papers/lux-omnichain-yield/sections/08-economic-analysis.tex

\section{Economic Analysis}
\label{sec:economics}
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\subsection{Capital Efficiency Comparison}

We compare three modes of holding ETH across a bridge:

\begin{definition}[Capital Efficiency Ratio]
\begin{equation}\label{eq:capital_efficiency}
  \eta = \frac{\text{APY}_{\text{bridge token}}}{\text{APY}_{\text{native}}}
\end{equation}
where $\text{APY}_{\text{native}}$ is the yield available on the source chain without bridging.
\end{definition}

\begin{table}[h]
\centering
\small
\caption{Capital efficiency comparison: ETH across a bridge from Ethereum to Lux. Native stETH APY = 3.5\% (Lido, Q1 2025).}
\label{tab:capital_efficiency}
\begin{tabular}{@{}lrrr@{}}
\toprule
\textbf{Token} & \textbf{APY} & \textbf{$\eta$} & \textbf{Source} \\
\midrule
WETH (Ethereum) & 0.0\% & 0.00 & Idle \\
LETH (Teleport) & 0.0\% & 0.00 & 1:1 wrapped \\
stETH (Lido) & 3.5\% & 1.00 & Native stake \\
yLETH (Teleport) & 5.5\% & 1.57 & Stake + restake \\
yLETH + DeFi & 11.5--21.5\% & 3.3--6.1 & Full composition \\
\bottomrule
\end{tabular}
\end{table}

\noindent
yLETH achieves $\eta > 1.0$ because the vault captures restaking yield that individual stakers may not access, and bridge fee income from xLUX adds yield not available to direct stETH holders.

\subsection{The xLUX Flywheel}\label{sec:flywheel}

On Lux Network, the stakeholder vault for the Teleport OmnichainRouter is LiquidLUX (xLUX). All protocol fees flow to xLUX holders:

\begin{equation}\label{eq:xlux_yield}
  r_{\text{xLUX}} = \frac{\sum_{p \in \mathcal{P}} f_p \cdot V_p}{\text{totalStaked}_{\text{LUX}}}
\end{equation}

where $\mathcal{P} = \{\text{bridge, DEX, lending, perps, NFT AMM}\}$ is the set of fee-generating protocols, $f_p$ is the fee rate for protocol $p$, and $V_p$ is the volume processed by protocol $p$.

\begin{figure}[h]
\centering
\begin{tikzpicture}[
  node distance=1.6cm,
  box/.style={draw, rounded corners=3pt, minimum width=2.2cm, minimum height=0.7cm, font=\scriptsize, align=center, thick},
  arr/.style={-{Stealth[length=2mm]}, thick, color=black!70},
]
  \node[box] (bridge) {Teleport\\Bridge};
  \node[box, right=1.2cm of bridge] (dex) {Lightspeed\\DEX};
  \node[box, below=1.2cm of bridge] (lending) {Lux\\Lending};
  \node[box, below=1.2cm of dex] (perps) {LPX\\Perps};
  \node[box, below right=0.2cm and -0.3cm of lending] (nft) {LSSVM\\NFT AMM};

  \node[box, fill=black!8, minimum width=2.5cm, below=3.5cm of bridge, xshift=1.0cm] (xlux) {\textbf{xLUX}\\fee aggregation};

  \draw[arr] (bridge.south) -- ++(0,-0.5) -| (xlux.north west);
  \draw[arr] (dex.south) -- ++(0,-0.5) -| (xlux.north east);
  \draw[arr] (lending.south) -- ++(0,-0.3) -| (xlux.west);
  \draw[arr] (perps.south) -- ++(0,-0.3) -| (xlux.east);
  \draw[arr] (nft.south) -- ++(0,-0.15) -| (xlux.south);

  \node[box, fill=black!8, below=1.0cm of xlux] (stakers) {\textbf{LUX stakers}\\receive yield};
  \draw[arr] (xlux) -- (stakers);

