diff --git a/lectures/_toc.yml b/lectures/_toc.yml
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@@ -55,7 +55,7 @@ parts:
- file: unpleasant
- file: money_inflation_nonlinear
- file: laffer_adaptive
- #- file: french_rev
+ - file: french_rev
- file: ak2
- caption: Stochastic Dynamics
numbered: true
diff --git a/lectures/french_rev.md b/lectures/french_rev.md
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+---
+jupytext:
+ text_representation:
+ extension: .md
+ format_name: myst
+ format_version: 0.13
+ jupytext_version: 1.16.1
+kernelspec:
+ display_name: Python 3 (ipykernel)
+ language: python
+ name: python3
+---
+
+
+# Inflation During French Revolution
+
+
+## Overview
+
+This lecture describes some monetary and fiscal features of the French Revolution
+described by {cite}`sargent_velde1995`.
+
+We use matplotlib to replicate several of the graphs that they used to present salient patterns.
+
+
+
+## Fiscal Situation and Response of National Assembly
+
+
+In response to a motion by Catholic Bishop Talleyrand,
+the National Assembly confiscated and nationalized Church lands.
+
+But the National Assembly was dominated by free market advocates, not socialists.
+
+The National Assembly intended to use earnings from Church lands to service its national debt.
+
+To do this, it began to implement a ''privatization plan'' that would let it service its debt while
+not raising taxes.
+
+Their plan involved issuing paper notes called ''assignats'' that entitled bearers to use them to purchase state lands.
+
+These paper notes would be ''as good as silver coins'' in the sense that both were acceptable means of payment in exchange for those (formerly) church lands.
+
+Finance Minister Necker and the Constituants planned
+to solve the privatization problem **and** the debt problem simultaneously
+by creating a new currency.
+
+They devised a scheme to raise revenues by auctioning
+the confiscated lands, thereby withdrawing paper notes issued on the security of
+the lands sold by the government.
+
+ This ''tax-backed money'' scheme propelled the National Assembly into the domain of monetary experimentation.
+
+Records of their debates show
+how members of the Assembly marshaled theory and evidence to assess the likely
+effects of their innovation.
+
+They quoted David Hume and Adam Smith and cited John
+Law's System of 1720 and the American experiences with paper money fifteen years
+earlier as examples of how paper money schemes can go awry.
+
+
+### Necker's plan and how it was tweaked
+
+Necker's original plan embodied two components: a national bank and a new
+financial instrument, the ''assignat''.
+
+
+Necker's national
+bank was patterned after the Bank of England. He proposed to transform the *Caisse d'Escompte* into a national bank by granting it a monopoly on issuing
+notes and marketing government debt. The *Caisse* was a
+discount bank founded in 1776 whose main function was to discount commercial bills
+and issue convertible notes. Although independent of the government in principle,
+it had occasionally been used as a source of loans. Its notes had been declared
+inconvertible in August 1788, and by the time of Necker's proposal, its reserves
+were exhausted. Necker's plan placed the National Estates (as the Church lands
+became known after the addition of the royal demesne) at the center of the financial
+picture: a ''Bank of France'' would issue a $5\%$ security mortgaged on the prospective
+receipts from the modest sale of some 400 millions' worth of National Estates in
+the years 1791 to 1793.
+```{note}
+ Only 170 million was to be used initially
+to cover the deficits of 1789 and 1790.
+```
+
+
+By mid-1790, members of the National Assembly had agreed to sell the National
+Estates and to use the proceeds to service the debt in a ``tax-backed money'' scheme
+```{note}
+Debt service costs absorbed
+ over 60\% of French government expenditures.
+```
+
+The government would issue securities with which it would reimburse debt.
+
+The securities
+were acceptable as payment for National Estates purchased at auctions; once received
+in payment, they were to be burned.
+
+```{note}
+The appendix to {cite}`sargent_velde1995` describes the
+auction rules in detail.
