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Jul 1, 2023
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MAINT: review workflows and actions #251

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666d470
MAINT: review workflows and actions
mmcky Jun 28, 2023
7b6fbad
Fix Syntax for PDF
mmcky Jul 1, 2023
55d296b
Update cache and publish workflows
mmcky Jul 1, 2023
9b8ba8e
enable some mathjax macros
mmcky Jul 1, 2023
2693476
FIX: fail the cache on execution issue
mmcky Jul 1, 2023
9843226
FIX: heavy_tails
mmcky Jul 1, 2023
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15 changes: 1 addition & 14 deletions .github/workflows/cache.yml
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Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -18,23 +18,10 @@ jobs:
python-version: "3.10"
environment-file: environment.yml
activate-environment: quantecon
# - name: Install latex dependencies
# run: |
# sudo apt-get -qq update
# sudo apt-get install -y \
# texlive-latex-recommended \
# texlive-latex-extra \
# texlive-fonts-recommended \
# texlive-fonts-extra \
# texlive-xetex \
# latexmk \
# xindy \
# dvipng \
# cm-super
- name: Build HTML
shell: bash -l {0}
run: |
jb build lectures --path-output ./
jb build lectures --path-output ./ -W --keep-going
- name: Upload "_build" folder (cache)
uses: actions/upload-artifact@v2
with:
Expand Down
44 changes: 22 additions & 22 deletions .github/workflows/ci.yml
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Original file line number Diff line number Diff line change
Expand Up @@ -15,22 +15,22 @@ jobs:
python-version: "3.10"
environment-file: environment.yml
activate-environment: quantecon
# - name: Graphics Support
# run: |
# sudo apt-get -qq update && sudo apt-get install -y graphviz
# - name: Install latex dependencies
# run: |
# sudo apt-get -qq update
# sudo apt-get install -y \
# texlive-latex-recommended \
# texlive-latex-extra \
# texlive-fonts-recommended \
# texlive-fonts-extra \
# texlive-xetex \
# latexmk \
# xindy \
# dvipng \
# cm-super
- name: Graphics Support #TODO: Review if graphviz is needed
run: |
sudo apt-get -qq update && sudo apt-get install -y graphviz
- name: Install latex dependencies
run: |
sudo apt-get -qq update
sudo apt-get install -y \
texlive-latex-recommended \
texlive-latex-extra \
texlive-fonts-recommended \
texlive-fonts-extra \
texlive-xetex \
latexmk \
xindy \
dvipng \
cm-super
- name: Display Conda Environment Versions
shell: bash -l {0}
run: conda list
Expand All @@ -45,12 +45,12 @@ jobs:
name: build-cache
path: _build
# Build Assets (Download Notebooks and PDF via LaTeX)
# - name: Build PDF from LaTeX
# shell: bash -l {0}
# run: |
# jb build lectures --builder pdflatex --path-output ./ -n --keep-going
# mkdir -p _build/html/_pdf
# cp -u _build/latex/*.pdf _build/html/_pdf
- name: Build PDF from LaTeX
shell: bash -l {0}
run: |
jb build lectures --builder pdflatex --path-output ./ -n --keep-going
mkdir -p _build/html/_pdf
cp -u _build/latex/*.pdf _build/html/_pdf
- name: Build Download Notebooks (sphinx-tojupyter)
shell: bash -l {0}
run: |
Expand Down
44 changes: 22 additions & 22 deletions .github/workflows/publish.yml
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -19,19 +19,19 @@ jobs:
python-version: "3.10"
environment-file: environment.yml
activate-environment: quantecon
# - name: Install latex dependencies
# run: |
# sudo apt-get -qq update
# sudo apt-get install -y \
# texlive-latex-recommended \
# texlive-latex-extra \
# texlive-fonts-recommended \
# texlive-fonts-extra \
# texlive-xetex \
# latexmk \
# xindy \
# dvipng \
# cm-super
- name: Install latex dependencies
run: |
sudo apt-get -qq update
sudo apt-get install -y \
texlive-latex-recommended \
texlive-latex-extra \
texlive-fonts-recommended \
texlive-fonts-extra \
texlive-xetex \
latexmk \
xindy \
dvipng \
cm-super
- name: Display Conda Environment Versions
shell: bash -l {0}
run: conda list
Expand All @@ -46,15 +46,15 @@ jobs:
name: build-cache
path: _build
# Build Assets (Download Notebooks and PDF via LaTeX)
# - name: Build PDF from LaTeX
# shell: bash -l {0}
# run: |
# jb build lectures --builder pdflatex --path-output ./ -n --keep-going
# - name: Copy LaTeX PDF for GH-PAGES
# shell: bash -l {0}
# run: |
# mkdir -p _build/html/_pdf
# cp -u _build/latex/*.pdf _build/html/_pdf
- name: Build PDF from LaTeX
shell: bash -l {0}
run: |
jb build lectures --builder pdflatex --path-output ./ -n --keep-going
- name: Copy LaTeX PDF for GH-PAGES
shell: bash -l {0}
run: |
mkdir -p _build/html/_pdf
cp -u _build/latex/*.pdf _build/html/_pdf
- name: Build Download Notebooks (sphinx-tojupyter)
shell: bash -l {0}
run: |
Expand Down
5 changes: 5 additions & 0 deletions lectures/_config.yml
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Original file line number Diff line number Diff line change
Expand Up @@ -60,6 +60,11 @@ sphinx:
binderhub_url : https://mybinder.org # The URL of the BinderHub (e.g., https://mybinder.org)
colab_url : https://colab.research.google.com
thebe : false # Add a thebe button to pages (requires the repository to run on Binder)
mathjax3_config:
tex:
macros:
"argmax" : "arg\\,max"
"argmin" : "arg\\,min"
mathjax_path: https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js
rediraffe_redirects:
index_toc.md: intro.md
Expand Down
72 changes: 20 additions & 52 deletions lectures/cagan_adaptive.md
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Original file line number Diff line number Diff line change
Expand Up @@ -11,10 +11,6 @@ kernelspec:
name: python3
---