  \node[font=\scriptsize, right=0.3cm of stakers, text width=2.5cm, align=left] {Higher APY\\$\to$ more staking\\$\to$ deeper liquidity\\$\to$ more volume};
\end{tikzpicture}
\caption{The xLUX fee aggregation flywheel. Fees from all Lux protocols compound into xLUX yield, attracting more staking, deepening liquidity, and generating more fees.}
\label{fig:flywheel}
\end{figure}

\begin{theorem}[Flywheel Convergence]\label{thm:flywheel}
Under the assumption that total protocol volume $V = \sum_p V_p$ is an increasing function of total liquidity $\mathcal{L}$, and liquidity is an increasing function of xLUX APY $r_{\text{xLUX}}$, the system converges to a stable equilibrium. Formally, if:
\begin{align}
  V &= g(\mathcal{L}), \quad g' > 0, \quad g'' < 0 \label{eq:fly1} \\
  \mathcal{L} &= h(r_{\text{xLUX}}), \quad h' > 0, \quad h'' < 0 \label{eq:fly2} \\
  r_{\text{xLUX}} &= \frac{f \cdot V}{\text{totalStaked}} \label{eq:fly3}
\end{align}
where $g$ and $h$ are concave increasing functions, then there exists a unique fixed point $(V^*, \mathcal{L}^*, r^*)$ satisfying all three equations.
\end{theorem}

\begin{proof}
Define $\Phi: \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ by $\Phi(r) = f \cdot g(h(r)) / \text{totalStaked}$. Since $g$ and $h$ are concave and increasing, $\Phi$ is concave and increasing. For sufficiently small $r$, $\Phi(r) > r$ (the flywheel generates more yield than required). For large $r$, the concavity of $g \circ h$ ensures $\Phi(r) < r$ (diminishing returns on volume growth). By the intermediate value theorem, there exists a unique $r^*$ with $\Phi(r^*) = r^*$.
\end{proof}

\subsection{Fee Flow Quantification}

\begin{table}[h]
\centering
\small
\caption{Projected annual fee flow to xLUX at varying TVL levels. Assumes: bridge fee 10 bps, DEX fee 5 bps, lending spread 50 bps, perps fee 8 bps, NFT AMM 200 bps. Velocity: bridge 2x/year, DEX 50x/year, lending 3x/year, perps 100x/year, NFT 10x/year. Stakeholder share: 90\%.}
\label{tab:fee_flow}
\begin{tabular}{@{}rrrr@{}}
\toprule
\textbf{Total TVL} & \textbf{Annual Fees} & \textbf{xLUX Share} & \textbf{xLUX APY\textsuperscript{*}} \\
\midrule
\$100M & \$8.2M & \$7.4M & 7.4\% \\
\$500M & \$41M & \$37M & 7.4\% \\
\$1B & \$82M & \$74M & 7.4\% \\
\$5B & \$410M & \$369M & 7.4\% \\
\bottomrule
\multicolumn{4}{l}{\textsuperscript{*}\scriptsize Assumes staked LUX = TVL (simplified). Real APY varies with staking ratio.}
\end{tabular}
\end{table}

\subsection{Yield vs.\ Risk Frontier}

The portfolio of yield strategies traces an efficient frontier in risk-return space. Let $\bm{r} = (r_1, \ldots, r_m)$ be the vector of strategy APYs and $\bm{\Sigma}$ be the covariance matrix of strategy returns. The mean-variance optimal allocation is:

\begin{equation}\label{eq:frontier}
  \max_{\bm{w}} \bm{w}^T \bm{r} - \frac{\lambda}{2} \bm{w}^T \bm{\Sigma} \bm{w}
  \quad \text{s.t.} \quad \bm{w} \in \mathcal{W}
\end{equation}

where $\lambda$ is the risk aversion parameter and $\mathcal{W}$ is the feasible set defined by the tier constraints~(\ref{eq:allocation}) and the single-strategy cap.

\begin{remark}
The governor (DAO) sets $\lambda$ based on the chain's risk appetite. A conservative chain (e.g., institutional-focused) uses high $\lambda$, concentrating in Tier 1. An aggressive chain uses low $\lambda$, allocating more to Tier 2/3 for higher yield.
\end{remark}

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