+```
+The Estates available for sale were thought to be worth about 2,400
+million, while the exactable debt (essentially fixed-term loans, unpaid arrears,
+and liquidated offices) stood at about 2,000 million. The value of the land was
+sufficient to let the Assembly retire all of the exactable debt and thereby eliminate
+the interest payments on it. After lengthy debates, in August 1790, the Assembly set the denomination
+and interest rate structure of the debt.
+
+
+```{note} Two distinct
+aspects of monetary theory help in thinking about the assignat plan. First, a system
+beginning with a commodity standard typically has room for a once-and-for-all emission
+of (an unbacked) paper currency that can replace the commodity money without generating
+inflation. \citet{Sargent/Wallace:1983} describe models with this property. That
+commodity money systems are wasteful underlies Milton Friedman's (1960) TOM:ADD REFERENCE preference
+for a fiat money regime over a commodity money. Second, in a small country on a
+commodity money system that starts with restrictions on intermediation, those restrictions
+can be relaxed by letting the government issue bank notes on the security of safe
+private indebtedness, while leaving bank notes convertible into gold at par. See
+Adam Smith and Sargent and Wallace (1982) for expressions of this idea. TOM: ADD REFERENCES HEREAND IN BIBTEX FILE.
+```
+
+
+```{note}
+The
+National Assembly debated many now classic questions in monetary economics. Under
+what conditions would money creation generate inflation, with what consequences
+for business conditions? Distinctions were made between issue of money to pay off
+debt, on one hand, and monetization of deficits, on the other. Would *assignats* be akin
+to notes emitted under a real bills regime, and cause loss of specie, or would
+they circulate alongside specie, thus increasing the money stock? Would inflation
+affect real wages? How would it impact foreign trade, competitiveness of French
+industry and agriculture, balance of trade, foreign exchange?
+```
+
+## Data Sources
+
+This notebook uses data from three spreadsheets:
+
+ * datasets/fig_3.ods
+ * datasets/dette.xlsx
+ * datasets/assignat.xlsx
+
+```{code-cell} ipython3
+import numpy as np
+import pandas as pd
+import matplotlib.pyplot as plt
+plt.rcParams.update({'font.size': 12})
+```
+
+
+## Figure 1
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Ratio of debt service to taxes, Britain and France"
+ name: fig1
+---
+
+# Read the data from the Excel file
+data1 = pd.read_excel('datasets/dette.xlsx', sheet_name='Debt', usecols='R:S', skiprows=5, nrows=99, header=None)
+data1a = pd.read_excel('datasets/dette.xlsx', sheet_name='Debt', usecols='P', skiprows=89, nrows=15, header=None)
+
+# Plot the data
+plt.figure()
+plt.plot(range(1690, 1789), 100 * data1.iloc[:, 1], linewidth=0.8)
+
+date = np.arange(1690, 1789)
+index = (date < 1774) & (data1.iloc[:, 0] > 0)
+plt.plot(date[index], 100 * data1[index].iloc[:, 0], '*:', color='r', linewidth=0.8)
+
+# Plot the additional data
+plt.plot(range(1774, 1789), 100 * data1a, '*:', color='orange')
+
+# Note about the data
+# The French data before 1720 don't match up with the published version
+# Set the plot properties
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().set_facecolor('white')
+plt.gca().set_xlim([1688, 1788])
+plt.ylabel('% of Taxes')
+
+plt.tight_layout()
