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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
%Alternate title? Constellations in Orbit
%\author{William King}
\institute{Washington State University}

\begin{document}

\maketitle

\begin{abstract}
    %Justification.
    Over the last decades, new technology has made low earth orbits (LEOs) more accessible, and 
    the resulting increase in LEO satellites has increased the risk of collision.
    %Discuss pollution externality
    Orbital operations produce an externality through the creation of debris during launch, 
    operation, and collisions which contributes to the risk of destruction.
    %Discuss debris propagation
    This effect is compounded as debris in orbit generates more debris through collisions with objects in orbit,
    possibly leading to a runaway effect called kessler syndrome.
    %Describe contribution
    This paper develops a dynamic model of satellite operation incorporating two effects not considered 
    in previous models: complementary network-like effects between satellites within 
    the same operator's fleet (called a constellation) and collision avoidance efficiencies realized within constellations.
    %Describe the state of the results
    The primary result is a preliminary model and the resulting analysis of the difference in satellite 
    survival rates between constellations and and the societal fleet.
\end{abstract}

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

\jel{Q29, Q58, L25}

\textbf{Acknowledgments:} I am the sole author and have received no contributions from others as of yet. 
This paper has been approved for dual submission in Econs 529 and Econs 594 by the instructors. 

\newpage
\tableofcontents

\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
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 unforeseen collision.
The defining feature of pollution in orbit is that it self-propagates 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 
may take place over various timescales.

% ---------------
%Discuss how various definitions of kessler syndrome
%    have been proposed in the economics literature to match the models.
%Not sure if the following contributes much given the previous paragraph.
%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.

% ---------------
Orbits may be divided into three primary groups, 
Low Earth Orbit (LEO),
Medium Earth Orbit (MEO), and High Earth Orbit (HEO) where Geostationary Earth Orbit (GEO) 
considered a particular classification of HEO. 
While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
a variety of new empirical studies and theoretical models published.

% ---------------
%Allocative efficiency

Macauley  provided the first evidence of sub-optimal behavior in orbit
by estimating the welfare loss due to the current method of assigning GEO slots to operators\autocite{Macauley_1998}.
The potential losses due to anti-competitive behavior were highlighted by Adilov et al ,
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\autocite{Adilov2019}.

The primary concern expressed in many of the published papers is whether or not orbits will be overused
due to their common-pool nature, and which policies may prevent kessler syndrome.
On this topic, Adilov, Alexander, and Cunningham examine pollution
using a two-period salop model, incorporating the effects of launch debris on 
survival into the second period\autocite{adilov_alexander_cunningham_2015}. 
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  in
the context of an infinite period dynamic model. 
%Potential Edit
Their approach is defined by the assumption that there are 
numerous operators in a free entry environment who 
can each launch a single, identical constellation\autocite{RaoRondina2020}.
Rao, Burgess, and Kaffine use this model to estimate that achieving socially optimal 
behavior through orbital use fees could increase the value generated by the 
space industry by a factor of four\autocite{Rao2020}.


% ---------------
%In addition to analyzing the allocative results, a significant area of interest is 
%what impact various policy interventions can have.
%The policies and methods used to analyze their impact have been widely varied.

%Other topics of interest include
%Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris)
%and cleanup.


% ---------------
My %FP
objective is to explore the effects from organizing satellites into constellations 
on satellite launch decisions and operation.
%I %FP
%do this by extending Rao and Rondina's dynamic satellite operators model\autocite{RaoRondina2020} 
%to account for non-symmetric constellation sizes and 
%incorporate the effects of both economies of scale as satellites in constellations complement each other and 
%collision avoidance efficiencies where satellites are less likely to collide with constellation members.
Although not explored in this paper, I %FP
hope to lay the groundwork for an 
analysis regarding pigouvian taxation as a solution to the externality of orbital debris.
%Explain what the article does.
The primary results of this paper are: 
preliminary development of the extended dynamic model, 
characterization of the general solutions to both the constellation operators' problems and
the fleet planner's problem, 
and an analysis of survival rates within constellations and the entire fleet.

