Stuff I had worked on previously but not touched recently. Cleanup before resuming work.

CurrentWriting/sections/04_ConstellationOperator.tex

@@ -56,7 +56,7 @@ First, find the single optimality condition
         \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 
+    0 =&  -\der{F}{x^i_t}{} + \beta \nabla V^i_{t+1} \cdot \vec a_t 
     \label{EQ:SimplifiedOptimalityCondition}
 \end{align}
 
@@ -92,7 +92,7 @@ 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 \right) 
+    \frac{1}{\beta} A^{-1} \left(\nabla \vec V^i_t - \vec u^i_t \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 

CurrentWriting/sections/05_SocialPlanner.tex

@@ -14,18 +14,21 @@ The Social (Fleet) Planner's problem can be written in the belman form as:
          + \Gamma \sum^N_{i=1} x^i_t
 \end{align}
 
-The resulting $N$ optimality conditions are:
+\subsubsection{Euler Equation}
+First find the $N$ optimality conditions:
 \begin{align}
-    0 =& -\der{F(x^i_t)}{s^i_t}{}  
+    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
-            \nabla_{\vec s_{t}, D_{t}}[\vec s_{t+1} ~ D_{t+1}]
+            \parder{}{x^I_t}{}[\vec s_{t+1} ~ D_{t+1}]
             \right] 
-        ~~\forall~~i \\
+        ~~\forall~~i
+\end{align}
+Which in vector form is:
+\begin{align}
     0 =& -\vec f +\beta \left[B\cdot \nabla W_{t+1} \right]
 \end{align}
-
 And the $N+1$ envelope conditions are:
 \begin{align}
     \nabla_{\vec s_{t}, D_{t}} W(\vec s_t, D_t) =& 
@@ -37,5 +40,10 @@ And the $N+1$ envelope conditions are:
     \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}
+
 
 \end{document}