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45 lines
1.8 KiB
TeX
45 lines
1.8 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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With the laws of motion introduced in sections \cref{SEC:Laws}, we can now describe
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the optimization problem facing each constellation operator in general terms.
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Actual functional specifications are described in \cref{SEC:Computation} on computation.
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Each operator recieve per-period benefits
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-- such as profits for firms and warfighting capability for militaries --
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from their constellation
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according to $u^i(S_t,D_t)$, 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 a general cost function $F(x)$.
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These satellites will become operational in the subsequent period.
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Thus the $M$-period (possibly infinite), problem is:
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\begin{align}
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\max_{\{x_t^i\}^M}&~
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\left[ \sum^M_{t=0} \beta^t u^i(S_t, D_t) - F(x^i_t) \right] \\
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&\text{subject to:}\\
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& s^j_{t+1} = R^j(S_t, D_t) s^j_t + x^j_t ~~~ \forall j \\
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& D_{t+1} = (1-\delta + g) D_t
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+ \gamma \sum^N_{i=1} \left( 1-R^i(S_t, D_t) \right) s^i_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} % Not sure how much help a new header is.
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The inifinite period version of the problem above can be rewritten in the bellman form as
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\begin{align}
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V^i(S_t, x^{\sim i}_t, D_t) = \max_{x^i_t} u^i(S_t, D_t) -F(x^i_t)
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+ \beta \left[ V^i(S_{t+1}, 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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One important point is 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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\end{document}
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