Merge branch 'main' into review-inequality

Author: mmcky

lectures/cagan_adaptive.md

@@ -16,13 +16,13 @@ kernelspec:
 ## Introduction
 
 
-This lecture is a sequel or prequel to another lecture {doc}`monetarist theory of  price levels <cagan_ree>`.
+This lecture is a sequel or prequel to the lecture {doc}`cagan_ree`.
 
-We'll use linear algebra to do some experiments with  an alternative "monetarist" or  "fiscal" theory of  price levels".
+We'll use linear algebra to do some experiments with  an alternative "monetarist" or  "fiscal" theory of  price levels.
 
-Like the model in this lecture {doc}`monetarist theory of  price levels <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.
+Like the model in {doc}`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}`monetarist theory of  price levels <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.  
+Instead of the "perfect foresight" or "rational expectations" version of the model in {doc}`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:
 
@@ -36,7 +36,7 @@ It combines these components:
 
 Our model stays quite close to Cagan's original specification.  
 
-As in the {doc}`present values <pv>` and {doc}`consumption smoothing<cons_smooth>` lectures, the only linear algebra operations that we'll be  using are matrix multiplication and matrix inversion.
+As in the lectures {doc}`pv` and {doc}`cons_smooth`, the only linear algebra operations that we'll be  using are matrix multiplication and matrix inversion.
 
 To facilitate using  linear matrix algebra as our principal mathematical tool, we'll use a finite horizon version of
 the model.
@@ -54,7 +54,7 @@ Let
 * $\pi_0^*$ 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
+The demand for real balances $\exp\left(m_t^d-p_t\right)$ is governed by the following  version of the Cagan demand function
 
 $$  
 m_t^d - p_t = -\alpha \pi_t^* \: , \: \alpha > 0 ; \quad t = 0, 1, \ldots, T .
@@ -88,7 +88,7 @@ $$ (eq:adaptexpn)
 As exogenous inputs into the model, we take initial conditions $m_0, \pi_0^*$
 and a money growth sequence $\mu = \{\mu_t\}_{t=0}^T$.  
 
-As endogenous outputs of our model we want to find sequences $\pi = \{\pi_t\}_{t=0}^T, p = \{p_t\}_{t=0}^T$ as functions of the endogenous inputs.
+As endogenous outputs of our model we want to find sequences $\pi = \{\pi_t\}_{t=0}^T, p = \{p_t\}_{t=0}^T$ as functions of the exogenous inputs.
 
 We'll do some mental experiments by studying how the model outputs vary as we vary
 the model inputs.
@@ -278,7 +278,7 @@ $$ (eq:notre)
 This outcome is typical in models in which adaptive expectations hypothesis like equation {eq}`eq:adaptexpn` appear as a
 component.  
 
-In this lecture {doc}`monetarist theory of the price level <cagan_ree>`, we studied a version of the model that replaces hypothesis {eq}`eq:adaptexpn` with
+In {doc}`cagan_ree` we studied a version of the model that replaces hypothesis {eq}`eq:adaptexpn` with
 a "perfect foresight" or "rational expectations" hypothesis.
 
 
@@ -296,7 +296,7 @@ import matplotlib.pyplot as plt
 Cagan_Adaptive = namedtuple("Cagan_Adaptive", 
                         ["α", "m0", "Eπ0", "T", "λ"])
 
-def create_cagan_model(α, m0, Eπ0, T, λ):
+def create_cagan_adaptive_model(α, m0, Eπ0, T, λ):
     return Cagan_Adaptive(α, m0, Eπ0, T, λ)
 ```
 +++ {"user_expressions": []}
@@ -314,7 +314,7 @@ m0 = 1
 μ0 = 0.5
 μ_star = 0
 
-md = create_cagan_model(α=α, m0=m0, Eπ0=μ0, T=T, λ=λ)
+md = create_cagan_adaptive_model(α=α, m0=m0, Eπ0=μ0, T=T, λ=λ)
 ```
 +++ {"user_expressions": []}
 
@@ -431,7 +431,7 @@ $$
      \end{cases}
 $$
 
-Notice that  we studied exactly this experiment  in a rational expectations version of the model in this lecture {doc}`monetarist theory of the price level <cagan_ree>`.
+Notice that  we studied exactly this experiment  in a rational expectations version of the model in {doc}`cagan_ree`.
 
 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.
 