+++ {"user_expressions": []}



# A Fiscal Theory of Price Level with Adaptive Expectations

## Introduction
Expand All @@ -27,29 +23,23 @@ from collections import namedtuple
import matplotlib.pyplot as plt
```

+++ {"user_expressions": []}

<!-- #region -->

This lecture is a sequel or prequel to this lecture {doc}`fiscal theory of the price level <cagan_ree>`.

We'll use linear algebra to do some experiments with an alternative "fiscal theory of the price level".

Like the model in this lecture {doc}`fiscal theory of the price level <cagan_ree>`, the model asserts that when a government persistently spends more than it collects in taxes and prints money to finance the shortfall, it puts upward pressure on the price level and generates persistent inflation.

Instead of the "perfect foresight" or "rational expectations" version of the model in this lecture {doc}`fiscal theory of the price level <cagan_ree>`, our model in the present lecture is an "adaptive expectations" version of a model that Philip Cagan {cite}`Cagan` used to study the monetary dynamics of hyperinflations.

It combines these components:

Instead of the "perfect foresight" or "rational expectations" version of the model in this lecture {doc}`fiscal theory of the price level <cagan_ree>`, our model in the present lecture is an "adaptive expectations" version of a model that Philip Cagan {cite}`Cagan` used to study the monetary dynamics of hyperinflations.
* a demand function for real money balances that asserts that the logarithm of the quantity of real balances demanded depends inversely on the public's expected rate of inflation

It combines these components:
* an **adaptive expectations** model that describes how the public's anticipated rate of inflation responds to past values of actual inflation

* a demand function for real money balances that asserts that the logarithm of the quantity of real balances demanded depends inversely on the public's expected rate of inflation

* an **adaptive expectations** model that describes how the public's anticipated rate of inflation responds to past values of actual inflation

* an equilibrium condition that equates the demand for money to the supply

* an exogenous sequence of rates of growth of the money supply
* an equilibrium condition that equates the demand for money to the supply

* an exogenous sequence of rates of growth of the money supply

Our model stays quite close to Cagan's original specification.

Expand All @@ -62,13 +52,13 @@ the model.

Let

* $ m_t $ be the log of the supply of nominal money balances;
* $\mu_t = m_{t+1} - m_t $ be the net rate of growth of nominal balances;
* $p_t $ be the log of the price level;
* $\pi_t = p_{t+1} - p_t $ be the net rate of inflation between $t$ and $ t+1$;
* $\pi_t^*$ be the public's expected rate of inflation between $t$ and $t+1$;
* $T$ the horizon -- i.e., the last period for which the model will determine $p_t$
* $\pi_0^*$ public's initial expected rate of inflation between time $0$ and time $1$.
* $ m_t $ be the log of the supply of nominal money balances;
* $\mu_t = m_{t+1} - m_t $ be the net rate of growth of nominal balances;
* $p_t $ be the log of the price level;
* $\pi_t = p_{t+1} - p_t $ be the net rate of inflation between $t$ and $ t+1$;
* $\pi_t^*$ be the public's expected rate of inflation between $t$ and $t+1$;
* $T$ the horizon -- i.e., the last period for which the model will determine $p_t$
* $\pi_0^*$ public's initial expected rate of inflation between time $0$ and time $1$.


