error in 05_socialPlanner, but need to leave for the day.

CurrentWriting/sections/01_LawsOfMotion.tex

@@ -16,6 +16,11 @@ Assuming satellites live indefinitely, these facts give us the following law of
 constellation $i$.
 \begin{align}
     S^i_{t+1} = \left( 1 - l^i(\{s^j_t\}, D_t)\right)s^i_t + x^i_t
+    %Couple of Notes:
+    %  This does not allow for natural decay of satellites.
+    %  Nor does it include a deorbit decision.
+    %  
+    %  
 \end{align}
 Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by collisions.
 Note that it is reasonable to assume that the loss of satellites to collisions should be

CurrentWriting/sections/02_KesslerSyndrome.tex

@@ -89,13 +89,13 @@ defined as the stock and debris levels such that:
     \kappa = \left\{ \{s^j_t\}, D_t : 
             D_{t+1} - D_{t} \geq \varepsilon > 0 \right\}
 \end{align}
-Note that the debris level is in a $\epsilon$-kessler region when it is in a proto-kesslerian region
-for all future periods.
+%Note that the debris level is in a $\epsilon$-kessler region when it is in a proto-kesslerian region
+%for all future periods.
 This even simpler to compute than the phase diagram, and can be used to generate a topological view
 of proto-kesslerian regions of degre $\varepsilon$.
-These are both easier to interpret and various approaches could be used to analyze how debris levels 
-transition between them.
-%what would the integral of gradients weighted by the dividing line give? just a thought.
+%These are both easier to interpret and various approaches could be used to analyze how debris levels 
+%transition between them.
+%%%what would the integral of gradients weighted by the dividing line measure? just a thought.
 %Other thoughts
 % proto-kesslerian paths, paths that pass into a proto kesslerian region.
 In order to capture the cyclic behavior that $\epsilon$-kessler regions miss, we can define a type of 

CurrentWriting/sections/04_ConstellationOperator.tex

@@ -28,7 +28,7 @@ Thus the $M$-period (possibly infinite), problem is:
 % 
 
 \subsection{Infinite Period (Bellman) Equation}
-The problem above can be rewritten in the bellman form as
+The inifinite period version of the problem above can be rewritten in the bellman form as
 \begin{align}
     V^i(\vec s_t, \vec x^{\sim i}_t, D_t) = \max_{x^i_t} u^i(\vec s_t, D_t) -F(x) 
     + \beta \left[ V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) \right]
@@ -42,61 +42,72 @@ To solve for the policy function, we have a variety of methods available.
 Due to the computational method chosen later, I'm going to examine the conditions
 for the existence of an euler equation.
 
+
 \subsubsection{Euler Equation}
-First, find the single optimality condition
+Appendix \cref{Appendix} contains more details on the math involved. 
+What follows is just a sketch of the applied method in matrix notation.
+
+As there is only one choice variable, we get a single optimality condition. 
+It can be written in various formats, with the latter matching the appendix the best.
 \begin{align}
-    0 =& \parder{}{x^i_t}{} u^i(\vec s_t, D_t) -\parder{}{x^i_t}{}F(x) 
-    + \beta \left[ \parder{}{x^i_t}{} 
-        V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
-    \right] \\
+%    0 =& \parder{}{x^i_t}{} u^i(\vec s_t, D_t) -\parder{}{x^i_t}{}F(x) 
+%    + \beta \left[ \parder{}{x^i_t}{} 
+%        V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
+%    \right] \\
     0 =&  -\der{F}{x^i_t}{} 
     + \beta \left[  
+        \nabla_{x^i_t} [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
+        \cdot
         \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}} 
             V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
-        \cdot
-        \nabla_{x^i_t} [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
     \right] \label{EQ:OptimalityCondition}\\
-    0 =&  -\der{F}{x^i_t}{} + \beta \nabla V^i_{t+1} \cdot \vec a_t 
-    \label{EQ:SimplifiedOptimalityCondition}
+    0 =&  -\der{F}{x^i_t}{} + \beta \vec a(\vec s_t,D_t) \cdot \nabla V^i_{t+1} 
+    \label{EQ:SimplifiedOptimalityCondition}\\
+        =& - f_{x_t} + \beta \vec a_t \cdot \nabla V^i_{t+1}
 \end{align}
 
