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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
\author{William King}
\institute{Washington State University}

\begin{document}

\maketitle

\begin{abstract}
    Over the last 10 years new technology has make low earth orbits (LEOs) more accessible, and 
    the resulting increase in LEO satellites has increased the risk of collision.
    Because debris in orbit generates more debris through collisions with objects in orbit 
    and the debris created during launch and operation imposes a negative externality
    on other operators,
    optimal use of orbits is believed to not occur under free entry.
    This paper develops a dynamic model of satellite operation incorporating two effects not considered 
    in previous models.
    The first effect is complementarity between same-purpose, single operator fleets (called constellations).
    The second effect is collision avoidance efficiencies that exist within constellations.
    The primary result is a theoretical model and the resulting analysis of the difference in survival ratios between 
    constellation operators and society.
\end{abstract}

\keywords{Orbits, Pollution, Economies of Scale, Externality  }

\jel{Q29, Q58, L25}

\newpage

% ---------------------------------------------------------------------------------------
\section{Introduction}
% Motivating Example (ESA - SpaceX)
In September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
adjustment maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
such coordination will become more difficult as the size and number of 
single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.

% Background on issues of congestion and pollution
% Kessler Syndrome
In spite of the fact that there is a lot of maneuvering room in outer space,
%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is. 
%I mean, you may think it’s a long way down the road to the chemist, 
%but that’s just peanuts to space.''\cite{DouglasAdams}}
the repeated interactions of periodic orbits make collisions probable.
Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
Congestion effects are primarily derived from avoiding collisions between artificial satellites.
Pollution in orbit consists of debris, both natural and man-made, which increases
the probability of an unforseen collision.
The defining dynamic of pollution in orbit is that it self-propogates as debris collides with itself
and orbiting satellites to generate more debris.
This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
that with sufficient mass in orbit (through satellite launches), the debris generating process
could undergo a runaway effect rendering various orbital regions unusable.
This cascade of collisions is often known as Kessler syndrome and theoretically 
may take place over various timescales.

% ---------------
Orbits may be divided into three primary groups, 
Low Earth Orbit (LEO, less than 2,400km in altitude\autocite{FAA2020}), 
Medium Earth Orbit (MEO), and High Earth Orbit (HEO) with Geostationary Earth Orbit (GEO) 
considered a particular classification of orbit. 
While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
a variety of new empircal studies and theoretical models published.
In general, three primary, related topics appear in the literature: 
Allocative Efficiency, Externality Mitigation, and Economic vs Physical Kessler Syndromes.

% ---------------
Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits 
physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
benefit of new launches negative before the physical kessler syndrome occurs.

% ---------------
%Allocative efficiency
The primary concern is to establish wether or not orbits will be overused
due to their common-pool nature, and if allocation procedures are efficient.
The earliest theoretical model I have found, due to Adilov, Alexander, and 
Cunningham \autocite{adilov_alexander_cunningham_2015}, examines pollution
using a two-period salop model, incorporating the effects of launch debris on 
survival into the second period. 
They find that the social planner generates debris and launches at lower rates 
than a free entry market.
This same result was found by Rao and Rondina \autocite{RaoRondina2020} in
the context of an infinite period dynamic model. 
They approach the problem in the case where numerous operators in a free entry environment
can each launch a single, identical satellite.


% ---------------
In addition to analyzing the allocative results, a significant area of interest is 
what impact various policy interventions can have.
The policies analyzed and methods used have been widely varied.
Macauley \autocite{Macauley_1998} provided the first evidence of suboptimal behavior in orbit
by estimating the welfare lose due to the current method of assigning GEO slots to operators.
The potential losses due to anti-competitive behavior was highlighted by Adilov et al \autocite{Adilov2019},
who have analyzed the opportunities for strategic 
``warehousing'' of non-functional satellites as a means of increasing competitive advantage by 
denying operating locations to competitors in GEO.
Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris)
and cleanup.
Finally, Rao and Rondina \autocite{RaoRondina2020b} estimate that achieving socially optimal 
behavior through orbital use fees could increase the value generated by the space industry by a factor of 4.


% ---------------
This paper's objective is to devlop a dynamic model which incoporates 
complementary effects of constellations as well as collision avoidance efficiencies of constellations,
thus addressing a gap in the current literature.
In addition, I examine if there exists a negative externality related to changes in stock size, and
establish a condition related to average behavior that describes this externality.
Finally, I lay foundations for the derivation of profit maximizing launch rules.

