralized exchanges and LSTs is the principal agent problem where what is at stake is not stake directly but the present discounted value of future fees from stakers. Therefore, we make a distinction between stake at risk and franchise value at risk. Solo validators secure Ethereum with their stake. Intermediaries secure Ethereum with their franchise value.

Then dollar stake by channel $j \in \{solo, LST, CEX\}$ is:

$$
V_j = \lambda_j M
$$

For solo staking, economic security contribution is:

$$
V_{solo}=\lambda_{solo}M
$$

For an intermediary $k \in \{LST, CEX\}$, suppose it earns fee rate $m_j$ on gross staking yield $r_s$. Its annual fee revenue is:

$$
F_k = m_k r_s V_k
$$

If future fee revenue is discounted at rate $\delta_k$, then a simple perpetuity value is:

$$
PV_k = \frac{F_k}{\delta_k}
$$

So:

$$
PV_k = \frac{m_k r_s \lambda_k M}{\delta_k}
$$

Then total effective economic security $\Omega$ is:

$$
\Omega = \lambda_{solo}M + \frac{m_{LST}r_s\lambda_{LST}M}{\delta_{LST}} + \frac{m_{CEX}r_s\lambda_{CEX}M}{\delta_{CEX}}
$$

Or factoring out $M$:

$$
\Omega = M\left(\lambda_{solo}+\frac{m_{LST}r_s\lambda_{LST}}{\delta_{LST}}+\frac{m_{CEX}r_s\lambda_{CEX}}{\delta_{CEX}}\right)
$$

The intuition is solo validators have the full stake at risk whereas LSTs and CEXs have mainly their future fee stream at risk. Therefore one dollar of solo-staked ETH contributes more direct economic security than one dollar of delegated ETH controlled by an intermediary.

If you want a one-third attack version:

$$
CTC = \frac{1}{3} \Omega \cdot D
$$

So Ethereum wants to:

$$
\max \frac{M}{3}\left(\lambda_{solo}+\frac{m_{LST}r_s\lambda_{LST}}{\delta_{LST}}+\frac{m_{CEX}r_s\lambda_{CEX}}{\delta_{CEX}}\right) \cdot D
$$

A problem with this optimization function is that it assumes we want to maximize the amount of security we have but it could be the case the we only want to pay for the minimum amount of security required to keep the chain "safe." We do not want to overpay for security, aka "minimum viable issuance."

Another optimization function could be:

$$
\min_k I(k)
$$

subject to:

$$
\begin{align}
\lambda > \overline{\lambda} \\
\Omega > \overline{\Omega} \\
\ D > \overline{D}
\end{align}
$$

In words, we want at least a specific staking ratio $\overline{\lambda}$, we want a minimum amount of effective economic security $\overline{\Omega}$ and a minimum amount of decentralization quality $\overline{D}$. Or we can we can make some combination of the two. Perhaps add a solo staker viability constraint.
## Issuance curves
TBD