The demand for real balances $\exp\left(\frac{m_t^d}{p_t}\right)$ is governed by the following version of the Cagan demand function
Expand All @@ -91,8 +81,6 @@ $$ (eq:eqfiscth1)
Taking the difference between equation {eq}`eq:eqfiscth1` at time $t+1$ and at time
$t$ gives



$$
\pi_t = \mu_t + \alpha \pi_{t+1}^* - \alpha \pi_t^*
$$ (eq:eqpipi)
Expand All @@ -112,15 +100,10 @@ As endogenous outputs of our model we want to find sequences $\pi = \{\pi_t\}_{t
We'll do some mental experiments by studying how the model outputs vary as we vary
the model inputs.

<!-- #endregion -->

<!-- #region -->
## Representing key equations with linear algebra

We begin by writing the equation {eq}`eq:adaptexpn` adaptive expectations model for $\pi_t^*$ for $t=0, \ldots, T$ as



$$
\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 & 0 \cr
-\lambda & 1 & 0 & \cdots & 0 & 0 \cr
Expand Down Expand Up @@ -153,12 +136,11 @@ $$ (eq:eq1)

where the $(T+2) \times (T+2) $matrix $A$, the $(T+2)\times (T+1)$ matrix $B$, and the vectors $\pi^* , \pi_0, \pi_0^*$
are defined implicitly by aligning these two equations.
<!-- #endregion -->

<!-- #region -->
Next we write the key equation {eq}`eq:eqpipi` in matrix notation as

$$ \begin{bmatrix}
$$
\begin{bmatrix}
\pi_0 \cr \pi_1 \cr \pi_1 \cr \vdots \cr \pi_T \end{bmatrix}
= \begin{bmatrix}
\mu_0 \cr \mu_1 \cr \mu_2 \cr \vdots \cr \mu_T \end{bmatrix}
Expand All @@ -185,21 +167,18 @@ $$ (eq:eq2)
where the $(T+1) \times (T+2)$ matrix $C$ is defined implicitly to align this equation with the preceding
equation system.



## Harvesting returns from our matrix formulation


We now have all of the ingredients we need to solve for $\pi$ as
a function of $\mu, \pi_0, \pi_0^*$.

Combine equations {eq}`eq:eq1`and {eq}`eq:eq2` to get

$$
\begin{align*}
\begin{aligned}
A \pi^* & = (1-\lambda) B \pi + \pi_0^* \cr
& = (1-\lambda) B \left[ \mu + C \pi^* \right] + \pi_0^*
\end{align*}
\end{aligned}
$$

which implies that
Expand Down Expand Up @@ -228,7 +207,6 @@ Knowing these, we can then quickly calculate the associated sequence $p$ of the
from equation {eq}`eq:eqfiscth1`.

Let's fill in the details for this step.
<!-- #endregion -->

Since we now know $\mu$ it is easy to compute $m$.

Expand Down Expand Up @@ -290,8 +268,6 @@ $$

which is just $\pi^*$ with the last element dropped.



## Forecast errors

Our computations will verify that
Expand Down Expand Up @@ -416,30 +392,25 @@ $$ (eq:suffcond)
The source of this condition is the following string of deductions:

$$
\begin{align*}
\begin{aligned}
\pi_{t}&=\mu_{t}+\alpha\pi_{t+1}^{*}-\alpha\pi_{t}^{*}\\\pi_{t+1}^{*}&=\lambda\pi_{t}^{*}+(1-\lambda)\pi_{t}\\\pi_{t}&=\frac{\mu_{t}}{1-\alpha(1-\lambda)}-\frac{\alpha(1-\lambda)}{1-\alpha(1-\lambda)}\pi_{t}^{*}\\\implies\pi_{t}^{*}&=\frac{1}{\alpha(1-\lambda)}\mu_{t}-\frac{1-\alpha(1-\lambda)}{\alpha(1-\lambda)}\pi_{t}\\\pi_{t+1}&=\frac{\mu_{t+1}}{1-\alpha(1-\lambda)}-\frac{\alpha(1-\lambda)}{1-\alpha(1-\lambda)}\left(\lambda\pi_{t}^{*}+(1-\lambda)\pi_{t}\right)\\&=\frac{\mu_{t+1}}{1-\alpha(1-\lambda)}-\frac{\lambda}{1-\alpha(1-\lambda)}\mu_{t}+\frac{\lambda-\alpha(1-\lambda)}{1-\alpha(1-\lambda)}\pi_{t}
\end{align*}
\end{aligned}
$$

By assuring that the coefficient on $\pi_t$ is less than one in absolute value, condition {eq}`eq:suffcond` assures stability of the dynamics of $\{\pi_t\}$ described by the last line of our string of deductions.

The reader is free to study outcomes in examples that violate condition {eq}`eq:suffcond`.
<!-- #endregion -->