%Contribution statement
%Adds to raoRondina2020 and adilov2018 in extedning to more diverse situations.
This work is most closely related to Rao and Rondina's model\autocite{RaoRondina2020} and the
dynamic model developed by Adilov et all \autocite{adilov_alexander_cunningham_2018}.
%Similarities
% - Rao
%   - Law of debris:
%   - law of motion for stocks
% - Adilov
%   - law of Debris
%   - constellations
%Differences
% - Rao
%   - constellation
%   - avoicance efficiencies
% - Adilov
%   - Allows for non-firm participants
%   - avoidance efficiencies
It is distinguished from the two models mentioned previously by accounting for
collision avoidance efficiencies where satellites are less likely to collide with constellation members, 
as neither of the mentioned models accounts for this behavior.
Additionally, it differs from Rao et al's model in that it allows constellations to be of different sizes.
Adilov et al permit constellations, but assume that all constellation operators are profit maximizing firms.
I explicitly provide a way to account for non-commercial space activities, such as military satellites.
One key similarity of all three models is the form of the intertemporal laws of motion of both constellation
sizes and debris.
For debris, this involves accounting for existing debris, debris from launches, and debris from collisions.
In the case of the fleet or constellation sizes, they all account for loss due to collisions 
and additions through launches.


% ---------------
%TODO: Needs rewritten after everything else.
The paper is organized as follows. 
In section \ref{Model} %describes the mathematical organization of the model 
the underlying mathematical model is given for both constellation operators and a societal fleet planner.
Section \ref{Analysis} %Examines marginal survival rate.
examines how externalities generated by operating satellite constellations differ between 
constellation operators and fleet planners.
It also examines various definitions of kessler syndrome and how that might be examined in this model.
The paper concludes in section \ref{Conclusion}, %concludes with a discussion of potential extensions and
%topics which have not yet been addressed.
with a discussion of outstanding issues, limitations to the model, and some areas of future interest.
The appendix \ref{APX:Derivations} contains mathematical derivations.





% ---------------------------------------------------------------------------------------
\section{Model}\label{Model}
%Intuitive description
This infinite period, dynamic model is an extension of Rao and Rondina's working paper\autocite{RaoRondina2020}
to include how operators deal with constellations.
In summary, each constellation operator has a utility function and a loss function that depend
on the number of satellites in the constellation, the total number of satellites in the societal fleet,
and the amount of debris in orbit.
The loss function describes the degradation and destruction of satellites within the constellation, 
and plays a critical role in the laws of motion of the satellite.
The utility function is used to describe how increases in constellation size affect utility production, given
the fleet size and debris levels.

\subsection{Model Description}
For a given  set of orbits that interact regularly (an orbital ``shell''), I %FP
assume there are $N$ operators, 
each of which has the potential to launch and operate a satellite
constellation consisting of some endogenously chosen number of identical satellites.

% -------------------
Each constellation $i$ is described by the number of satellites 
in period $t$, where this satellite stock is denoted by $s^i_t$.
Each operator of the constellation $i$ chooses the number of launches $x^i_t$ in each time period $t$.
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 by collisions at the rate $l^i(s^i_t,S_t,D_t) \in (0,1)$.
This includes collisions both within and without constellations.
I %FP
assume that:
\begin{align}
    \parder{l^i}{D_t}{} >& 0 \\
    \parder{l^i}{S_t}{} >& 
        \der{l^i}{s^i_t}{}  = \parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}  > 0 \label{EQ:xx}
\end{align}
Equation \ref{EQ:xx} represents one of the key distinctions from previous dynamic models, in that
the marginal risk of collision from adding a satellite to one's own constellation is
lower than the marginal risk of collision from other operators adding satellites.
The effects due to collision avoidance efficiencies within constellations will be examined in section \ref{Analysis}.
For any numerical examination, this assumption requires that:
\begin{align}
    0 > \parder{l^i}{s^i_t}{} > -\parder{l^i}{S_t}{}
\end{align}
This functional assumption, as described in \cref{EQ:xx}, is justified by the fact that when adding
satellites to a constellation, an operator can choose to place the satellites in orbits that will 
have nearly zero probability of colliding with another satellite in the constellation.
Operators who experience a collision between two of their own satellites experience
a higher cost than if one satellite collides with the satellite of another operator,
thus we would expect more care to be given to the internal organization of constellations.
Consequent to this ex-ante optimal organization within constellations, 
the majority of collisions observed should occur between satellites of different constellations 
and not within the same constellation.


Between the launch rate and destruction rate, I %FP
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}
where next period satellite stock equals the surviving satellite stock plus the total number of launches.



% -------------------
The level of debris in each period is represented by $D_t$, and is assumed to pose a latent risk.
In particular, I %FP we can
assume that once debris is created, the risk it provides is only avoidable 
through not launching future satellites.
%\footnote{This is one important extension as avoiding debris reduces the operational lifetime
% of satellites and may affect optimal taxation.
In addition to naturally occurring debris, new 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 proportional 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.
The parameters $\delta, M,$ and $m$ are assumed to be exogenously determined and non-stochastic.


