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@ -2,7 +2,120 @@
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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Introduction goes here
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\subsection{testing}
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This is a subsection of the introduction.
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With the laws of motion introduced in sections \cref{asdf}, we can now describe
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the optimization problem facing each constellation operator.
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Each operator recieve utility in each period per
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their per period utility $u^i(\vec s_t,D_t)$, which depends
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on the current sizes of constellations and the level of debris.
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In addition, the operator pays for the launch of $x^i_t$ satellites
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according to the cost function $F(x)$.
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These satellites will become operational in the next period.
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Thus the $M$-period (possibly infinite), problem is:
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\begin{align}
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\max_{\{\vec x_t\}^M}&~
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E\left[ \sum^M_{t=0} \beta^t u^i(\vec s_t, D_t) - F(x^i_t) \right] \\
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&\text{subject to:}\\
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& s^i_{t+1} = (1-l^i(\vec s_t, D_t))s^i_t +x^i_t ~~~ \forall i \\
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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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%Assumptions
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% - Identical launch costs
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% - Identical debris production from destruction.
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%
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\subsection{Infinite Period (Bellman) Equation}
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The problem above can be rewritten in the bellman form as
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\begin{align}
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V^i(\vec s_t, \vec x^{\sim i}_t, D_t) = \max_{x^i_t} u^i(\vec s_t, D_t) -F(x)
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+ \beta \left[ V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) \right]
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\end{align}
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where $x^{\sim i}_t$ represents the launch decisions of all the other constellation
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operators.
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This implies that the policy function is a best response function, allowing for
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a nash equilibrium interpretation of the result.
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To solve for the policy function, we have a variety of methods available.
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Due to the computational method chosen later, I'm going to examine the conditions
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for the existence of an euler equation.
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\subsubsection{Euler Equation}
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First, find the single optimality condition
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\begin{align}
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0 =& \parder{}{x^i_t}{} u^i(\vec s_t, D_t) -\parder{}{x^i_t}{}F(x)
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+ \beta \left[ \parder{}{x^i_t}{}
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V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1})
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\right] \\
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0 =& -\der{F}{x^i_t}{}
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+ \beta \left[
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\nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}}
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V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1})
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\cdot
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\nabla_{x^i_t} [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
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\right] \label{EQ:OptimalityCondition}\\
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0 =& -\der{F}{x^i_t}{} + \beta \nabla V^i_t \cdot \vec a_t
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\label{EQ:SimplifiedOptimalityCondition}
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\end{align}
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Second, the $2N$\footnote{recall that $N$ is the number of constellations.}
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envelope conditions can also be found:
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\begin{align}
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\nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} V^i(\vec s_t, \vec x^{\sim i}_t, D_t)
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=& \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} u^i(\vec s_t, D_t) \notag \\
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&- \der{}{x}{}F(x^i(\vec s_t, \vec x^{\sim i}_t, D_t)) \cdot
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\nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} x^i(\vec s_t, \vec x^{\sim i}_t, D_t) \notag\\
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&+ \beta \left[
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\nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} }
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V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1})
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\cdot
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\nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
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[ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
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\right] \label{EQ:EnvelopeConditions}
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\\
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\nabla \vec V^i_t =& \vec u^i - \vec f + \beta A \cdot \nabla \vec V^i_{t+1}
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\label{EQ:SimplifiedEnvelopeConditions}
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\end{align}
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When interpreting this, note that
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$$
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\nabla \vec V^i_{t+1} = \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} }
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V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1})
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$$
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is a $2N \times 1$ vector of first derivatives but
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$$
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A = \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
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[ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
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$$
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is a $2N \times 2N$ matrix of first derivatives.
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By solving for $\vec V^i_{t+1}$ as a function of $\vec V^i_{t}$ we get the
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intertemporal condition:
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\begin{align}
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\frac{1}{\beta} A^{-1} \left(\nabla \vec V^i_t - \vec u^i +\vec f \right)
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= \nabla \vec V^i_{t+1}
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\end{align}
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Thus one crucial condition for the existence of a solution is that $A^{-1}$ exists for
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all values the laws of motion and choice functions can take.
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% \subsection{Existence}
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% I need to do some more diving into conditions for existence.
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% Of particular concern is that the way I have specified the debris may lead to
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% non-convergence.
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%
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Finally, to construct the euler equation, we take
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\cref{EQ:SimplifiedOptimalityCondition}
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and iterate it forward $2N-1$ times.
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By substituting
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\cref{EQ:SimplifiedEnvelopeConditions}
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into each iteration enough times
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you get a system that defines $\nabla V^i_t$
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By substituting this defined value of $\nabla V^i_t$ into
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\cref{EQ:SimplifiedOptimalityCondition}
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one final time, we get a function that fully determines the policy function.
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\cref{EQ:SimplifiedOptimalityConditions,EQ:SimplifiedEnvelopeConditions}
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\end{document}
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youainti
5 years ago