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In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation.
I've applied this to my model and extended the results.
This approach allows us to construct a elasticity of survival and satellite additions, i.e. an elasticity
of risk.
%I should probably look up how to analyze changes in risk level and quantitative representations etc.

% Marginal survival.
The survival rate for a constellation $i$ is defined as $R^i = 1-l^i(\cdot)$, i.e. 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.

To extend this definition to all fleets, we can measure the total number of 
satellites that survive. 
This can be calculated as the weighted sum of survival rates.
\begin{align}
    R =& \frac{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_t}
\end{align}

\subsubsection{Marginal Survival Rates}

We can find the marginal survival rate with respect to a given constellation $s^i_t$ as:

\begin{align}
    \parder{R}{s^i_t}{} =& \parder{}{s^i_t}{}\frac{\sum_{j=1}^n s^j_t R^i}{\sum_{j=1}^n s^i_t} \\
        =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)^2 
           \left[ 
                \left(\sum^n_{j=1}s^j_t\right) \left(\parder{}{s^i_t}{}\sum^n_{j=1} s^j_t R^j\right)
                - \sum^n_{j=1} s^j_t R^j
           \right] \\
        =& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right) 
           \left[ 
                \left(\sum^n_{j \neq i} s^j_t \parder{R^j}{s^i_t}{}\right) 
                + \left( R^i + s^i_t \parder{R^i}{s^i_t}{}\right)
                - R 
           \right] 
        \\
        =&  
                \left(\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}\right) 
                + \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right)
        \\
        \parder{R}{s^i_t}{} 
        =&
        \sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}
        + \left(\frac{ R^i - R  }{\sum_{j=1}^n s^j_t}\right) \label{EQ:MarginalSurvivalRelation}
\end{align}
This can also be written in differential form as
\begin{align}
        d{R} 
        =&
        \sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) d{R^j}
        + \left(\frac{ R^i - R  }{\sum_{j=1}^n s^j_t}\right) d{s^i_t}\label{EQ:differentialSurvivalRelation}
\end{align}

From \cref{EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation},
we can see that the fleetwide marginal survival rate
is made up of two components.
\begin{itemize}
    \item $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$
    represents the effect on each satellite constellation, and is always negative because
    $\parder{R^j}{s^i_t}{} < 0$ by assumption.
    Thus each constellations' survival rate will decrease as satellites are added to 
    any constellation.
    \item $\frac{ R^i - R  }{\sum_{j=1}^n s^j_t}$,
    represents the effect of averaging out marginal survival rates.
    Intuitively, when a constellation has a higher survival rate
    than the fleet's survival rate, adding a satellite to that fleet contributes
    less colision risk than if it were given to another 
    Note that it is positive but only when $R^i > R$.
    Additionally, it disappears quickly as the total number of satellites increase.
    Thus when there are a large number of satellites in orbit, regardless of who 
    owns them, it is almost certain that any increase in satellite stocks will 
    lead to a reduction in the survival rate.
    \footnote{I believe Rao makes this an assumption, I show it is a result}
\end{itemize}

Consequently, we can see that in many cases, the marginal survival rate will be negative.

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