% -------------------
%Describe the situation in which operators operate
Satellite operators -- whether commercial, governmental, research, or hobbyist\footnote{
    Notable examples of hobby satellites are the amateur (HAM) radio OSCAR satellites} --
expect to receive some utility from satellite operation. 
Because there are both firm and non-firm operators, we cannot denote this utility as 
exclusively profit utility nor consumption utility.
Firms, such as television or internet providers experience this utility as profit, while 
government, research institutions, or hobbyists operating satellites will experience this utility as
consumption of the service provided.
The choice of terminology acknowledges that the utility derived from orbit use is neither exclusively 
productive nor consumptive,
and there may be interference between productive commercial and consumptive non-commercial operations.

Mathematically, this is represented by time-separable utility function:
\begin{align}
    u^i(s^i_t, S_t, D_t) 
\end{align}
For simplicity, each constellation produces utility such that it is not affected by 
the size of any other given constellation.
In the case that the constellation operator is a profit maximizing firm, this implies that
they are a monopolist in their market.
The period utility function may incorporate the effects of orbital congestion ($S_t$) or debris ($D_t$), 
accounting for their effect in producing value to the operator.
Productive economies of scale within a constellation appear when 
$\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$,
and represents situations such as those of satellite-based internet providers
that require a minimum number of satellites in the constellation to provide a given level of service.


%Adilov et al analyzed the effects of competition between operators in launch decisions \autocite{Adilov2019}.
%A similar approach could be used, but would add significant complexity to the model.


% ---------------------------------------------
\subsection{Constellation Operator's Program}
%The aforementioned 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 develop the euler equation
%in a generalizable way. 

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 who's welfare is addressed are the satellite operators themselves.
Each operator faces the following problem:
\input{./includes/Appendix_constellation_program}

% ---------------------------------------------
\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 $W(\{s^i_t\},S_t,D_t) = \sum^N_{i=1} V^i(s^i_t,S_t,D_t)$
as satellite debris rarely poses a threat to the welfare of those on earth.

%\begin{align}
%    W(\{s^i_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.
%The derivation of the euler equation, and conditions on it's existence are
%outlined in Appendix \ref{APX:Derivations:Fleet}.

\input{./includes/Appendix_planner_program}


% ---------------------------------------------------------------------------------------
\section{Analysis}\label{Analysis}



\subsection{Survival Ratios}\label{Survival}

In line with basic theories of common-pool resources, 
we expect there to be a negative externality incurred on other constellations 
when a constellation increases their own satellite stock (resource usage). 
This externality comes from two effects, congestion and pollution.
Congestion, due to size of the societal fleet, may affect the utility achieved by other satellite operators
and it increases the probability of a satellite on satellite collision.
Pollution, the debris in all future periods, increase the rate of degradation and destruction
of satellites.
When a constellation operator increases their satellite stock, the other operators 
experience a loss of welfare through both congestion and pollution.
One way to measure the effects of satellite operations is through survival rates.

% Marginal survival.
The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, the proportion of satellites 
that were not lost (degraded nor destroyed) between period $t$ and $t+1$.
Thus the marginal survival rate represents the additional loss of
satellites due to a slightly larger constellation or fleet stock.

Mathematically the survival rates for a constellation and for society's fleet are defined as:
\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} \label{EQ:socsurv}
\end{align}
In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites 
in period $t+1$ that survived from period $t$.

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 weighted, average marginal survival rate 
across constellation operators.
The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:Survival_Direct}.
Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
and specifying which firm will be compared to society.
In spite of this, conditions on the average effects can be developted as follows.

The marginal survival rate of the fleet is greater than the weighted, arithmetic mean of marginal survival rates 
of the constellations when:
\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}{}
        \leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
    \sum_{i=1}^N  R_i  - R \leq& 0\\
    \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\\
    \sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)]  \leq& 0 \label{EQ:ii}
\end{align}
which is true if every constellation has at least one satellite.
As any constellation of interest has at least one satellite
and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$, 
we conclude that the marginal survival rate of the entire satellite fleet is lower
than the weighted arithmetic mean of marginal survival rates across constellations.
Note that it is possible for some constellations to have a lower marginal survival rate than the fleet,
but the survival rate for many operators must be higher than the societal rate.
Consequently, we would expect many operators to underestimate the impact of their behaviors on others
if they use their own observed or expected risk factors to estimate the risk they impose on others.
%%%Note on this section:
%%% So there is probably more insight into how to define survival rates in regards to geometric or harmonic
%%% means.
%%% The societal survival rate I chose is a simple and straightforward way of analyzing the issue,
%%% but there are probably other ways to define a fleet survival rate.
%%% I am interested in looking at weighted geometric or harmonic means as well.

