\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}