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finished fixing references

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youainti 5 years ago
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17
CurrentWriting/assets/preambles/References.bib
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@ -686,4 +686,21 @@ JEL Classification Nos.: H4, Q2},
primaryclass = {astro-ph.EP}, primaryclass = {astro-ph.EP},
} }
@Article{Maliar2019,
author = {Lilia Maliar and Serguei Maliar and Pablo Winant},
title = {Will Artificial Intelligence Replace Computational Economists Any Time Soon?},
journal = {Center for Economic Policy Research: Monetary Economics and Fluctuations},
year = {2019},
}
@Article{White1990,
author = {Kurt Hornik and Maxwell Stinchcombe and Halbert White},
title = {Universal Approximation of an Unknown Mapping and its Derivatives using Multilayer Feedforward Networks},
journal = {Neural Networks},
year = {1990},
volume = {3},
pages = {551-560},
month = jan,
}
@Comment{jabref-meta: databaseType:bibtex;} @Comment{jabref-meta: databaseType:bibtex;}

12
CurrentWriting/sections/04_ConstellationOperator.tex
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@ -2,7 +2,7 @@
\graphicspath{{\subfix{Assets/img/}}} \graphicspath{{\subfix{Assets/img/}}}
\begin{document} \begin{document}
With the laws of motion introduced in sections \cref{asdf}, we can now describe With the laws of motion introduced in sections \cref{SEC:Laws}, we can now describe
the optimization problem facing each constellation operator. the optimization problem facing each constellation operator.
Each operator recieve utility in each period per Each operator recieve utility in each period per
@ -44,8 +44,9 @@ for the existence of an euler equation.
\subsubsection{Euler Equation} \subsubsection{Euler Equation}
Appendix \cref{Appendix} contains more details on the math involved. Appendix \cref{APX:Derivations:EulerEquations} contains more details
What follows is just a sketch of the applied method in matrix notation. on the math involved.
What follows is just a sketch of the method in matrix notation.
As there is only one choice variable, we get a single optimality condition. 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. It can be written in various formats, with the latter matching the appendix the best.
@ -68,7 +69,8 @@ It can be written in various formats, with the latter matching the appendix the
As there are $N$ constellations we get $N$ satellite stocks, As there are $N$ constellations we get $N$ satellite stocks,
$N-1$ decisions $x^{\sim i}$, $N-1$ decisions $x^{\sim i}$,
and $1$ debris state for a total of $2N$ state variables\footnote{recall that $N$ is the number of constellations.}. 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: Thus there are $2N$ envelope conditions to be found:
\begin{align} \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} V^i(\vec s_t, \vec x^{\sim i}_t, D_t)
@ -126,7 +128,7 @@ To solve this symbolicly involves inverting a $6 \times 6$ matrix.
As matrix inversion has approximately an $O(n^3)$ computational complexity, As matrix inversion has approximately an $O(n^3)$ computational complexity,
this becomes unsustainable very quickly. this becomes unsustainable very quickly.
Section \cref{SectionOnComputational} describes how to address this issue to generate Section \cref{SEC:Computation} describes how to address this issue to generate
these euler equations using features of modern programming languages and linear algebra these euler equations using features of modern programming languages and linear algebra
libraries. libraries.

3
CurrentWriting/sections/05_SocialPlanner.tex
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@ -28,7 +28,8 @@ Some particular features of the model include:
\end{itemize} \end{itemize}
\subsubsection{Euler Equation} \subsubsection{Euler Equation}
In accordance with Appendix \cref{Appendix}, find the $N$ optimality conditions: In accordance with Appendix \cref{APX:Derivations:EulerEquations},
we find the $N$ optimality conditions:
\begin{align} \begin{align}
0 =& -\der{F(x^i_t)}{x^i_t}{} 0 =& -\der{F(x^i_t)}{x^i_t}{}
+ \beta \left[ + \beta \left[

9
CurrentWriting/sections/07_ComputationalApproach.tex
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@ -3,7 +3,7 @@
\begin{document} \begin{document}
The computational approach I have decided to take is an application of The computational approach I have decided to take is an application of
\cite{MALIAR2018}, where the policy function is approximated using a \cite{Maliar2019}, where the policy function is approximated using a
neural network. neural network.
The approach uses the fact that the euler equation implicitly defines the The approach uses the fact that the euler equation implicitly defines the
@ -17,7 +17,7 @@ allowing one to find $x(\dot)$ as the solution to a minimization problem.
By choosing a neural network as the functional approximation, we are able to By choosing a neural network as the functional approximation, we are able to
use the fact that a NN with a single hidden layer can be used to approximate use the fact that a NN with a single hidden layer can be used to approximate
functions arbitrarily well functions arbitrarily well
under certain conditions\footnote{FIND REFERENCE, SEE MALIAR}. under certain conditions \cref{White1990}.
We can also We can also
take advantage of the significant computational and practical improvements take advantage of the significant computational and practical improvements
currently revolutionizing Machine Learning. currently revolutionizing Machine Learning.
@ -74,9 +74,8 @@ and laws of motion functions, retuning a $k$-period transition function.
The second step is to generate functions that represent the optimality conditions. The second step is to generate functions that represent the optimality conditions.
By taking the appropriate derivatives with respect to the laws of motion and By taking the appropriate derivatives with respect to the laws of motion and
utility functions, this can be constructed explicitly. utility functions, this can be constructed explicitly.
Once these two functions are completed, they can be combined to create Once these two functions are completed, they can be combined to create
the euler equations, as described in appendix \ref{appx??}. the euler equations, as described in appendix \ref{APX:Derivations:EulerEquations}.
%%% Is it FaFCCs or recursion that allows this to occur? %%% Is it FaFCCs or recursion that allows this to occur?
%%% I believe both are ways to approach the problem. %%% I believe both are ways to approach the problem.
@ -107,7 +106,7 @@ selecting from that distribution.
One key question is how to handle the case of heterogeneous agents. One key question is how to handle the case of heterogeneous agents.
When the laws of motion depend on other agents' decisions, as is the case When the laws of motion depend on other agents' decisions, as is the case
described in \ref{lawsOFMotion}, intertemporal iteration may described in \ref{SEC:Laws}, intertemporal iteration may
require knowing the other agents best response function. require knowing the other agents best response function.
I believe I can model this in the constellation operator's case I believe I can model this in the constellation operator's case
by solving for the policy functions of each class of operator by solving for the policy functions of each class of operator

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