papers/lux-tokenomics/sections/06-appchain-economic-coupling.tex

\section{Appchain Economic Coupling}
\label{sec:appchain}

\subsection{Appchain Registration and Fees}

Creating a appchain on Lux Network requires a one-time creation fee $F_{\text{create}}$ and ongoing validator staking. The creation fee is denominated in LUX and burned:
\begin{equation}
    F_{\text{create}} = F_{\text{base}} \cdot (1 + \delta \cdot |\mathcal{K}|)
\end{equation}
where $F_{\text{base}} = 1{,}000$ LUX and $\delta = 0.001$ provides mild congestion pricing as the number of appchains $|\mathcal{K}|$ grows.

\subsection{Cross-Appchain Value Flow}

Each appchain contributes to the primary network economy through multiple channels:

\begin{enumerate}[label=(\arabic*)]
    \item \textbf{Validator staking}: Appchain validators must stake LUX on the primary network.
    \item \textbf{Fee tax}: A fraction $\tau_k \in [0.01, 0.05]$ of appchain fees flows to the primary network burn.
    \item \textbf{Warp messaging fees}: Cross-appchain messages pay LUX fees on the primary network.
    \item \textbf{Appchain token backing}: Appchains may optionally back their native tokens with LUX reserves.
\end{enumerate}

\begin{definition}[Economic Coupling Coefficient]
The economic coupling coefficient for appchain $k$ is:
\begin{equation}
    \kappa_k = \frac{\tau_k \cdot F_k + \text{Stake}_k + \text{WarpFees}_k}{\text{TotalEconomicActivity}_k}
\end{equation}
measuring the fraction of appchain economic activity that accrues to the primary network.
\end{definition}

\subsection{Optimal Tax Rate}

\begin{theorem}[Revenue-Maximizing Tax Rate]
The appchain fee tax rate $\tau_k^*$ that maximizes total primary network revenue, accounting for the elasticity of appchain transaction demand, is:
\begin{equation}
    \tau_k^* = \frac{1}{1 + \epsilon_k}
\end{equation}
where $\epsilon_k > 0$ is the price elasticity of demand for transactions on appchain $k$.
\end{theorem}

\begin{proof}
Primary network revenue from appchain $k$ is $\mathcal{R}_k(\tau_k) = \tau_k \cdot F_k(\tau_k)$ where $F_k(\tau_k) = F_k^0 \cdot (1 + \tau_k)^{-\epsilon_k}$ models demand elasticity. Taking the first-order condition:
\begin{equation}
    \frac{\partial \mathcal{R}_k}{\partial \tau_k} = F_k - \tau_k \cdot \epsilon_k \cdot \frac{F_k}{1 + \tau_k} = 0
\end{equation}
Solving yields $\tau_k^* = 1 / (1 + \epsilon_k)$. For typical blockchain transaction elasticities $\epsilon_k \in [0.5, 2.0]$, this gives $\tau_k^* \in [0.33, 0.67]$. In practice, we cap $\tau_k \leq 0.05$ to encourage appchain adoption, accepting suboptimal revenue extraction.
\end{proof}