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49 lines
2.4 KiB
TeX
49 lines
2.4 KiB
TeX
\begin{align}
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W(\{s^i_t\},D_t) =& \max_{\{x^i_t\}^N_{i=1} \geq 0}
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~~\left( \sum^N_{i=1} u^i(s^i_t,S_t,D_t)\right) - FX_t
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+ \beta W(\{s^i_{t+1}\}, D_{t+1}) \\
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\text{Subject To:}& \notag\\
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D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} s^i_t l^i(s^i_t,S_t,D_t) \right) + g(D_t) \\
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s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
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S_t =&\sum_{i=1}^N s^i_t \\
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X_t =&\sum_{i=1}^N x^i_t
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\end{align}
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Solving for the euler equation follows the steps laid out in
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the section
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% appendix section \ref{APX:Derivations:Constellation}
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for constellation operators.
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\subsubsection{Characterizing solutions}
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The $N+1$ Envelope Conditions are:
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\begin{align}
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\parder{W}{s_t^i}{} =& \sum^N_{j=1} \der{u^j}{s_t^i}{}
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+ \beta \left[ \sum^N_{j=1} \parder{W}{s_{t+1}^j}{} \parder{s_{t+1}^j}{s_t^i}{}
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+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{s_t^i}{} \right]
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~~~ \forall i \in \{1,\dots,N\} \\
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\parder{W}{D_t}{} =& \sum^N_{j=1} \der{u^j}{D_t}{}
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+ \beta \left[ \sum^N_{j=1} \parder{W}{s_{t+1}^j}{} \parder{s_{t+1}^j}{D_t}{}
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+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{} \right] \\
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\nabla W_t - \sum^N_{j=1} \nabla u^j_t =& \beta B \cdot \nabla W_{t+1}
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\end{align}
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Assuming $B$ is non-singular, we again find that:
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\begin{align}
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\nabla W_{t+1} =& (\beta B)^{-1} (\nabla W_t - \sum^N_{j=1} \nabla u^j_t) \label{EQ:viii}
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\end{align}
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The $N$ Optimality Conditions are:
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\begin{align}
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0 =& -F + \beta \left[ \sum^N_{j=1} \parder{W}{s^j_{t+1}}{} \parder{s^j_{t+1}}{x^i_t}{}
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+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{x^i_t}{}\right]
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~~~ \forall i \in \{1,\dots,N\} \label{EQ:ix}\\
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\frac{F}{\beta} \vect{1} =& C \nabla W_{t+1}
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% = C(\beta B)^{-1} (\nabla W_t - \sum^N_{j=1} \nabla u^j_t) \label{EQ:viii}
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\end{align}
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Where $C$ is a $N \times N+1$ matrix.
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Iterating \cref{EQ:ix} one period forward (from $t+1$ to $t+2$) for $i=1$ and and substituting in \cref{EQ:viii}
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twice provides the final equation for a system of $N+1$ equations for $\nabla W_t$.
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Finally, iterating \cref{EQ:ix} one period backward (from $t+1$ to $t$) for all $i$, and substituting the
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previously found values for $\nabla W_t$ into these optimality conditions defines the system of euler equations
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that characterize $\{x^i_t\}$.
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