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

\begin{document}
With the laws of motion introduced in sections \cref{SEC:Laws}, we can now describe
the optimization problem facing each constellation operator in general terms. 
Actual functional specifications are described in \cref{SEC:Computation} on computation.

Each operator recieve per-period benefits 
-- such as profits for firms and warfighting capability for militaries --
from their constellation
according to $u^i(\{s^j_t\},D_t)$, which depends 
on the current sizes of constellations and the level of debris.
In addition, the operator pays for the launch of $x^i_t$ satellites
according to a general cost function $F(x)$.
These satellites will become operational in the subsequent period.

Thus the $M$-period (possibly infinite), problem is:
\begin{align}
    \max_{\{\vec x_t\}^M}&~ 
        E\left[ \sum^M_{t=0} \beta^t u^i(\vec s_t, D_t) - F(x^i_t) \right] \\
    &\text{subject to:}\\
    & s^i_{t+1} = (1-l^i(\vec s_t, D_t))s^i_t +x^i_t ~~~  \forall i \\
    & 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}
%Assumptions
% - Identical launch costs
% - Identical debris production from destruction.
% 

%\subsection{Infinite Period (Bellman) Equation} % Not sure how much help a new header is.
The inifinite period version of the problem above can be rewritten in the bellman form as
\begin{align}
    V^i(\vec s_t, \vec x^{\sim i}_t, D_t) = \max_{x^i_t} u^i(\vec s_t, D_t) -F(x) 
    + \beta \left[ V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) \right]
\end{align}
where $x^{\sim i}_t$ represents the launch decisions of all the other constellation
operators.
One important point is that the policy function is a best response function, allowing for
a nash equilibrium interpretation of the result.

\end{document}