-Second, the $2N$\footnote{recall that $N$ is the number of constellations.} 
-envelope conditions can also be found:
+As there are $N$ constellations we get $N$ satellite stocks, 
+$N-1$ decisions $x^{\sim i}$,
+and $1$ debris state for a total of $2N$ state variables\footnote{recall that $N$ is the number of constellations.}.
+Thus there are $2N$ envelope conditions to be found:
 \begin{align}
+%    \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} V^i(\vec s_t, \vec x^{\sim i}_t, D_t) 
+%    =& \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} u^i(\vec s_t, D_t) \notag \\
+%    &+ \beta \left[ 
+%        \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} }
+%            V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
+%        \cdot
+%        \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
+%            [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
+%    \right] \label{EQ:EnvelopeConditions}
+%    \\
     \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} V^i(\vec s_t, \vec x^{\sim i}_t, D_t) 
-    =& \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t} u^i(\vec s_t, D_t) \notag \\
-    &+ \beta \left[ 
-        \nabla_{\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} }
-            V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
-        \cdot
-        \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
-            [ \vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1} ]
-    \right] \label{EQ:EnvelopeConditions}
-    \\
-    \nabla \vec V^i_t =& \vec u^i 
-        + \beta A \cdot \nabla \vec V^i_{t+1} 
+    =
+    \nabla \vec V^i_t 
+    = \vec u^i 
+        + \beta B_t \cdot \nabla \vec V^i_{t+1} 
         \label{EQ:SimplifiedEnvelopeConditions}
 \end{align}
-When interpreting this, note that 
-$$
-\nabla \vec V^i_{t+1} = \nabla_{[\vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1}] }
-    V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
-$$
-is a $2N \times 1$ vector of first derivatives but 
-$$
-A = \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
-    [ \vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1} ]
-$$
-is a $2N \times 2N$ matrix of first derivatives.
+%When interpreting this, note that 
+% $$
+% \nabla \vec V^i_{t+1} = \nabla_{[\vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1}] }
+%     V^i(\vec s_{t+1}, \vec x^{\sim i}_{t+1}, D_{t+1}) 
+% $$
+% is a $2N \times 1$ vector of first derivatives but 
+% $$
+% A = \nabla_{\vec s_t, \vec x^{\sim i}_t, D_t}
+%     [ \vec s_{t+1}~ \vec x^{\sim i}_{t+1}~ D_{t+1} ]
+% $$
+% is a $2N \times 2N$ matrix of first derivatives.
 
-By solving for $\vec V^i_{t+1}$ as a function of $\vec V^i_{t}$ we get the 
-intertemporal condition:
-\begin{align}
-    \frac{1}{\beta} A^{-1} \left(\nabla \vec V^i_t - \vec u^i_t \right) 
-    = \nabla \vec V^i_{t+1} 
-\end{align}
-Thus one crucial condition for the existence of a solution is that $A^{-1}$ exists for 
-all values the laws of motion and choice functions can take.
+% By solving for $\vec V^i_{t+1}$ as a function of $\vec V^i_{t}$ we get the 
+% intertemporal condition:
+% \begin{align}
+%     \frac{1}{\beta} A^{-1} \left(\nabla \vec V^i_t - \vec u^i_t \right) 
+%     = \nabla \vec V^i_{t+1} 
+% \end{align}
+% Thus one crucial condition for the existence of a solution is that $A^{-1}$ exists for 
+% all values the laws of motion and choice functions can take.
 
 % \subsection{Existence}
 % I need to do some more diving into conditions for existence.
@@ -104,16 +115,19 @@ all values the laws of motion and choice functions can take.
 % non-convergence.
 % 
 
-Finally, to construct the euler equation, we take
-\cref{EQ:SimplifiedOptimalityCondition}
-and iterate it forward $2N-1$ times.
-By substituting 
-\cref{EQ:SimplifiedEnvelopeConditions} 
-into each iteration enough times
-you get a system that defines $\nabla V^i_t$
-By substituting this defined value of $\nabla V^i_t$ into 
-\cref{EQ:SimplifiedOptimalityCondition}
-one final time, we get a function that fully determines the policy function.
+To finish constructing the euler equation, we would use the intertemporal 
+transition function \cref{EQ:SimplifiedEnvelopeConditions} and iterated
+versions of \cref{EQ:OptimalityCondition,EQ:SimplifiedOptimalityCondition}
+to construct the $2N+1$ euler equations.\footnote{Double check numbers}
+Note that for even a small number of agents -- e.g. 3 -- this iterated substitution
+becomes relatively complex, requiring caculating an iterated intertemporal tranisition
+function and laws of motion 6 times.
+To solve this symbolicly involves inverting a $6 \times 6$ matrix. 
+As matrix inversion has approximately an $O(n^3)$ computational complexity,
+this becomes unsustainable very quickly.
 
