diff --git a/CurrentWriting/sections/01_LawsOfMotion.tex b/CurrentWriting/sections/01_LawsOfMotion.tex index 1c2efbb..a3b9454 100644 --- a/CurrentWriting/sections/01_LawsOfMotion.tex +++ b/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 diff --git a/CurrentWriting/sections/02_KesslerSyndrome.tex b/CurrentWriting/sections/02_KesslerSyndrome.tex index 2ec52ca..c88cc8e 100644 --- a/CurrentWriting/sections/02_KesslerSyndrome.tex +++ b/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 diff --git a/CurrentWriting/sections/04_ConstellationOperator.tex b/CurrentWriting/sections/04_ConstellationOperator.tex index b8c2672..68fcf8a 100644 --- a/CurrentWriting/sections/04_ConstellationOperator.tex +++ b/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} diff --git a/CurrentWriting/sections/05_SocialPlanner.tex b/CurrentWriting/sections/05_SocialPlanner.tex index c6cf8d9..fa7d2fc 100644 --- a/CurrentWriting/sections/05_SocialPlanner.tex +++ b/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}