%TODO2: Some more analysis can be done by comparing the case of avoidance efficiencies vs non-efficiencies.


%\subsubsection{Average Effects}
%TODO2: Review and rewrite this section, including discussing the implications

%As we are analyzing survival rates, a geometric mean is better used to describe average effects.
%By weighting the geometric mean with constellation sizes, we get:
%\begin{align}
%    R_G = \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
%\end{align}
%The marginal effect is assumed to be negative, thus 
%\begin{align}
%    0 > \parder{R_G}{s^i_t}{} =& \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
%     \left[  \parder{}{s^i_t}{} \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right] \\
%    0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t 
%                                \left( \ln(1-l^i) 
%                                        - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
%                                        - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
%                                \right) 
%                                - \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
%    0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t 
%                                \left( \ln(R_i) 
%                                        - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
%                                        - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
%                                \right) 
%                                - \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
%    0 > &  \ln R_i  - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
%           - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \sum^N_{j=1} \frac{s_t^j}{S_t} \ln(R_j)  \\
%    0 > &  \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
%           - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G  \\
%    \ln \frac{R_G}{R_i} =& \ln R_G - \ln R_i  > - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
%           - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}  
%\end{align}

%Welfare
% TODO3: Develop overarching results.

% ---------------------------------------------------------------------------------------
\subsection{Kessler Syndrome}\label{Kessler}
%Current plan: Explain the kessler region in this model
%Rao's physical approach
%Adilov's economic approach
Rao and Rondina\autocite{RaoRondina2020} interpret their model in terms of a physical 
kessler syndrome, while Adilov et al\autocite{adilov_alexander_cunningham_2018}
develop the concept of an economic kessler syndrome.
Generalizing Rao's approach, I %FP 
define the kessler region as the set of states such that 
the debris stock will tend to infinity, and kessler syndrome as when the state is in
the kessler region.
Formally, the kessler region is:
\begin{align}
    \vartheta_1 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow 
        \lim_{t \rightarrow \infty} D_{t+1} = \infty  \right\}
\end{align}
I suspect, but have not been able to prove, that an equivalent condition is:
\begin{align}
    \vartheta_2 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
        \parder{(D_{t+1}-D_t)}{D_t}{} > 0 \right\}
\end{align}
If the assumption holds, then a condition for a physical kessler region in this model is:
\begin{align}
    \vartheta_2 = 
        \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
        m\parder{X_t(\{s^i_t\},D_t)}{D_t}{}
        + M\cdot \left( \sum^N_{i=1} \parder{l^i}{D_t}{} \right)
        + g(D_t)  > \delta \right\}
\end{align}

Adilov et al\autocite{adilov_alexander_cunningham_2018} define an economic kessler syndrome 
(and thus kessler region) along the lines of
\begin{align}
    \vartheta_3 = \left\{ (\{s^i_t\},D_t) :  X_t(\{s^i_t\},D_t) = 0 \right\}
\end{align}
This represents the conditions under which adding satellites to the orbit becomes unprofitable.
They are able to establish conditions under which an economic kessler syndrome precedes a 
physical kessler syndrome.
Some modification of the conditions are required to get them to match the terminology in this
model, but I have not yet completed that work.
The benefit of this definition is that the euler equation defining $X_t(\cdot)$
can be searched for the states that imply $X_t = 0, \forall t$
\footnote{I have yet to conduct such a search, but plan on doing so as part of a numerical simulation.}.



% ---------------------------------------------------------------------------------------
%\subsection{Numerical Model}\label{Numerical}
% 2-firm model: Symmetric
% 2-firm model: asymetric sizes or payoffs.


