Correcting errors regarding envelope conditions

CurrentWriting/Main.tex

@@ -57,7 +57,7 @@ I hope to write a section clearly explaining assumptions, caveats, and shortcomi
 Needs completed.
 \subsection{Derivations}
 \subsubsection{Marginal Survival Rates}\label{APX:Derivations:SurvivalRates}
-\subfile{sections/appedicies/apx_01_MarginalSurvivalRates} 
+%\subfile{sections/appedicies/apx_01_MarginalSurvivalRates} 
 
 
 \end{document}

CurrentWriting/sections/04_ConstellationOperator.tex

@@ -65,8 +65,6 @@ 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}) 
@@ -75,25 +73,26 @@ envelope conditions can also be found:
             [ \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} 
+    \nabla \vec V^i_t =& \vec u^i 
+        + \beta A \cdot \nabla \vec V^i_{t+1} 
         \label{EQ:SimplifiedEnvelopeConditions}
 \end{align}
 When interpreting this, note that 
 $$
-\nabla \vec V^i_{t+1} = \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} }
+\nabla \vec V^i_{t+1} = \nabla_{[\vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1}] }
     V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
 $$
 is a $2N \times 1$ vector of first derivatives but 
 $$
 A = \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
-    [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
+    [ \vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1} ]
 $$
 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) 
+    \frac{1}{\beta} A^{-1} \left(\nabla \vec V^i_t - \vec u^i \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 
@@ -116,6 +115,5 @@ 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}

CurrentWriting/sections/05_SocialPlanner.tex

@@ -2,7 +2,40 @@
 \graphicspath{{\subfix{Assets/img/}}}
 
 \begin{document}
-Introduction goes here
+The Social (Fleet) Planner's problem can be written in the belman form as:
-\subsection{testing}
+\begin{align}
-This is a subsection of the introduction.
+    W(\vec s_t, D_t) =& \max_{\vec 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 \\
+     & s^i_{t+1} = (1-l^i(\vec s_t, D_t))s^i_t +x^i_t ~~~  \forall i \notag \\
+     & 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}
+
+The resulting $N$ optimality conditions are:
+\begin{align}
+    0 =& -\der{F(x^i_t)}{s^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}]
+            \right] 
+        ~~\forall~~i \\
+    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) =& 
+        \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}
+
 \end{document}