The paper is organized as follows. 
Section \ref{Model} describes the mathematical organization of the model 
for the cases of independent constellation operators and a social planner 
operating the same constellations.
%It also includes a brief digression into the free entry conditions.
Section \ref{Comparisons} evaluates the differences between the 
constellation operators and social planner models, particularly 
the difference between marginal survival rates .
%Of particular interest is the difference in launch rates and marginal survival rates.
%Section \ref{Kessler} ...
Section \ref{Conclusion} concludes with a discussion of potential extensions and
topics which have not yet been addressed.

% ---------------------------------------------------------------------------------------
\section{Model}\label{Model}
%Intuitive description
The dynamic model is an extension of Rao and Rondina's working paper \autocite{RaoRondina2020},
specifically their non-stochastic model.
For a given orbital shell (a set of orbits that interact regularly), I assume there are $N$ operators, 
each of which has the potential to launch and operate a satellite
constellation consisting of some endogenosly chosen number of identical satellites.
These satellites are not only identical within a constellation, but across constellations.


% -------------------
Each constellation operator has a personal satellite stock $s^i_t$ in each period, and chooses the 
number of launches in that time period $x^i_t$.
For simplicity, each launch is assumed to have a fixed cost $F$.
In the aggregate, the satellite stock and launches for each period are represented by:
\begin{align}
    S_t =&\sum_{i=1}^N s^i_t \\
    X_t =&\sum_{i=1}^N x^i_t 
\end{align}

% -------------------
Satellites in a constellation are damaged or destroyed at the rate $l^i(s^i_t,S_t,D_t)$,
which is assumed to be increasing in $s^i_t$, $S_t$, and $D_t$ (debris, see below). 
One key difference from the previous models of Rao and Redina \autocite{RaoRondina2020} and 
Adilov et al \autocite{adilov_alexander_cunningham_2018} is that this model allows the rate of 
collision within constellations and between constellations to be different.
This reflects the assumption that an operator can and will put more effort into protecting the satellites within
the constellation from each other.
One example of how this can be acomplished is that while choosing the orbits for a constellation, 
it is possible for an operator to chose a set of trajectories that best meet their needs and 
minimizes the risk of collision within the constellation. 
Mathematically this is represented by the inclusion of $s^i_t$ in  $l^i$.
Together with the launch rate, we obtain a law of motion for both constellation-level 
and society-level satellite stocks.
\begin{align}
    s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
    S_{t+1} =& X_t + \sum^N_{i=1} \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t
\end{align}

% -------------------
The level of debris in each period is represented by $D_t$, and is assumed to pose a latent risk.
In particular, it is assumed that once debris is created, the risk it provides is only avoidable 
through not launching future satellites.
In addition to natually occuring debris, debris is generated through the following three mechanisms.
\begin{itemize}
    \item At launch, various processes can shed debris. 
        Examples include leftover rocket stages, explosions during launch and deployment,
        and slag from solid rocket boosters.
    \item When destroyed, satellites will fragment and produce debris.
    \item Debris can collide with other debris, forming more but smaller debris.
\end{itemize}
This provides the following law of debris dynamics.
\begin{align}
    D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t)
\end{align}
where $\delta$ represents the decay of debris -- through reentering the atmosphere -- for a given shell,
$M$ represents the debris generated from each collision, 
$m$ represents the debris generated from each launch, 
and $g(D_t)$ represents the new fragments from debris colliding with other debris.


% -------------------
Each constellation $i \in {1,\dots,N}$ produces value for their operator at each period according to the function:
\begin{align}
    u^i(s^i_t, S_t, D_t) = u^i(s^i_t)
\end{align}
For computational simplicity, it is assumed that benefits provided are wholely dependent on the number
of satellites in operation. 
The approach presented in the appendix is generalizable to the case where benefits are conditional on
the total satellite and debris stocks.
Complementarity within a constellation appears when $\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$.

% ---------------------------------------------
\subsection{Constellation Operator's Program}
The aformentioned aspects of the model form the following bellman equation for each constellation operator.
\begin{align}
    V^i(s^i_t,S_t,D_t) =& \max_{x^i_t \geq 0} ~~ u^i(s^i_t) - Fx^i_t + \beta V^i(s^i_{t+1}, S_{t+1}, D_{t+1}) \\
    \text{Subject To:}& \notag\\
    D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t) \\
    s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
    S_t =&\sum_{i=1}^N s^i_t \\
    X_t =&\sum_{i=1}^N x^i_t  % Is this also a state variable?
\end{align}
The system of envelope conditions is linear and can be written as a matrix equation.
In Appendix \ref{APX:Derivations:Constellation} I begin development of the euler equation
in a generalizable way. 
Unfortunately repeated errors in the mathematics has prevented me from achieving more in the 
launch rate analysis to date.

