\documentclass[../Main.tex]{subfiles}
\graphicspath{{\subfix{Assets/img/}}}

\begin{document}
The Social (Fleet) Planner's problem can be written in the bellman form as:
\begin{align}
    W(S_t, D_t) =& \max_{X_t} \left[
      \sum^N_{i=1} \left( u^i(S_t, D_t) - F(x^i_t) \right)
    + \beta \left[ W(S_{t+1}, D_{t+1}) \right]\right] \notag \\
     &\text{subject to:} \notag \\
     & s^i_{t+1} = (R^i(S_t, D_t)) s^i_t +x^i_t ~~~  \forall i \notag \\
     & D_{t+1} = (1-\delta + g)D_t  
         + \gamma \sum^N_{i=1} \left(1-R^i(\vec s_t, D_t)\right) s^i_t
         + \Gamma \sum^N_{i=1} x^i_t
\end{align}
%Some particular features of the model include:
%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.
Although the social planner controls each constellation, note that they do not reap additional
collision avoidance efficiencies.
One justification is that no social planner could concieve of every future use of an orbit
and consequentally constellations will be designed sequentially.
This prevents intra-constellation benefits to be achieved across the entire fleet.

\end{document}
