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379 lines
16 KiB
TeX
379 lines
16 KiB
TeX
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%----------------------------------------------------------------------------------------
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% TITLE PAGE
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%----------------------------------------------------------------------------------------
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\title[Orbits]{Dynamic Launch Decision for Satellite Constellation Operators}
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%Constellations in orbit
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\author{Will King} % Your name
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\institute[WSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
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{
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Washington State University \\ % Your institution for the title page
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\medskip
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\textit{william.f.king@wsu.edu} % Your email address
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}
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\date{\today} % Date, can be changed to a custom date
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\begin{document}
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\begin{frame}
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\titlepage % Print the title page as the first slide
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\end{frame}
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\section{Introduction}
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\begin{frame}
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\frametitle{Introduction}
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\begin{block}{ESA -- Sep. 2019}
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For the first time ever, ESA has performed a 'collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic \autocite{EsaTweet}
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\end{block}
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In 1978,Donald Kessler and Burton Cour-Palais identified a potential threat to the new frontier of Earth Orbit.
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They suggested that if there are enough objects in orbit, debris colliding with other debris and artificial
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satellites could create debris at an increasing rate, leading to an uncontrollable cascade of collisions,
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now termed kessler syndrome \autocite{Kessler1978}.
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My goal is to evaluate how the organization of satellite operations into ``constellations'' affects
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pollution dynamics and the incentives of operators to deviate from socially optimal behaviors.
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\end{frame}
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\begin{frame}
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\frametitle{Overview}
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\tableofcontents
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\end{frame}
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%----------------------------------------------------------------------------------------
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% PRESENTATION SLIDES
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%----------------------------------------------------------------------------------------
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\section{Literature}
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\begin{frame}
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\frametitle{Literature}
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\begin{itemize}
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\item \autocite{Macauley_1998} : Estimates the welfare loss due to inefficient allocation
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of geostationary orbit slots.
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\item \autocite{adilov_alexander_cunningham_2015} : Two period model evaluating launch decisions.
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\item \autocite{adilov_alexander_cunningham_2018} : Develop an economic Kessler syndrome
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where pollution is sufficient to halt launches.
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\item \autocite{RaoRondina2020} : A widely cited working paper developing the
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first dynamic model of orbit allocations. Originates in Rao's dissertation from 2015.
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\item \autocite{Adilov2019} : Develops a dynamic model evaluating competitive interactions
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between firms.
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\item \autocite{Rao2020} : Estimates the impact of implementing satellite taxes on future
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profitability of the satellite industry.
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\end{itemize}
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\end{frame}
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%----------------------------------------------------------------------------------------
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\section{Model}
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\begin{frame}
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\frametitle{High level description}
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This model is the first dynamic model to incorporate effects from organization as constellations.
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These effects enter in two forms:
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\begin{enumerate}
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\item Economies of scale in value production.
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\item Collision avoidance efficiencies from constellation planning.
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\end{enumerate}
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Key features of this model are:
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\begin{itemize}
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\item The assumption that each constellation creates utility without
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competitive interactions (i.e. monopolistically).
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\item Each satellite within a constellation is considered identical.
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Only the number of satellites contributes to the value produced.
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\end{itemize}
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These features simplify computation significantly.
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\end{frame}
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%------------------------------------------------
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\begin{frame}
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\frametitle{Mathematical Terms}
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\begin{tabular}{| p{0.17\linewidth} | p{0.2\linewidth} p{0.5\linewidth} | }
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\hline
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Symbol & Details & Description \\
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\hline
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$N$ & $N>0$ & Number of constellations \\
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\hline
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$s^i_t$ & $i \in \{1,\dots,N\}$ & Satellite stock of $i$ in $t$ \\
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\hline
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$x^i_t$ & Ditto & Launches of satellites in $t$ by $i$ \\
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\hline
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$S_t$ & & Total number of satellites in $t$ \\
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\hline
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$D_t$ & $D_t \geq 0$ & Level of debris in $t$ \\
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\hline
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$m,M$ & $m>0,M>0$ & Debris generated from launches and collisions respectively \\
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\hline
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$g(D_t)$ & & Debris generated from collisions with debris \\
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\hline
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$\delta$ & $\delta \in (0,1)$ & Decay rate of debris \\
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\hline
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$l^i(s^i_t,S_t,D_t)$ & $l^i() \in (0,1)$ & Rate of satellite loss in $i$ due to collisions \\
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\hline
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$u^i(s^i_t,S_t,D_t)$ & & Utility generated by satellite stock $s^i_t$ given $S_t,D_t$.\\
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\hline
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\end{tabular}
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\end{frame}
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%------------------------------------------------
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\subsection{Constellation Operator's Problem}
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\begin{frame}
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\frametitle{Constellation Operator's Problem}
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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 ~~~ X_t =\sum_{i=1}^N x^i_t
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\end{align}
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\end{frame}
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%------------------------------------------------
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\begin{frame}[allowframebreaks]
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\frametitle{Solving Constellation's Problem}
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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}}{} - \der{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] \label{EQ:env1}\\
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\parder{V^i}{S_{t}}{} - \der{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] \label{EQ:env2}\\
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\parder{V^i}{D_{t}}{} - \der{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] \label{EQ:env3} \\
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\nabla V^i_t - \nabla u_t^i =& \beta A \nabla V^i_{t+1}
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\end{align}
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The optimality conditions is:
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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 period 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:opt1}\\
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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:opt2}\\
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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:opt3}
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\end{align}
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Thus by iterating \cref{EQ:env1,EQ:env2,EQ:env3} to match \cref{EQ:opt1,EQ:opt2,EQ:opt3},
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we can simplify from 9 equations with 9 unknowns to 3 equations with 3 unknowns
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allowing us to solve for $\nabla_{[s^i_t,S_t,D_t]} V_t$
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in terms of derivatives of the utility function and derivatives of the laws of motion.
