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119 lines
5.3 KiB
TeX
119 lines
5.3 KiB
TeX
%Given the following bellman equation
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\begin{align}
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V^i(s^i_t,S_t,D_t) =& \max_{x^i_t \geq 0} ~~ u^i(s^i_t,S_t,D_t) - Fx^i_t + \beta V^i(s^i_{t+1}, S_{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) \label{law_motion:debris}\\
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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 \label{law_motion:private_stock}\\
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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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Where $V^i(\cdot)$ represents the value function for the constellation $i$ and $\beta$ represents a common
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discount factor across operators.
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\subsubsection{Characterizing solutions}
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These give the optimality condition:
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\begin{align}
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\frac{F}{\beta} =& \parder{V^i}{s^i_{t+1}}{}
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+ \parder{V^i}{S_{t+1}}{}
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+ m\parder{V^i}{D_{t+1}}{}
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\end{align}
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Iterating both forward and backward one condition gives the system
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\begin{align}
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\frac{F}{\beta} =& \parder{V^i}{s^i_{t}}{}
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+ \parder{V^i}{S_{t}}{}
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+ m\parder{V^i}{D_{t}}{} \label{EQ:vi}\\
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\frac{F}{\beta} =& \parder{V^i}{s^i_{t+1}}{}
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+ \parder{V^i}{S_{t+1}}{}
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+ m\parder{V^i}{D_{t+1}}{} \label{EQ:vii}\\
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\frac{F}{\beta} =& \parder{V^i}{s^i_{t+2}}{}
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+ \parder{V^i}{S_{t+2}}{}
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+ m\parder{V^i}{D_{t+2}}{} \label{EQ:viii}
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\end{align}
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The general envelope conditions are:
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\begin{align}
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\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{}=& \beta\left[
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\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{s^i_t}{}
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+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{s^i_t}{}
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+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{s^i_t}{}
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\right] \\
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\parder{V^i}{S_{t}}{} - \parder{u^i}{S_t}{} =& \beta\left[
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\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{S_t}{}
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+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{S_t}{}
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+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{S_t}{}
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\right] \\
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\parder{V^i}{D_{t}}{} - \parder{u^i}{D_t}{} =& \beta\left[
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\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{D_t}{}
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+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{D_t}{}
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+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{}
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\right]
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\end{align}
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Note the linearity of the equations.
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This allows us to rewrite the system of envelope conditions as the following matrix expression.
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\begin{align}
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\beta
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\left[
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\begin{matrix}
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\parder{s^i_{t+1}}{s^i_t}{} & \parder{S_{t+1}}{s^i_t}{} & \parder{D_{t+1}}{s^i_t}{} \\
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\parder{s^i_{t+1}}{S_t}{} & \parder{S_{t+1}}{S_t}{} & \parder{D_{t+1}}{S_t}{} \\
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\parder{s^i_{t+1}}{D_t}{} & \parder{S_{t+1}}{D_t}{} & \parder{D_{t+1}}{D_t}{}
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\end{matrix}
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\right]
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\left[
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\begin{matrix}
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\parder{V^i}{s^i_{t+1}}{} \\
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\parder{V^i}{S_{t+1}}{} \\
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\parder{V^i}{D_{t+1}}{}
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\end{matrix}
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\right]
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=&
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\left[
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\begin{matrix}
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\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{} \\
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\parder{V^i}{S_{t}}{} - \parder{u^i}{S_t}{}\\
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\parder{V^i}{D_{t}}{} - \parder{u^i}{D_t}{}
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\end{matrix}
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\right] \\
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\beta A \nabla_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i =& \nabla_{[s^i_t,S_t,D_t]} V^i - \nabla_{[s^i_t,S_t,D_t]} u^i \\
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\beta A \nabla V^i_{t+1} =& \nabla V^i_t - \nabla u_t^i ~~\text{for conciseness}
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\end{align}
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Solving for $\nabla V^i_{t+1}$, we get
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\begin{align}
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\nabla V^i_{t+1} =& (\beta A)^{-1} (\nabla V^i_t - \nabla u_t^i) \label{EQ:iv}
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\end{align}
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By iterating \eref{EQ:iv} one period, we get:
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\begin{align}
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\beta A \nabla V^i_{t+2} =& \nabla V^i_{t+1} - \nabla u_{t+1}^i \\
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\nabla V^i_{t+2} =& (\beta A)^{-1}
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\left( (\beta A)^{-1} (\nabla V^i_t - \nabla u_t^i)- \nabla u_{t+1}^i \right) \label{EQ:v}
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\end{align}
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With \cref{EQ:iv,EQ:v} substituted into the system of \cref{EQ:vi,EQ:vii,EQ:viii}, we can
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now solve for the optimal, functional form of $\nabla_{[s^i_t,S_t,D_t]} V^i$.
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Substituting this back into \cref{EQ:vi} gives the euler equation for the optimal launch function
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$x^i_t(s^i_t,S_t,D_t)$.
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\subsubsection{Conditions for existence of a solution}
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For any given set of functional forms $l^i(\cdot),g(\cdot)$ and coefficients $m,M$,
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one must verify that $A$ is invertible for all
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values of the state and choice variables $s^i_t,S_t,D_t$, and $x^i_t$.
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For the laws of motion \cref{law_motion:debris,law_motion:private_stock}, the matrix $A$ above is:
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\begin{align}
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\left[
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\begin{matrix}
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1- l^i(\cdot) - s^i_t \parder{l^i}{s^i_t}{}
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& 1-l^i(\cdot) - s^i_t \parder{l^i}{s^i_t}{} - \sum_{j=1}^N s^j_t \parder{l^j}{S_t}{}
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& M\left[\parder{l^i}{s^i_t}{} + \sum^N_{j=1} \parder{l^i}{S_t}{} \right] \\
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- s^i_t \parder{l^i}{S_t}{}
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& - \sum_{j=1}^N s^j_t \parder{l^j}{S_t}{}
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& M \sum^N_{j=1} \parder{l^i}{S_t}{} \\
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- s^i_t \parder{l^i}{D_t}{}
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& - \sum_{j=1}^N s^j_t \parder{l^j}{D_t}{}
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& (1-\delta) + M \sum^N_{j=1} \parder{l^i}{D_t}{} + \parder{g}{D_t}{} \\
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\end{matrix}
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\right]
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\end{align}
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