\documentclass[../Main.tex]{subfiles}
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\begin{document}
With the laws of motion introduced in sections \cref{asdf}, we can now describe
the optimization problem facing each constellation operator.

Each operator recieve utility in each period per 
their per period utility $u^i(\vec s_t,D_t)$, 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 the cost function $F(x)$.
These satellites will become operational in the next 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}
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.
This implies that the policy function is a best response function, allowing for
a nash equilibrium interpretation of the result.

To solve for the policy function, we have a variety of methods available.
Due to the computational method chosen later, I'm going to examine the conditions
for the existence of an euler equation.

\subsubsection{Euler Equation}
First, find the single optimality condition
\begin{align}
    0 =& \parder{}{x^i_t}{} u^i(\vec s_t, D_t) -\parder{}{x^i_t}{}F(x) 
    + \beta \left[ \parder{}{x^i_t}{} 
        V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
    \right] \\
    0 =&  -\der{F}{x^i_t}{} 
    + \beta \left[  
        \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}} 
            V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
        \cdot
        \nabla_{x^i_t} [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
    \right] \label{EQ:OptimalityCondition}\\
    0 =&  -\der{F}{x^i_t}{} + \beta \nabla V^i_t \cdot \vec a_t 
    \label{EQ:SimplifiedOptimalityCondition}
\end{align}

Second, the $2N$\footnote{recall that $N$ is the number of constellations.} 
envelope conditions can also be found:
\begin{align}
    \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} V^i(\vec s_t, \vec x^{\sim i}_t, D_t) 
    =& \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} u^i(\vec s_t, D_t) \notag \\
    &- \der{}{x}{}F(x^i(\vec s_t, \vec x^{\sim i}_t, D_t)) \cdot
        \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} x^i(\vec s_t, \vec x^{\sim i}_t, D_t) \notag\\
    &+ \beta \left[ 
        \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} }
            V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
        \cdot
        \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
            [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
    \right] \label{EQ:EnvelopeConditions}
    \\
    \nabla \vec V^i_t =& \vec u^i - \vec f + \beta A \cdot \nabla \vec V^i_{t+1} 
        \label{EQ:SimplifiedEnvelopeConditions}
\end{align}
When interpreting this, note that 
$$
\nabla \vec V^i_{t+1} = \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} }
    V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
$$
is a $2N \times 1$ vector of first derivatives but 
$$
A = \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
    [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
$$
is a $2N \times 2N$ matrix of first derivatives.

By solving for $\vec V^i_{t+1}$ as a function of $\vec V^i_{t}$ we get the 
intertemporal condition:
\begin{align}
    \frac{1}{\beta} A^{-1} \left(\nabla \vec V^i_t - \vec u^i +\vec f \right) 
    = \nabla \vec V^i_{t+1} 
\end{align}
Thus one crucial condition for the existence of a solution is that $A^{-1}$ exists for 
all values the laws of motion and choice functions can take.

% \subsection{Existence}
% I need to do some more diving into conditions for existence.
% Of particular concern is that the way I have specified the debris may lead to
% non-convergence.
% 

Finally, to construct the euler equation, we take
\cref{EQ:SimplifiedOptimalityCondition}
and iterate it forward $2N-1$ times.
By substituting 
\cref{EQ:SimplifiedEnvelopeConditions} 
into each iteration enough times
you get a system that defines $\nabla V^i_t$
By substituting this defined value of $\nabla V^i_t$ into 
\cref{EQ:SimplifiedOptimalityCondition}
one final time, we get a function that fully determines the policy function.

\cref{EQ:SimplifiedOptimalityConditions,EQ:SimplifiedEnvelopeConditions}

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