+plt.show()
+
+#plt.savefig('frfinfig1.pdf', dpi=600)
+#plt.savefig('frfinfig1.jpg', dpi=600)
+```
+
+
+TO TEACH TOM: By staring at {numref}`fig1` carefully
+
+
+## Figure 2
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Government Expenditures and Tax Revenues in Britain"
+ name: fig2
+---
+
+# Read the data from Excel file
+data2 = pd.read_excel('datasets/dette.xlsx', sheet_name='Militspe', usecols='M:X', skiprows=7, nrows=102, header=None)
+
+# Plot the data
+plt.figure()
+plt.plot(range(1689, 1791), data2.iloc[:, 5], linewidth=0.8)
+plt.plot(range(1689, 1791), data2.iloc[:, 11], linewidth=0.8, color='red')
+plt.plot(range(1689, 1791), data2.iloc[:, 9], linewidth=0.8, color='orange')
+plt.plot(range(1689, 1791), data2.iloc[:, 8], 'o-', markerfacecolor='none', linewidth=0.8, color='purple')
+
+# Customize the plot
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().tick_params(labelsize=12)
+plt.xlim([1689, 1790])
+plt.ylabel('millions of pounds', fontsize=12)
+
+# Add text annotations
+plt.text(1765, 1.5, 'civil', fontsize=10)
+plt.text(1760, 4.2, 'civil plus debt service', fontsize=10)
+plt.text(1708, 15.5, 'total govt spending', fontsize=10)
+plt.text(1759, 7.3, 'revenues', fontsize=10)
+
+
+plt.tight_layout()
+plt.show()
+
+# Save the figure as a PDF
+#plt.savefig('frfinfig2.pdf', dpi=600)
+```
+
+
+## Figure 3
+
+
+
+
+```{code-cell} ipython3
+# Read the data from the Excel file
+data1 = pd.read_excel('datasets/fig_3.xlsx', sheet_name='Sheet1', usecols='C:F', skiprows=5, nrows=30, header=None)
+
+data1.replace(0, np.nan, inplace=True)
+```
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Government Spending and Tax Revenues in France"
+ name: fr_fig3
+---
+# Plot the data
+plt.figure()
+
+plt.plot(range(1759, 1789, 1), data1.iloc[:, 0], '-x', linewidth=0.8)
+plt.plot(range(1759, 1789, 1), data1.iloc[:, 1], '--*', linewidth=0.8)
+plt.plot(range(1759, 1789, 1), data1.iloc[:, 2], '-o', linewidth=0.8, markerfacecolor='none')
+plt.plot(range(1759, 1789, 1), data1.iloc[:, 3], '-*', linewidth=0.8)
+
+plt.text(1775, 610, 'total spending', fontsize=10)
+plt.text(1773, 325, 'military', fontsize=10)
+plt.text(1773, 220, 'civil plus debt service', fontsize=10)
+plt.text(1773, 80, 'debt service', fontsize=10)
+plt.text(1785, 500, 'revenues', fontsize=10)
+
+
+
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.ylim([0, 700])
+plt.ylabel('millions of livres')
+
+plt.tight_layout()
+plt.show()
+
+#plt.savefig('frfinfig3.jpg', dpi=600)
+```
+
+
+TO TEACH TOM: By staring at {numref}`fr_fig3` carefully
+
+```{code-cell} ipython3
+# Plot the data
+plt.figure()
+
+plt.plot(np.arange(1759, 1789, 1)[~np.isnan(data1.iloc[:, 0])], data1.iloc[:, 0][~np.isnan(data1.iloc[:, 0])], '-x', linewidth=0.8)
+plt.plot(np.arange(1759, 1789, 1)[~np.isnan(data1.iloc[:, 1])], data1.iloc[:, 1][~np.isnan(data1.iloc[:, 1])], '--*', linewidth=0.8)
+plt.plot(np.arange(1759, 1789, 1)[~np.isnan(data1.iloc[:, 2])], data1.iloc[:, 2][~np.isnan(data1.iloc[:, 2])], '-o', linewidth=0.8, markerfacecolor='none')
+plt.plot(np.arange(1759, 1789, 1)[~np.isnan(data1.iloc[:, 3])], data1.iloc[:, 3][~np.isnan(data1.iloc[:, 3])], '-*', linewidth=0.8)
+
+plt.text(1775, 610, 'total spending', fontsize=10)
+plt.text(1773, 325, 'military', fontsize=10)
+plt.text(1773, 220, 'civil plus debt service', fontsize=10)
+plt.text(1773, 80, 'debt service', fontsize=10)
+plt.text(1785, 500, 'revenues', fontsize=10)
