added work on Constellation Operator problem

CurrentWriting/sections/04_ConstellationOperator.tex

@@ -2,7 +2,120 @@
 \graphicspath{{\subfix{Assets/img/}}}
 
 \begin{document}
-Introduction goes here
+With the laws of motion introduced in sections \cref{asdf}, we can now describe
-\subsection{testing}
+the optimization problem facing each constellation operator.
-This is a subsection of the introduction.
+
+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}
+
 \end{document}