%The resulting euler equation is:
%\begin{align}
%    F \det(A) =&  
%        [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{1} \\
%        &+ 2 [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{2} \notag\\
%        &+ m [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{3} \notag
%\end{align} % TODO: This could also be changed to a matrix form with a row-vector.
%Where $A$ is the matrix of partial derivatives of the laws of motion corresponding to
%the envelope conditions. 
%The matrix $A$ and the derivation of the euler equation 
%is described in Appendix \ref{APX:Derivations:Constellation}
% ---------------------------------------------------------------------------------------
%\subsubsection{Free Entry}
%Operators are assumed to enter the market as long as the value of entering is above 0.

% ---------------------------------------------
\subsection{Social Planner's Program}

The social planner (or fleet planner to use Rao and Rondina's terminology), is tasked with 
maximizing the sum of the operators' benefits $W(\{s^i_t\},S_t,D_t) = \sum^N_{i=1} V^i(s^i_t,S_t,D_t)$.
%Crucial assumption. $\beta^i =\beta \forall i$
Often, in polluting environments, there is an ambient population that is harmed by pollution.
Very rarely does satellite debris pose a hazard to those on earth, thus in this model
the only population whom's welfare is addressed are the satellite operators themselves.

\begin{align}
    W(\{s^i_t\},S_t,D_t) =& \max_{\{x^i_t\}^N_{i=1} \geq 0}
            ~~ \left(\sum^N_{i=1}  u^i(s^i_t,S_t,D_t)\right) - FX_t 
            + \beta W(\{s^i_{t+1}\}, S_{t+1}, D_{t+1}) \\
    \text{Subject To:}& \notag\\
    D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t) \\
    s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
    S_t =&\sum_{i=1}^N s^i_t \\
    X_t =&\sum_{i=1}^N x^i_t 
\end{align}

%Goal: Add the euler equation.
Due to the aformentioned errors, I have not begun a derivation of the optimal launch rate
for the social planner at this point.
I expect it to be solvable using the same approach as for the constellation operators
outlined in Appendix \ref{APX:Derivations:Constellation}.

% ---------------------------------------------------------------------------------------
%\section{Convergence Properties}\label{Convergence}

% ---------------------------------------------------------------------------------------
\section{Comparisons}\label{Comparisons}

% Marginal survival.
In line with theory on common-pool resources, we expect there to be a negative externality
incurred by increasing the satellite stock. 
The details of this externality can be observed in the marginal suvival rate.
Define the survival rate for a constellation and the society to be:
\begin{align}
    R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
    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}
\end{align}
The marginal survival rates when a given constellation $i$ changes size are:
\begin{align}
    \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) 
        =  - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\
    \parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N 
        \left(  [1-l^i(s^i_t,S_t,D_t)] + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right) 
        - \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}{(S_t)^2} \\
    =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i}
\end{align}
Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the average marginal survival rate 
across constellation operators.
The derivation of equation \ref{EQ:i} is in Appendix \ref{APX:Derivations:Survival}.
Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
cannot proceede further without specifying the functional loss forms $l^i(\cdot)$
and specifying which firm the comparison is with.
In spite of this, conditions on the average effects can be specified as follows.
Society's marginal survival rate is greater than the average marginal survival rate when:
% NOTE: Should I do this using absolute value arguments? I don't think so.
\begin{align}
    \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}
        \geq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
    \sum_{i=1}^N  R_i  - R \geq& 0\\
    \sum_{i=1}^N S_t [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)]  \geq& 0\\
    \sum_{i=1}^N (S_t - s^i_t) [1- l^i(s^i_t,S_t,D_t)]  \geq& 0 \label{EQ:ii}
\end{align}
Which is always true as $S_t > s^i_t$ and $l^i(\cdot) \in [0,1]$ for all $i$.
%As we are discussing arithemetic means of rates, it makes sense. 
%A geometric mean might behave differently.
If a single constellation makes up the whole stock of satellites, then \eref{EQ:ii} reduces
to a tautology.
As $\parder{R_i}{s^i_t}{} < 0$ from \eref{EQ:iii} and the assumptions on collision mechanics, we see
that the average marginal survival rate acts as a lower bound on the marginal societal survival rate.
Assuming that survival rates are not increased by adding another satellite i.e. $\parder{R}{s^i_t}{}<0$ then gives
us the following bounds on societal rates $\sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{}<\parder{R}{s^i_t}{}<0$
%By way of interpretation, this means adding a satellite to a constellation has a larger impact 
%on the society's survival rate than on the average survival rate across constellations.
%This result is consistent with previous results establishing a negative externality
This suggests that some operators experience marginal changes to their own satellite 
stocks much more intensely than society as a whole does.