+
+
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.ylim([0, 700])
+plt.ylabel('millions of livres')
+
+plt.tight_layout()
+plt.show()
+
+#plt.savefig('frfinfig3_ignore_nan.jpg', dpi=600)
+```
+
+
+## Figure 4
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Military Spending in Britain and France"
+ name: fig4
+---
+# French military spending, 1685-1789, in 1726 livres
+data4 = pd.read_excel('datasets/dette.xlsx', sheet_name='Militspe', usecols='D', skiprows=3, nrows=105, header=None).squeeze()
+years = range(1685, 1790)
+
+plt.figure()
+plt.plot(years, data4, '*-', linewidth=0.8)
+
+plt.plot(range(1689, 1791), data2.iloc[:, 4], linewidth=0.8)
+
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().tick_params(labelsize=12)
+plt.xlim([1689, 1790])
+plt.xlabel('*: France')
+plt.ylabel('Millions of livres')
+plt.ylim([0, 475])
+
+plt.tight_layout()
+plt.show()
+
+#plt.savefig('frfinfig4.pdf', dpi=600)
+```
+
+TO TEACH TOM: By staring at {numref}`fig4` carefully
+
+## Figure 5
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Index of real per capital revenues, France"
+ name: fig5
+---
+# Read data from Excel file
+data5 = pd.read_excel('datasets/dette.xlsx', sheet_name='Debt', usecols='K', skiprows=41, nrows=120, header=None)
+
+# Plot the data
+plt.figure()
+plt.plot(range(1726, 1846), data5.iloc[:, 0], linewidth=0.8)
+
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().set_facecolor('white')
+plt.gca().tick_params(labelsize=12)
+plt.xlim([1726, 1845])
+plt.ylabel('1726 = 1', fontsize=12)
+
+plt.tight_layout()
+plt.show()
+
+# Save the figure as a PDF
+#plt.savefig('frfinfig5.pdf', dpi=600)
+```
+
+TO TEACH TOM: By staring at {numref}`fig5` carefully
+
+## Rise and Fall of the *Assignat*
+
+
+
+ We have partitioned Figures~\ref{fig:fig7}, \ref{fig:fig8}, and \ref{fig:fig9}
+ into three periods, corresponding
+to different monetary regimes or episodes. The three clouds of points in
+Figure~\ref{fig:fig7}
+ depict different real balance-inflation relationships. Only the cloud for the
+third period has the inverse relationship familiar to us now from twentieth-century
+hyperinflations. The first period ends in the late summer of 1793, and is characterized
+by growing real balances and moderate inflation. The second period begins and ends
+with the Terror. It is marked by high real balances, around 2,500 millions, and
+roughly stable prices. The fall of Robespierre in late July 1794 begins the third
+of our episodes, in which real balances decline and prices rise rapidly. We interpret
+these three episodes in terms of three separate theories about money: a ``backing''
+or ''real bills'' theory (the text is Adam Smith (1776)),
+a legal restrictions theory (TOM: HERE PLEASE CITE
+Keynes,1940, AS WELL AS Bryant/Wallace:1984 and Villamil:1988)
+and a classical hyperinflation theory.%
+```{note}
+According to the empirical definition of hyperinflation adopted by {cite}`Cagan`,
+beginning in the month that inflation exceeds 50 percent
+per month and ending in the month before inflation drops below 50 percent per month
+for at least a year, the *assignat* experienced a hyperinflation from May to December
+1795.
+```
+We view these
+theories not as competitors but as alternative collections of ``if-then''
+statements about government note issues, each of which finds its conditions more
+nearly met in one of these episodes than in the other two.