+Section \cref{SectionOnComputational} describes how to address this issue to generate
+these euler equations using features of modern programming languages and linear algebra
+libraries.
 
 \end{document}

CurrentWriting/sections/05_SocialPlanner.tex

@@ -2,7 +2,7 @@
 \graphicspath{{\subfix{Assets/img/}}}
 
 \begin{document}
-The Social (Fleet) Planner's problem can be written in the belman form as:
+The Social (Fleet) Planner's problem can be written in the bellman form as:
 \begin{align}
     W(\vec s_t, D_t) =& \max_{\vec x_t} \left[
        \left(\sum^N_{i=1} u^i(\vec s_t, D_t) - F(x^i_t) \right)
@@ -13,9 +13,22 @@ The Social (Fleet) Planner's problem can be written in the belman form as:
          + \gamma \sum^N_{i=1} l^i(\vec s_t, D_t)
          + \Gamma \sum^N_{i=1} x^i_t
 \end{align}
+Some particular features of the model include:
+\begin{align}
+    \item The single period welfare function consists only of constellation operators.
+        Although satellites do deorbit and occasionally pose a risk to humans living on the
+        earth's surface\footnote{Skylab fell in Australia, with some pieces landing near towns.}
+        modeling this risk properly would require adding a deorbit decisions, 
+        including uncontrolled deorbits.
+    \item Although the social planner controls each constellation, they do not reap additional
+        collision avoidance efficiencies.
+        One justification is that no social planner could concieve of every use of orbit
+        at any single point in time, and thus constellations are added sequentially.
+        This allows only the intra-constellation benefits to be achived.
+\end{align}
 
 \subsubsection{Euler Equation}
-First find the $N$ optimality conditions:
+In accordance with Appendix \cref{Appendix}, find the $N$ optimality conditions:
 \begin{align}
     0 =& -\der{F(x^i_t)}{x^i_t}{}  
         + \beta \left[
@@ -27,17 +40,17 @@ First find the $N$ optimality conditions:
 \end{align}
 Which in vector form is:
 \begin{align}
-    0 =& -\vec f +\beta \left[B\cdot \nabla W_{t+1} \right]
+    0 =& -\vec f_x +\beta \left[B\cdot \nabla W_{t+1} \right]
 \end{align}
-And the $N+1$ envelope conditions are:
+Similarly, the $N+1$ envelope conditions are:
 \begin{align}
-    \nabla_{\vec s_{t}, D_{t}} W(\vec s_t, D_t) =& 
-        \sum^N_{i=1} \nabla_{\vec s_{t}, D_{t}} u^i(\vec s_t, D_t) 
-        %- \der{}{x^i_t}{}F(x^i_t) \nabla_{\vec s_{t}, D_{t}}x^i_t %This equals zero due to the envelope theorem
-        \notag \\
-        &+ \beta \left[ \nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1})
-            \cdot \nabla_{\vec s_{t}, D_{t}} [\vec s_{t+1} ~ D_{t+1}]
-    \right] \\
+%    \nabla_{\vec s_{t}, D_{t}} W(\vec s_t, D_t) =& 
+%        \sum^N_{i=1} \nabla_{\vec s_{t}, D_{t}} u^i(\vec s_t, D_t) 
+%        %- \der{}{x^i_t}{}F(x^i_t) \nabla_{\vec s_{t}, D_{t}}x^i_t %This equals zero due to the envelope theorem
+%        \notag \\
+%        &+ \beta \left[ \nabla_{\vec s_{t+1}, D_{t+1}} W(\vec s_{t+1}, D_{t+1})
+%            \cdot \nabla_{\vec s_{t}, D_{t}} [\vec s_{t+1} ~ D_{t+1}]
+%    \right] \\
     \nabla W_t =& \vec U + \beta \left[C \cdot \nabla W_{t+1} \right] 
 \end{align}
 Which gives us the iteration format
@@ -45,5 +58,8 @@ Which gives us the iteration format
     \nabla W_{t+1} =& (\beta C)^{-1} \cdot \left(\nabla W_t - \vec U \right)
 \end{align}
 
+Thus two iterations of the optimality condition are needed, but only to provide $N+1$ binding conditions.
+This lets us discard $N-1$ of the conditions from the second iteration of the optimality condition.
+% NEed to explain better. Not quite true.
 
 \end{document}