@@ -442,7 +442,7 @@ So by comparing outcomes across the two lectures, we can learn about consequence
 π_seq_1, Eπ_seq_1, m_seq_1, p_seq_1 = solve_and_plot(md, μ_seq_1)
 ```
 
-We invite the reader to compare outcomes with those under rational expectations studied in another lecture {doc}`monetarist theory of  price levels <cagan_ree>`.
+We invite the reader to compare outcomes with those under rational expectations studied in {doc}`cagan_ree`.
 
 Please note how the actual inflation rate $\pi_t$ "overshoots" its ultimate steady-state value at the time of the sudden reduction in the rate of growth of the money supply at time $T_1$.
 

lectures/geom_series.md

@@ -47,7 +47,7 @@ Among these are
 These and other applications prove the truth of the wise crack that
 
 ```{epigraph}
-"in economics, a little knowledge of geometric series goes a long way "
+"In economics, a little knowledge of geometric series goes a long way."
 ```
 
 Below we'll use the following imports:
@@ -171,7 +171,7 @@ The right side records bank $i$'s liabilities,
 namely, the deposits $D_i$ held by its depositors; these are
 IOU's from the bank to its depositors in the form of either checking
 accounts or savings accounts (or before 1914, bank notes issued by a
-bank stating promises to redeem note for gold or silver on demand).
+bank stating promises to redeem notes for gold or silver on demand).
 
 Each bank $i$ sets its reserves to satisfy the equation
 
@@ -573,7 +573,7 @@ Recall that $R = 1+r$ and $G = 1+g$ and that $R > G$
 and $r > g$ and that $r$ and $g$ are typically small
 numbers, e.g., .05 or .03.
 
-Use the Taylor series of $\frac{1}{1+r}$ about $r=0$,
+Use the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) of $\frac{1}{1+r}$ about $r=0$,
 namely,
 
 $$
@@ -641,7 +641,7 @@ $$
 Expanding:
 
 $$
-\begin{aligned} p_0 &=\frac{x_0(1-1+(T+1)^2 rg -r(T+1)+g(T+1))}{1-1+r-g+rg}  \\&=\frac{x_0(T+1)((T+1)rg+r-g)}{r-g+rg} \\ &\approx \frac{x_0(T+1)(r-g)}{r-g}+\frac{x_0rg(T+1)}{r-g}\\ &= x_0(T+1) + \frac{x_0rg(T+1)}{r-g}  \end{aligned}
+\begin{aligned} p_0 &=\frac{x_0(1-1+(T+1)^2 rg +r(T+1)-g(T+1))}{1-1+r-g+rg}  \\&=\frac{x_0(T+1)((T+1)rg+r-g)}{r-g+rg} \\ &= \frac{x_0(T+1)(r-g)}{r-g + rg}+\frac{x_0rg(T+1)^2}{r-g+rg}\\ &\approx \frac{x_0(T+1)(r-g)}{r-g}+\frac{x_0rg(T+1)}{r-g}\\  &= x_0(T+1) + \frac{x_0rg(T+1)}{r-g}  \end{aligned}
 $$
 
 We could have also approximated by removing the second term

lectures/heavy_tails.md

@@ -19,7 +19,7 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 ```{code-cell} ipython3
 :tags: [hide-output]
 
-!pip install --upgrade yfinance pandas_datareader interpolation
+!pip install --upgrade yfinance pandas_datareader
 ```
 
 We use the following imports.
@@ -31,7 +31,6 @@ import yfinance as yf
 import pandas as pd
 import statsmodels.api as sm
 
-from interpolation import interp
 from pandas_datareader import wb
 from scipy.stats import norm, cauchy
 from pandas.plotting import register_matplotlib_converters
@@ -602,7 +601,7 @@ def empirical_ccdf(data,
         fw = np.empty_like(aw, dtype='float64')
         for i, a in enumerate(aw):
             fw[i] = a / np.sum(aw)
-        pdf = lambda x: interp(data, fw, x)
+        pdf = lambda x: np.interp(x, data, fw)
         data = np.sort(data)
         j = 0
         for i, d in enumerate(data):

lectures/inequality.md

@@ -66,7 +66,11 @@ We will need to install the following packages
 ```{code-cell} ipython3
 :tags: [hide-output]
 
+<<<<<<< HEAD
 !pip install wbgapi plotly
+=======
+!pip install quantecon
+>>>>>>> main
 ```
 
