Removed social planner's references to euler equations.

CurrentWriting/sections/05_SocialPlanner.tex

@@ -4,7 +4,7 @@
 \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[
+    W(S_t, D_t) =& \max_{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 \\
@@ -21,44 +21,8 @@ The Social (Fleet) Planner's problem can be written in the bellman form as:
 %        including uncontrolled deorbits.
 Although the social planner controls each constellation, note that they do not reap additional
 collision avoidance efficiencies.
-this is because no social planner could concieve of every use of orbit
+One justification no social planner could concieve of every future use of an orbit
-at any single point in time, and thus constellations may be designed sequentially.
+and consequentally constellations may be designed sequentially.
-This allows only the intra-constellation benefits to be achived.
+This prevents intra-constellation benefits to be achieved across the entire fleet.
-
-\subsubsection{Euler Equation}
-In accordance with Appendix \cref{APX:Derivations:EulerEquations}, 
-we 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}