```{code-cell} ipython3
print(np.abs((λ - α*(1-λ))/(1 - α*(1-λ))))
```
+++ {"user_expressions": []}

```{code-cell} ipython3
print(λ - α*(1-λ))
```
+++ {"user_expressions": []}

Now we'll turn to some experiments.

<!-- #region -->

### Experiment 1

We'll study a situation in which the rate of growth of the money supply is $\mu_0$
Expand All @@ -460,14 +431,12 @@ Notice that we studied exactly this experiment in a rational expectations vers

So by comparing outcomes across the two lectures, we can learn about consequences of assuming adaptive expectations, as we do here, instead of rational expectations as we assumed in that other lecture.


```{code-cell} ipython3
μ_seq_1 = np.append(μ0*np.ones(T1), μ_star*np.ones(T+1-T1))

# solve and plot
π_seq_1, Eπ_seq_1, m_seq_1, p_seq_1 = solve_and_plot(md, μ_seq_1)
```
+++ {"user_expressions": []}

We invite the reader to compare outcomes with those under rational expectations studied in this lecture {doc}`fiscal theory of the price level <cagan_ree>`.

Expand All @@ -494,4 +463,3 @@ The sluggish fall in inflation is explained by how anticipated inflation $\pi_t
# solve and plot
π_seq_2, Eπ_seq_2, m_seq_2, p_seq_2 = solve_and_plot(md, μ_seq_2)
```
+++ {"user_expressions": []}
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