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GEModelTools.lyx~
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GEModelTools.lyx~
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\begin_preamble
% general
\usepackage[titletoc]{appendix}
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% algorithm
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% tables
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<{\__siunitx_table_print:}%
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% figures
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% footnotes
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% links
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% code
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% manual
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\use_default_options true
\begin_modules
theorems-ams
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\end_header
\begin_body
\begin_layout Title
\series bold
The
\series default
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
lstinline{GEModelTools}
\end_layout
\end_inset
\series bold
Package v0.2
\end_layout
\begin_layout Author
Jeppe Druedahl
\begin_inset Newline newline
\end_inset
Emil Holst Partsch
\end_layout
\begin_layout Abstract
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
This note provides an overview of the
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
lstinline{GEModelTools}
\end_layout
\end_inset
package for solving general equilibrium models easily in Python using the
sequence-space method.
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{1mm}
\end_layout
\end_inset
\end_layout
\begin_layout Abstract
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
You can learn how to use the package following these steps:
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{1mm}
\end_layout
\end_inset
\end_layout
\begin_deeper
\begin_layout Enumerate
Read this document
\end_layout
\begin_layout Enumerate
Install the package
\end_layout
\begin_layout Enumerate
Run the example notebooks
\end_layout
\begin_layout Enumerate
Read the commented code for the example notebooks
\end_layout
\begin_layout Enumerate
Implement your own model
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{3mm}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\series bold
Course:
\series default
\begin_inset CommandInset href
LatexCommand href
name "Adv. Macro: Heterogenous Agent Models"
target "https://sites.google.com/view/numeconcph-advmacrohet/"
literal "false"
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
\series bold
Literature:
\series default
\size small
\begin_inset CommandInset citation
LatexCommand citet
key "boppart_exploiting_2018"
literal "false"
\end_inset
and
\begin_inset CommandInset citation
LatexCommand citet
key "auclert_using_2021"
literal "false"
\end_inset
\size default
.
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{3mm}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
noindent
\end_layout
\end_inset
\series bold
\size large
Structure:
\end_layout
\begin_layout Itemize
\begin_inset ERT
status open
\begin_layout Plain Layout
[]
\end_layout
\end_inset
\series bold
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Model-class"
plural "false"
caps "false"
noprefix "false"
\end_inset
\series default
describes the class of models considered
\end_layout
\begin_layout Itemize
\begin_inset ERT
status open
\begin_layout Plain Layout
[]
\end_layout
\end_inset
\series bold
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:GEModelClass"
plural "false"
caps "false"
noprefix "false"
\end_inset
\series default
explains the required user inputs and available methods
\end_layout
\begin_layout Itemize
\begin_inset ERT
status open
\begin_layout Plain Layout
[]
\end_layout
\end_inset
\series bold
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Efficient-computation-of"
plural "false"
caps "false"
noprefix "false"
\end_inset
\series default
explains how to efficiently compute the household Jacobian
\end_layout
\begin_layout Itemize
\begin_inset ERT
status open
\begin_layout Plain Layout
[]
\end_layout
\end_inset
\series bold
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Additional-features"
plural "false"
caps "false"
noprefix "false"
\end_inset
\series default
explains additional features
\end_layout
\begin_layout Itemize
\begin_inset ERT
status open
\begin_layout Plain Layout
[]
\end_layout
\end_inset
\series bold
Section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:Troubleshooting"
plural "false"
caps "false"
noprefix "false"
\end_inset
\series default
provides basic troubleshooting
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{3mm}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\series bold
\size large
Code:
\series default
\size default
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{1mm}
\end_layout
\end_inset
\begin_inset Newline newline
\end_inset
Package:
\begin_inset CommandInset href
LatexCommand href
name "github.com/NumEconCopenhagen/GEModelTools"
target "https://github.com/NumEconCopenhagen/GEModelTools"
literal "false"
\end_inset
\begin_inset Newline newline
\end_inset
Notebooks:
\begin_inset CommandInset href
LatexCommand href
name "github.com/NumEconCopenhagen/GEModelToolsNotebooks"
target "https://github.com/NumEconCopenhagen/GEModelToolsNotebooks"
literal "false"
\end_inset
\begin_inset ERT
status open
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\backslash
vspace{3mm}
\end_layout
\end_inset
\begin_inset Newline newline
\end_inset
\series bold
Requirements:
\series default
Rely on
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
lstinline{EconModel}
\end_layout
\end_inset
and
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
lstinline{ConSav}
\end_layout
\end_inset
.
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{1mm}
\end_layout
\end_inset
\begin_inset Newline newline
\end_inset
Packages:
\begin_inset Newline newline
\end_inset
\begin_inset CommandInset href
LatexCommand href
name "github.com/NumEconCopenhagen/EconModel"
target "https://github.com/NumEconCopenhagen/EconModel"
literal "false"
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset CommandInset href
LatexCommand href
name "github.com/NumEconCopenhagen/ConsumptionSaving"
target "https://github.com/NumEconCopenhagen/ConsumptionSaving"
literal "false"
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
vspace{1mm}
\end_layout
\end_inset
\begin_inset Newline newline
\end_inset
Notebooks:
\begin_inset Newline newline
\end_inset
\begin_inset CommandInset href
LatexCommand href
name "github.com/NumEconCopenhagen/EconModelNotebooks"
target "https://github.com/NumEconCopenhagen/EconModelNotebooks"
literal "false"
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset CommandInset href
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name "github.com/NumEconCopenhagen/ConsumptionSavingNotebooks"
target "https://github.com/NumEconCopenhagen/ConsumptionSavingNotebooks"
literal "false"
\end_inset
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\end_deeper
\begin_layout Standard
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status open
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\backslash
thispagestyle{empty}
\end_layout
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\backslash
addtocounter{page}{-1}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage newpage
\end_inset
\end_layout
\begin_layout Section
Model class
\begin_inset CommandInset label
LatexCommand label
name "sec:Model-class"
\end_inset
\end_layout
\begin_layout Standard
In this section, we describe the class of general equilibrium models with
heterogeneous agents the package is designed solve, and explain how to
use the
\emph on
sequence space method
\emph default
developed in
\begin_inset CommandInset citation
LatexCommand citet
key "auclert_using_2021"
literal "false"
\end_inset
to solve them.