 We will also use the following imports.
@@ -76,8 +80,11 @@ import pandas as pd
 import numpy as np
 import matplotlib.pyplot as plt
 import random as rd
+<<<<<<< HEAD
 import wbgapi as wb
 import plotly.express as px
+=======
+>>>>>>> main
 ```
 
 ## The Lorenz curve

lectures/inflation_history.md

@@ -39,7 +39,7 @@ Thus, in the US, the price level at $t$ is measured in dollars (month $t$ or yea
 
 Until the early 20th century, in many western economies, price levels fluctuated from year to year but didn't have much of a trend.  
 
-Often the price level ended a century near where they started.
+Often the price levels ended a century near where they started.
 
 Things were different in the 20th century, as we shall see in this lecture.
 
@@ -202,7 +202,7 @@ The graphs depict logarithms of price levels during the early post World War I y
 * Figure 3.3, Wholesale prices, Poland, 1921-1924 (page 44)
 * Figure 3.4, Wholesale prices, Germany, 1919-1924 (page 45)
 
-We have added logarithms of the exchange rates vis a vis the US dollar to each of the four graphs
+We have added logarithms of the exchange rates vis-à-vis the US dollar to each of the four graphs
 from chapter 3 of {cite}`sargent2013rational`.
 
 Data underlying our graphs appear in tables in an appendix to chapter 3 of {cite}`sargent2013rational`.
@@ -382,7 +382,7 @@ For each country, we'll plot two graphs.
 The first graph plots logarithms of 
 
 * price levels
-* exchange rates vis a vis US dollars
+* exchange rates vis-à-vis US dollars
 
 For each country, the scale on the right side of a graph will pertain to the price level while the scale on the left side of a graph will pertain to the exchange rate. 
 
@@ -392,7 +392,7 @@ For each country, the second graph plots a centered three-month moving average o
 
 The sources of our data are:
 
-* Table 3.3, $\exp p$
+* Table 3.3, retail price level $\exp p$
 * Table 3.4, exchange rate with US
 
 ```{code-cell} ipython3

lectures/prob_dist.md

@@ -155,74 +155,43 @@ Check that your answers agree with `u.mean()` and `u.var()`.
 
 #### Bernoulli distribution
 
-Another useful (and more interesting) distribution is the Bernoulli distribution
+Another useful distribution is the Bernoulli distribution on $S = \{0,1\}$, which has PMF:
 
-We can import the uniform distribution on $S = \{1, \ldots, n\}$  from SciPy like so:
-
-```{code-cell} ipython3
-n = 10
-u = scipy.stats.randint(1, n+1)
-```
+$$
+p(x_i)=
+\begin{cases}
+p & \text{if $x_i = 1$}\\
+1-p & \text{if $x_i = 0$}
+\end{cases}
+$$
 
+Here $x_i \in S$ is the outcome of the random variable.
 
-Here's the mean and variance
+We can import the Bernoulli distribution on $S = \{0,1\}$ from SciPy like so:
 
 ```{code-cell} ipython3
-u.mean(), u.var()
+p = 0.4 
+u = scipy.stats.bernoulli(p)
 ```
 
-The formula for the mean is $(n+1)/2$, and the formula for the variance is $(n^2 - 1)/12$.
-
 
-Now let's evaluate the PMF
+Here's the mean and variance:
 
 ```{code-cell} ipython3
-u.pmf(1)
+u.mean(), u.var()
 ```
 
-```{code-cell} ipython3
-u.pmf(2)
-```
+The formula for the mean is $p$, and the formula for the variance is $p(1-p)$.
 
 
-Here's a plot of the probability mass function:
+Now let's evaluate the PMF:
 
 ```{code-cell} ipython3
-fig, ax = plt.subplots()
-S = np.arange(1, n+1)
-ax.plot(S, u.pmf(S), linestyle='', marker='o', alpha=0.8, ms=4)
-ax.vlines(S, 0, u.pmf(S), lw=0.2)
-ax.set_xticks(S)
-plt.show()
-```
-
-
-Here's a plot of the CDF:
-
-```{code-cell} ipython3
-fig, ax = plt.subplots()
-S = np.arange(1, n+1)
-ax.step(S, u.cdf(S))
-ax.vlines(S, 0, u.cdf(S), lw=0.2)
-ax.set_xticks(S)
-plt.show()
-```
-
-
-The CDF jumps up by $p(x_i)$ and $x_i$.
-
-
-```{exercise}
-:label: prob_ex2
-
-Calculate the mean and variance for this parameterization (i.e., $n=10$)
-directly from the PMF, using the expressions given above.
-
-Check that your answers agree with `u.mean()` and `u.var()`. 
+u.pmf(0)
+u.pmf(1)
 ```
 
 
-
 #### Binomial distribution
 
 Another useful (and more interesting) distribution is the **binomial distribution** on $S=\{0, \ldots, n\}$, which has PMF