%Do it again using a geometric mean.

%Welfare
% TODO: The efforts to establish optimal launch rates is holding back this section.
Once optimal launch rates have been determined, an evaluation of the welfare effects of open access
policy can be evaluated, in line with much of the current literature.

% ---------------------------------------------------------------------------------------
%\section{Kessler Syndrome}\label{Kessler}
%Discuss the impacts of Kessler Syndrome (an numerical example where this will occur?)
%Raou and Rondina
%Adilov 
% TODO: Not enough material to discuss this yet. I think I'll need relative launch rate information.

% ---------------------------------------------------------------------------------------
%\section{Numerical Model}\label{Numerical}


% ---------------------------------------------------------------------------------------
\section{Concluding Remarks}\label{Conclusion}
The dynamic model developed in this paper provides insight into the incentives faced by
constellation operators in comparison with a social planner and, when completed, should provide
insight on how self-perpetuating externalities drive sub-optimal behavior.

At this point, major work remains in developing optimal launch rates and verifying if
the expected difference in optimal launch rates between individual operators and a social planner exist,
as occurs in other models.
In addition to the remaining work on fleshing out the model, the following extensions and applications of the
model will fill gaps in the literature or complement current work:
\begin{itemize}
    \item Asymmetric constellation sizes: What are the impacts on social welfare when a variety of 
        constellation sizes exist
    \item Policy interventions: Various policy proposals to reduce negative externalities have been proposed,
        including launch quotas, launch taxes, and orbit use fees \autocite{RaoRondina2020b}.
    \item Introduction of stochastics: There are various ways that stochastics can enter the model, from the scales
        determining debris generation to the per-period satellite collision rate. 
    \item Differentiation of satellites and launch methods: Different launch methods and satellite features can 
        affect the accumulation of debris.
    \item Richer satellite lifetimes: the current satellite lifetime of [launch, operate] could be extended 
        to include stages such as development and disposal. 
        In particular, a multiperiod develoment cycle with sunk costs incurred along the way may
        exacerbate problems where stable equilibria are overshot.
        This will allow for more policy interventions to be analyzed.
    \item Strategic behavior: Concerns include whether constellation network effects can be used to prevent new entrants 
        in the case of competition for a satellite services market.
\end{itemize}

While computationally complicated, the results so far imply that there is a defined difference between 
the risks faced at the constellation operator's level and the level of society as a whole. 
%While I expect there to be a socially suboptimal launch rate under open access, as 
Although not a common topic in economics, orbit use has properties that requires
current study in order to determine and drive optimal behavior, before there are no more viable orbits to use.

\newpage
\printbibliography

\newpage
\appendix
\section{Derivations} \label{APX:Derivations}
%\subsection{Useful Mathematical Notes}\label{APX:Derivations:Useful}
%To fill in with a set of useful mathematical notes for use throughout.
%\subsubsection{Useful Derivatives}


