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42 lines
1.5 KiB
TeX
42 lines
1.5 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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The Social (Fleet) Planner's problem can be written in the belman form as:
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\begin{align}
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W(\vec s_t, D_t) =& \max_{\vec x_t} \left[
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\left(\sum^N_{i=1} u^i(\vec s_t, D_t) - F(x^i_t) \right)
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+ \beta \left[ W(\vec s_{t+1}, D_{t+1}) \right]\right] \notag \\
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&\text{subject to:} \notag \\
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& s^i_{t+1} = (1-l^i(\vec s_t, D_t))s^i_t +x^i_t ~~~ \forall i \notag \\
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& D_{t+1} = (1-\delta)D_t + g(D_t)
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+ \gamma \sum^N_{i=1} l^i(\vec s_t, D_t)
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+ \Gamma \sum^N_{i=1} x^i_t
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\end{align}
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The resulting $N$ optimality conditions are:
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\begin{align}
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0 =& -\der{F(x^i_t)}{s^i_t}{}
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+ \beta \left[
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\nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1})
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\cdot
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\nabla_{\vec s_{t}, D_{t}}[\vec s_{t+1} ~ D_{t+1}]
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\right]
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~~\forall~~i \\
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0 =& -\vec f +\beta \left[B\cdot \nabla W_{t+1} \right]
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\end{align}
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And the $N+1$ envelope conditions are:
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\begin{align}
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\nabla_{\vec s_{t}, D_{t}} W(\vec s_t, D_t) =&
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\sum^N_{i=1} \nabla_{\vec s_{t}, D_{t}} u^i(\vec s_t, D_t)
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%- \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
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\notag \\
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&+ \beta \left[ \nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1})
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\cdot \nabla_{\vec s_{t}, D_{t}} [\vec s_{t+1} ~ D_{t+1}]
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\right] \\
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\nabla W_t =& \vec U + \beta \left[C \cdot \nabla W_{t+1} \right]
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\end{align}
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\end{document}
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