The starting point is a model with
\emph on
perfect foresight
\emph default
, where the
\emph on
non-linear transition path
\emph default
can be found given the initial distribution of agents and a sequence of
exogenous shocks.
Next, we show how to solve for the
\emph on
linearized impulse responses.
\emph default
These impulse responses are equal to those from a model with
\emph on
aggregate risk
\emph default
once it is linearized and certainty equivalence holds.
This implies that the sequence space method can be used to simulate time-series
data for aggregate variables and the distribution of agents.
Throughout this note a simple
\emph on
Heterogeneous Agent Neo-Classical
\emph default
(HANC) is used as an example.
\end_layout
\begin_layout Subsection
Model class
\end_layout
\begin_layout Standard
We consider economies where:
\end_layout
\begin_layout Enumerate
Time is discrete (index
\begin_inset Formula $t$
\end_inset
).
\end_layout
\begin_layout Enumerate
There is a continuum of households (index
\begin_inset Formula $i$
\end_inset
, when needed).
\end_layout
\begin_layout Enumerate
There is
\emph on
perfect foresight
\emph default
wrt.
all aggregate variables,
\begin_inset Formula $\boldsymbol{X}$
\end_inset
, indexed by
\begin_inset Formula $\mathcal{N}$
\end_inset
,
\begin_inset Formula
\[
\boldsymbol{X}=\{\boldsymbol{X}_{t}\}_{t=0}^{\infty}=\{\boldsymbol{X}^{j}\}_{j\in\mathcal{N}}=\{X_{t}^{j}\}_{t=0,j\in\mathcal{N}}^{\infty},
\]
\end_inset
where
\begin_inset Formula $\mathcal{N}=\mathcal{Z}\cup\mathcal{U}\cup\mathcal{O}$
\end_inset
, and
\begin_inset Formula $\mathcal{Z}$
\end_inset
are
\emph on
exogenous shocks,
\begin_inset Formula $\mathcal{U}$
\end_inset
\emph default
are
\emph on
unknowns
\emph default
,
\begin_inset Formula $\mathcal{O}$
\end_inset
are outputs, and
\begin_inset Formula $\mathcal{H\in\mathcal{O}}$
\end_inset
are
\emph on
targets.
\end_layout
\begin_layout Enumerate
The model structure is described in terms of a set of
\emph on
blocks
\emph default
indexed by
\begin_inset Formula $\mathcal{B}$
\end_inset
, where each block has inputs,
\begin_inset Formula $\mathcal{I}_{b}\subset\mathcal{N}$
\end_inset
, and outputs,
\begin_inset Formula $\mathcal{O}_{b}\subset\mathcal{O}$
\end_inset
, and there exists functions
\begin_inset Formula $h^{o}(\{\boldsymbol{X}^{i}\}_{i\in\mathcal{I}_{b}})$
\end_inset
for all
\begin_inset Formula $o\in\mathcal{O}_{b}$
\end_inset
.
\end_layout
\begin_layout Enumerate
The blocks are
\emph on
ordered
\emph default
such that (i) each output is
\emph on
unique
\emph default
to a block, (ii) the first block only have shocks and unknowns as inputs,
and (iii) later blocks only additionally take outputs of previous blocks
as inputs.
This implies the blocks can be structured as a
\emph on
directed acyclical graph
\emph default
(DAG).
\end_layout
\begin_layout Enumerate
The number of targets are equal to the number of unknowns, and an
\emph on
equilibrium
\emph default
implies
\begin_inset Formula $\boldsymbol{X}^{o}=0$
\end_inset
for all
\begin_inset Formula $o\in\mathcal{H}$
\end_inset
.
Equivalently, the model can be summarized by an
\emph on
target equation system
\emph default
from the unknowns and shocks to the targets,
\begin_inset Formula
\begin{equation}
\boldsymbol{H(U,Z)=0},\label{eq:H}
\end{equation}
\end_inset
and an
\emph on
auxiliary model equation
\emph default
to infer all variables
\begin_inset Formula
\begin{equation}
\boldsymbol{X=M(U,Z).}\label{eq:M}
\end{equation}
\end_inset
A
\emph on
steady state
\emph default
satisfy
\begin_inset Formula
\[
\boldsymbol{H(\boldsymbol{U}_{ss},\boldsymbol{Z}_{ss})}=0\text{ and \boldsymbol{X}_{ss}=\boldsymbol{M(}\boldsymbol{U_{ss}}\boldsymbol{,}\boldsymbol{Z}_{ss}\boldsymbol{)}.}
\]
\end_inset
\end_layout
\begin_layout Enumerate
The
\emph on
discretized household block
\emph default
can be written recursively as
\begin_inset Formula
\begin{align}