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55 lines
2.9 KiB
TeX
55 lines
2.9 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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In \cite{Kessler1978} the authors described and forecasted what has come to be
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known as ``kessler syndrome'', where debris collides with itself in such a way that
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the overall debris level grows exponentially.
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A few methods have been used to model this behavior in the economics literature.
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The first one I want to explain was developed by \cite{Adilov2018}.
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They characterize kessler syndrome as the point in time at which an orbit is
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unusable as each satellite launched will be destroyed within a single time period.
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In my notation, this is that $l^i(\{s^j_t\}, D_t) = 1$.
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The benefit of this approach is that it is algebraically simple.
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It was used in this role to show that firms will stop launching before
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orbits are rendered physically useless.
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Unfortunately, it does not convey the original intent of ``kessler syndrome'',
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i.e. a runaway pollution effect, but instead corresponds to the end result of kessler syndrome.
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The second common definition of ``kessler syndrome'' is due to \cite{RaoRondina}.
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They define it in terms of a ``kessler region'', the set of satellite stocks and the debris level
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such that:
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\begin{align}
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\kappa = \left\{ \{s^j_t\}, D_t :
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\lim_{k\rightarrow \infty} D_{t+k}\left(\{s^j_{t+k-1}\}, D_{t+k-1}, \{x^j\}\right) = \infty \right\}
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\end{align}
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\subsection{My approach to kessler syndrome}
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I propose to analyze kessler syndrome in a slightly more restricted fashion than \cite{RaoRondina}.
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I would define the $\epsilon$-kessler region as:
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\begin{align}
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\kappa = \left\{ \{s^j_t\}, D_t :
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\forall k \geq 0, D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
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\end{align}
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It is easily shown that this criteria is sufficient to guarantee Rao and Rondina's criteria.
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It has three primary benefits:
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\begin{itemize}
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\item % Can be solved for algebraically or numerically for a given, bounded state space.
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The kessler region can be numerically described within bounded state spaces.
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\item % This is what you would actually compute.
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In a Computational General Equilibrium Model, as most models of any complexity will be,
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you cannot check for divergence numerically.
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The condition given is a basic guarantee of the divergent behavior that is
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required for Kessler Syndrome and acknowledges computational limitations.
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\item Finally, a slow divergence is no divergence in the grand scheme of things.
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It is possible to have a mathematically divergent function, but one that is so slow,
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there is no noticable degree of debris growth before Sol enters a red giant phase.
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In this specification, it is possible to choose $\epsilon$ such that the divergent behavior
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has an impact on a meaningful timescale.
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% \item % Stochastic versions could might be describable as martigales.
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\end{itemize}
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\end{document}
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