\documentclass[../Main.tex]{subfiles}
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\begin{document}
The Social (Fleet) Planner's problem can be written in the bellman form as:
\begin{align}
    W(\vec s_t, D_t) =& \max_{\vec x_t} \left[
       \left(\sum^N_{i=1} u^i(\vec s_t, D_t) - F(x^i_t) \right)
    + \beta \left[ W(\vec s_{t+1}, D_{t+1}) \right]\right] \notag \\
     &\text{subject to:} \notag \\
     & s^i_{t+1} = (1-l^i(\vec s_t, D_t))s^i_t +x^i_t ~~~  \forall i \notag \\
     & D_{t+1} = (1-\delta)D_t + g(D_t) 
         + \gamma \sum^N_{i=1} l^i(\vec s_t, D_t)
         + \Gamma \sum^N_{i=1} x^i_t
\end{align}
Some particular features of the model include:
\begin{align}
    \item The single period welfare function consists only of constellation operators.
        Although satellites do deorbit and occasionally pose a risk to humans living on the
        earth's surface\footnote{Skylab fell in Australia, with some pieces landing near towns.}
        modeling this risk properly would require adding a deorbit decisions, 
        including uncontrolled deorbits.
    \item Although the social planner controls each constellation, they do not reap additional
        collision avoidance efficiencies.
        One justification is that no social planner could concieve of every use of orbit
        at any single point in time, and thus constellations are added sequentially.
        This allows only the intra-constellation benefits to be achived.
\end{align}

\subsubsection{Euler Equation}
In accordance with Appendix \cref{Appendix}, find the $N$ optimality conditions:
\begin{align}
    0 =& -\der{F(x^i_t)}{x^i_t}{}  
        + \beta \left[
            \nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1}) 
            \cdot
            \parder{}{x^I_t}{}[\vec s_{t+1} ~ D_{t+1}]
            \right] 
        ~~\forall~~i
\end{align}
Which in vector form is:
\begin{align}
    0 =& -\vec f_x +\beta \left[B\cdot \nabla W_{t+1} \right]
\end{align}
Similarly, the $N+1$ envelope conditions are:
\begin{align}
%    \nabla_{\vec s_{t}, D_{t}} W(\vec s_t, D_t) =& 
%        \sum^N_{i=1} \nabla_{\vec s_{t}, D_{t}} u^i(\vec s_t, D_t) 
%        %- \der{}{x^i_t}{}F(x^i_t) \nabla_{\vec s_{t}, D_{t}}x^i_t %This equals zero due to the envelope theorem
%        \notag \\
%        &+ \beta \left[ \nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1})
%            \cdot \nabla_{\vec s_{t}, D_{t}} [\vec s_{t+1} ~ D_{t+1}]
%    \right] \\
    \nabla W_t =& \vec U + \beta \left[C \cdot \nabla W_{t+1} \right] 
\end{align}
Which gives us the iteration format
\begin{align}
    \nabla W_{t+1} =& (\beta C)^{-1} \cdot \left(\nabla W_t - \vec U \right)
\end{align}

Thus two iterations of the optimality condition are needed, but only to provide $N+1$ binding conditions.
This lets us discard $N-1$ of the conditions from the second iteration of the optimality condition.
% NEed to explain better. Not quite true.

\end{document}