\subsection{Constellation Operator}\label{APX:Derivations:Constellation}
Given the following bellman equation
\begin{align}
    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}) \\
    \text{Subject To:}& \notag\\
    D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t) \\
    s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
    S_t =&\sum_{i=1}^N s^i_t \\
    X_t =&\sum_{i=1}^N x^i_t 
\end{align}
Giving the optimality condition:
\begin{align}
    \frac{F}{\beta} =& 2\parder{V^i}{S_{t+1}}{} 
                    + m\parder{V^i}{D_{t+1}}{} 
                    + \parder{V^i}{s^i_{t+1}}{}
\end{align}
Assuming $\parder{u^i}{S_t}{} = 0$ and $\parder{u^i}{D_t}{} = 0$, then the envelope conditions are:
\begin{align}
    \parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{}=& \beta\left[ 
                \parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{s^i_t}{}
                + \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{s^i_t}{}
                + \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{s^i_t}{}
                \right] \\
    \parder{V^i}{S_{t}}{} =& \beta\left[ 
                \parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{S_t}{}
                + \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{S_t}{}
                + \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{S_t}{}
                \right] \\
    \parder{V^i}{D_{t}}{} =& \beta\left[ 
                \parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{D_t}{}
                + \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{D_t}{}
                + \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{}
                \right] 
\end{align}
Note the linearity of the equations. 
This allows us to rewrite the system as the following matrix expression.
\begin{align}
    \beta
    \left[
        \begin{matrix}
            \parder{s^i_{t+1}}{s^i_t}{} & \parder{S_{t+1}}{s^i_t}{} & \parder{D_{t+1}}{s^i_t}{} \\
            \parder{s^i_{t+1}}{S_t}{} & \parder{S_{t+1}}{S_t}{} & \parder{D_{t+1}}{S_t}{} \\
            \parder{s^i_{t+1}}{D_t}{} & \parder{S_{t+1}}{D_t}{} & \parder{D_{t+1}}{D_t}{} 
        \end{matrix}
    \right] 
    \left[
        \begin{matrix}
            \parder{V^i}{s^i_{t+1}}{} \\ 
            \parder{V^i}{S_{t+1}}{} \\
            \parder{V^i}{D_{t+1}}{}  
        \end{matrix}
    \right] 
    =&
    \left[
        \begin{matrix}
            \parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{} \\ 
            \parder{V^i}{S_{t}}{} \\
            \parder{V^i}{D_{t}}{} 
        \end{matrix}
    \right] \\
    A  D_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i =& D_{[s^i_t,S_t,D_t]} V^i - b
\end{align}
The matrix $A$ above is equivalent to
\begin{align}
    \left[
        \begin{matrix}
            1- l^i(\cdot) - s^i_t \parder{l^i}{s^i_t}{} 
                & 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}{} 
                    & M\left[\parder{l^i}{s^i_t}{} + \sum^N_{j=1} \parder{l^i}{S_t}{} \right] \\
            - s^i_t \parder{l^i}{S_t}{} 
                & - \sum_{j=1}^N s^j_t \parder{l^j}{S_t}{} 
                    & M \sum^N_{j=1} \parder{l^i}{S_t}{} \\
            - s^i_t \parder{l^i}{D_t}{} 
                & - \sum_{j=1}^N s^j_t \parder{l^j}{D_t}{} 
                    & (1-\delta) + M \sum^N_{j=1} \parder{l^i}{D_t}{} + \parder{g}{D_t}{} \\
        \end{matrix}
    \right] 
\end{align}
Solving this directly is difficult. 
We can use the fact that $A^{-1} = \frac{\adj(A)}{\det{A}}$, assuming $A$ is invertible.
\begin{align}
    D_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i =& \frac{\adj(A)}{\beta \det(A)} (D_{[s^i_t,S_t,D_t]} V^i - b)
\end{align}
Using each entry from $D_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i$ in the optimality condition and the notation
$B|_{i,j}$ to represent the element $b_{i,j}$ from the matrix $B$, we get the condition:
\begin{align}
    \frac{F}{\beta} \beta \det(A) = F \det(A) =&  
                    [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{1} \\
                    &+ 2 [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{2} \notag\\
                    &+ m [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{3} \notag
\end{align}
A little work remains to develop the euler equation that characterizes the optimal launch decision.

Of course, for any given set of functional forms $l^i,g$, one must verify if $A$ is invertible.


\subsection{Fleet Planner}
\begin{align}
    W(\{s^i_t\},S_t,D_t) =& \max_{\{x^i_t\}^N_{i=1} \geq 0}
            ~~\left( \sum^N_{i=1}  u^i(s^i_t,S_t,D_t)\right) - FX_t 
            + \beta W(\{s^i_{t+1}\}, S_{t+1}, D_{t+1}) \\
    \text{Subject To:}& \notag\\
    D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t) \\
    s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
    S_t =&\sum_{i=1}^N s^i_t \\
    X_t =&\sum_{i=1}^N x^i_t 
\end{align}
This is expected to follow the constellation operator's results closely.



\subsection{Survival Rates}\label{APX:Derivations:Survival}
\begin{align}
    \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) 
        =  - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{}  \\
    \parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N 
        \left(  [1-l^i(s^i_t,S_t,D_t)] + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right) 
        - \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}{(S_t)^2} \\
    =& \sum_{i=1}^N \left[ \frac{S_t [1-l^i(s^i_t,S_t,D_t)]}{(S_t)^2}
        - \frac{ s^i_t[1-l^i(s^i_t,S_t,D_t)] }{(S_t)^2} \right] 
            +\sum_{i=1}^N \frac{ s^i_t S_t [ -\parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{}] }{(S_t)^2} \\
    =& \sum_{i=1}^N \left[ \frac{S_t - s^i_t}{(S_t)^2}[1-l^i(s^i_t,S_t,D_t)]  \right]
            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \\
    =& \sum_{i=1}^N \left[ \frac{1}{S_t}[1-l^i(s^i_t,S_t,D_t)]  \right] - \frac{R}{S_t}
            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \\
    =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
            +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} 
\end{align}


\end{document}