% ---------------------------------------------------------------------------------------
\section{Summary and Concluding Remarks}\label{Conclusion}
%Summary
%Restate topic and objective
Although significant work remains to describe the impacts of organizing satellites as constellations,
I have been able to achieve 
%model not complete
many of preliminary milestones.
%conditions for the existence of an euler equation
%   - kessler region analysis
Foremost among these is the section which characterizes the general euler equation and provides
a simple set of conditions for existence.
This has opened a possible numerical approach to determining the economic kessler region.
%survival rates R analysis
Additionally, we have identified some preliminary results constraining the fleet's marginal survival rate
to be less than the weighted arithmetic average of the constellations' marginal survival rate.
This result -- consistent with the assumptions on avoidance efficiencies -- highlights the nature
of the externality imposed by operating and launching satellites.
%In spite of this 


%Limitations
%Change the state space to include the quantities in each satellite constellation.
There are three primary limitations within the model. 
The first is the implicit assumption on $u(\cdot)$ that firms operating constellations
act monopolistically, i.e. they do not compete in the same market.
This is an unreasonable assumption as there are already firms attempting to compete in LEO
as satellite internet providers, most notably SpaceX's Starlink and OneWeb.
%Computational difficulty - I believe that algebraic solutions require either a very 
%simple model with strict assumptions or significante algebraic work. 
%Computational solutions depend on the accuracy of the chosen functional form.
The second primary limitation is that of computational difficulty, due to the large state space
of the model. 
Even the simple constellation operator's problem presented here requires intensive
algebra to define the euler equation.
The typical response to this issue is to use computational methods to estimate 
the value and policy functions for both the operators and the fleet planner, but this has the disadvantage 
of reducing generalizability.
%The model doesn't track individual satellite lifetimes.
%   - Agent-based modeling?
The third limitation is that the model doesn't track individual satellites through their lifetime, particularly
the decision to deorbit or park the satellite.
Thus I ignore satellite both ex-ante and ex-post heterogeneity, preventing the analysis of
how policies affect satellite disposal decisions.


%Policy Implications
%Discuss application to pigouvian taxation. 
%   - Does optimal taxation depend on 
%       - Avoidance efficiencies? This affects the externalities of congestion, and maybe pollution?
%       - Relation between constellation size and fleet size? A larger firm may internalize more of the externality.
%       - In-Network economies of scale? If the tax is targeted to affect marginal utility, this may become more difficult
%           with economies of scale in value production.
The ultimate goal of developing this model is to facilitate policy analyses geared towards optimizing
the productive use of orbits.
As previous work has suggested that taxation may be an appropriate policy response to encourage
optimal use, I hope to be able to address the following questions with this model, 
at least in specific (computational) cases:
\begin{enumerate}
    \item Do avoidance efficiencies affect the optimal tax schedule for a given constellation operator?
        E.g. one constellation may be able to almost completely eliminate the chance of a within constellation 
        collision, while another may not. Should they be taxed at different rates?
%    \item Does the optimal tax rate depend on the relative size of a constellation to the fleet?
        %As the case of the fleet planner is similar to having a single constellation 
        %in orbit, but having many constellations in orbit leads to pollution issues
        %Would a quota on operators give similar enough results to be an effective policy step?
    \item Do productive economies of scale require a non-linear tax schedule to optimize orbit use?
    \item How does the decay rate $\delta$ (which depends on constellation altitude) 
        affect the optimal tax schedule?
\end{enumerate}

%Future Research Implications
%Areas of interest
%   - Strategic behavior of firms: Preemptive entry
One concern, tangential to work by Adilov, et al\autocite{Adilov2019} is that there may be ways for firms 
to increase barriers to entry for competitors by holding more satellites in orbit.
If this is the case, it begs the question of whether this will move the satellite stock closer
to kessler syndrome through an increase in the fleet stock of satellites, or if 
the avoidance efficiencies are sufficient to move it farther from kessler syndrome.
This is a crucial question to answer as it could inform policies regarding launch quotas and 
taxation.

%Add stochastics 
%   - incorporate risk adversion
Finally, a glaring issue is that the model is deterministic, and thus doesn't include
risk aversion. 
The variety of satellite operators that currently exist include militaries operating 
intelligence and communications satellites. 
One would expect that the critical nature of these constellations would imply a high level
of risk aversion in these operators, making this an important area of study.

%TODO: Concluding paragraph?

%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 identifying 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, work on the following extensions and applications of the
%model can fill gaps in the literature or complement current work.
%Notable areas of interest for future research include:
%\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 multi-period development 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. 
%Although not a common topic in economics, orbit use has properties that requires
%current study in order to identify optimal behavior, inform policies, and prevent kessler syndrome
%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}
%\input{./includes/Appendix_constellation_program}

%\subsection{Fleet Planner}\label{APX:Derivations:Fleet}
%\input{./includes/Appendix_planner_program}

\subsection{Survival Rates}\label{APX:Derivations:Survival_Direct}
\input{./includes/Appendix_Survival_direct}

%\subsection{Survival Rates: Geometric Mean Analysis}\label{APX:Derivations:Survival_Geometric}
%\input{./includes/Appendix_Survival_geometric}

\end{document}