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Substituting $\nabla V_t$ into the equation below (\cref{EQ:opt1}) provides the euler equation
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that characterizes the policy function $x^i_t(s^i_T,S_t,D_t)$.
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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}}{} \notag
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\end{align}
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\end{frame}
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%------------------------------------------------
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\subsection{Social Planner's Problem}
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\begin{frame}
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\frametitle{Social Planner's Problem}
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We can address the social planner's problem in much the same way.
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\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 ~~~ X_t =\sum_{i=1}^N x^i_t
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\end{align}
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\end{frame}
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%------------------------------------------------
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\begin{frame}[allowframebreaks]
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\frametitle{Solving Planner's Problem}
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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\} \label{EQ:S:env1}\\
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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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\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:S:opt1}
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\end{align}
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Iterating \cref{EQ:S:opt1} one period forward (from $t+1$ to $t+2$) for $i=1$ and and substituting in the
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correctly iterated envelope conditions provides
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the final equation for a system of $N+1$ optimality conditions for $\nabla W_t$.
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Once again, iterating \cref{EQ:S:opt1} backwards from $t+1$ to $t$ and substituting in $\nabla W_t$ will allow
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you to find the $N$ euler equations characterizing the policy functions $\{x^i_t\}$.
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%----------------------------------------------------------------------------------------
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\section{Analysis}
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\subsection{Welfare Analysis}
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\end{frame}
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\begin{frame}
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\frametitle{Welfare analysis}
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A standard result in the models mentioned in the slide on previous work is that of how free
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entry or competitive use results in launching more than the socially optimal number of satellites.
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I suspect that result will hold true in this model.
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\textit{Unfortunately I have not been able to do more than these derivations.
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The welfare analysis will involve some numerical methods at some point as it gets very messy.}
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\end{frame}
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%------------------------------------------------
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\subsection{Survival Rates}
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\begin{frame}
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\frametitle{Survival Rates}
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One key analysis in \cite{RaoRondina2020} is about the survival rates of satellites.
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Define the survival rate for a constellation and the society to be:
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\begin{align}
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R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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R =& \frac{S_{t+1}- X_t}{S_t} = \frac{\sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] }{S_t}
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\end{align}
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\end{frame}
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\begin{frame}
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\frametitle{Survival Rates}
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The marginal survival rates when a given constellation $i$ changes size are:
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\begin{align}
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\parder{R_i}{s^i_t}{} =& -\left(\parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{} \right)
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\leq 0 \\
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\parder{R}{s^i_t}{} =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i}
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\end{align}
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\end{frame}
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\begin{frame}
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\frametitle{Survival Rates}
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Thus society's marginal survival rate is less than the weighted arithemetic mean of
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survival rates for individually growing constellations when:
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\begin{align}
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\sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}
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\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
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\sum_{i=1}^N R_i - R \leq& 0\\
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\sum_{i=1}^N [1- l^i(s^i_t,S_t,D_t)] - \sum_{i=1}^N s^i_t [1- l^i(s^i_t,S_t,D_t)] \leq& 0\\
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\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0
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\end{align}
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This condition is met as every constellation consists of at least one satellite.
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\end{frame}
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%------------------------------------------------
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\begin{frame}
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\subsection{Kessler Syndrome}
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\frametitle{Economic Kessler Syndrome}
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\cite{adilov_alexander_cunningham_2018} develop a description of economic kessler syndrom
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as when the debris and satellite stocks are such that it is not profitable to launch.
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Mathematically this is:
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\begin{align}
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\vartheta_3 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) = 0 \right\}
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\end{align}
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This definition has the benefit that it can be found through a numerical search directly on
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the euler equations developed previously.
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\textit{Again, I have not been able to implement this analysis.}
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\end{frame}
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%------------------------------------------------
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\section{Conclusion}
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\begin{frame}
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\frametitle{Conclusion}
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\textbf{Summary:}
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In this paper I have described a model and general set of euler equations describing
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the decisions facing satellite constellation operators.
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Additionally I have established that negative pollution externalities exist, consitent
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with other models.
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This model provides a basis for analyses of competitive and non-competitive
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interaction between constellation operators, and for the analysis of policy interventions.
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\textbf{Future Work:}
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There remains significant work to finalize the model, including exploring a numerical model,
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clarifying existence criteria, and verifying if constellation operators are likely to overuse orbits.
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\end{frame}
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%----------------------------------------------------------------------------------------
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\begin{frame}[allowframebreaks]
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\frametitle{References}
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\printbibliography
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\end{frame}
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\end{document}
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