Writing appendix on general multivariate notation and methods.

CurrentWriting/sections/apx_02_GeneralizedEuEqSteps.tex

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+\documentclass[../Main.tex]{subfiles}
+\graphicspath{{\subfix{Assets/img/}}}
+
+\begin{document}
+General Description of Defining Euler Equations with many choice 
+and state variables.
+
+Consider the following constrained Bellman Equation:
+
+\begin{align}
+    W(\theta_t) =& \max_{x_t} F(\theta_t,x_t) + \beta W(\theta_{t+1}) \\
+    \text{Subjet To:}\\
+    &\theta_{t+1} = G(\theta_t, x_t) %\\
+    %&a \leq x_t \leq b %%TODO: Add this back in later.
+\end{align}
+Where $\theta_t$ is a $m \times 1$ vector of state variables and
+$x_t$ is a $k\times 1$ vector of choice variables.
+
+The resulting optimality conditions in matrix form are
+\begin{align}
+    [0] =& \underset{k\times 1}{\parder{F(\theta_t,x_t)}{x_t}{} }
+        + \beta 
+            \underset{k \times m}{\parder{G(\theta_t,x_t)}{x_t}{} }
+            \cdot
+            \underset{m\times 1}{\parder{W(\theta_{t+1})}{\theta_{t+1}}{}} \\
+    [0] =& \vec f_{x_t} +\beta B_t \cdot \vec W_{\theta_{t+1}}
+\end{align}
+
+Similarly the envelope condations can be written as:
+\begin{align}
+    \underset{m\times 1}{\parder{W(\theta_{t})}{\theta_{t}}{}} 
+        =& \underset{m\times 1}{\parder{F(\theta_t,x_t)}{\theta_t}{}}
+        + \beta 
+            \underset{m \times m}{\parder{G(\theta_t,x_t)}{\theta_t}{} }
+            \cdot
+            \underset{m\times 1}{\parder{W(\theta_{t+1})}{\theta_{t+1}}{}} \\
+    \vec W_{\theta_t} =&  \vec f_{\theta_t} +\beta A_t \cdot \vec W_{\theta_{t+1}}
+\end{align}
+If $A_t$ is invertible, it gives the iteration condition
+\begin{align}
+    \vec W_{\theta_{t+1}} =& \beta^{-1} A_t^{-1} 
+        \cdot \left[ \vec W_{\theta_t} - \vec f_{\theta_t}\right] 
+\end{align}
+If $A_t$ is not invertible, you don't have a solution leading to the standard euler equation.
+But, assuming it is, we can now begin solving for the Euler Equation.
+
+The basic approach is to choose enough extra 
+
+The first problem we need to address is dimensionality conserns
+
+
+\end{document}