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