+
+
+
+
+
+## Figure 7
+
+
+## To Do for Zejin
+
+I want to tweak and consolidate the extra lines that Zejin drew on the beautiful **Figure 7**.
+
+I'd like to experiment in plotting the **six** extra lines all on one graph -- a pair of lines for each of our subsamples
+
+ * one for the $y$ on $x$ regression line
+ * another for the $x$ on $y$ regression line
+
+I'd like the $y$ on $x$ and $x$ on $y$ lines to be in separate colors.
+
+Once we are satisfied with this new graph with its six additional lines, we can dispense with the other graphs that add one line at a time.
+
+Zejin, I can explain on zoom the lessons I want to convey with this.
+
+
+
+Just to recall, to compute the regression lines, Zejin wrote a function that use standard formulas
+for a and b in a least squares regression y = a + b x + residual -- i.e., b is ratio of sample covariance of y,x to sample variance of x; while a is then computed from a = sample mean of y - \hat b *sample mean of x
+
+We could presumably tell students how to do this with a couple of numpy lines
+I'd like to create three additional versions of the following figure.
+
+To remind you, we focused on three subperiods:
+
+
+* subperiod 1: ("real bills period): January 1791 to July 1793
+
+* subperiod 2: ("terror:): August 1793 - July 1794
+
+* subperiod 3: ("classic Cagan hyperinflation): August 1794 - March 1796
+
+
+I can explain what this is designed to show.
+
+
+
+```{code-cell} ipython3
+def fit(x, y):
+
+ b = np.cov(x, y)[0, 1] / np.var(x)
+ a = y.mean() - b * x.mean()
+
+ return a, b
+```
+
+```{code-cell} ipython3
+# load data
+caron = np.load('datasets/caron.npy')
+nom_balances = np.load('datasets/nom_balances.npy')
+
+infl = np.concatenate(([np.nan], -np.log(caron[1:63, 1] / caron[0:62, 1])))
+bal = nom_balances[14:77, 1] * caron[:, 1] / 1000
+```
+
+```{code-cell} ipython3
+# fit data
+
+# reg y on x for three periods
+a1, b1 = fit(bal[1:31], infl[1:31])
+a2, b2 = fit(bal[31:44], infl[31:44])
+a3, b3 = fit(bal[44:63], infl[44:63])
+
+# reg x on y for three periods
+a1_rev, b1_rev = fit(infl[1:31], bal[1:31])
+a2_rev, b2_rev = fit(infl[31:44], bal[31:44])
+a3_rev, b3_rev = fit(infl[44:63], bal[44:63])
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7.pdf', dpi=600)
+```
+
+
+
+```{code-cell} ipython3
+# fit data
+
+# reg y on x for three periods
+a1, b1 = fit(bal[1:31], infl[1:31])
+a2, b2 = fit(bal[31:44], infl[31:44])
+a3, b3 = fit(bal[44:63], infl[44:63])
+
+# reg x on y for three periods
+a1_rev, b1_rev = fit(infl[1:31], bal[1:31])
+a2_rev, b2_rev = fit(infl[31:44], bal[31:44])
+a3_rev, b3_rev = fit(infl[44:63], bal[44:63])
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+
+# second subsample
+plt.plot(bal[34:44], infl[34:44], '+', color='red', label='terror')
+
+# third subsample # Tom tinkered with subsample period
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7.pdf', dpi=600)
+```
+
+
+The above graph is Tom's experimental lab. We'll delete it eventually.
+
+Zejin: below is the grapth with six lines in one graph. The lines generated by regressing y on x have the same color as the corresponding data points, while the lines generated by regressing x on y are all in green.
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+plt.plot(bal[1:31], a1 + bal[1:31] * b1, color='blue', linewidth=0.8)
+plt.plot(a1_rev + b1_rev * infl[1:31], infl[1:31], color='green', linewidth=0.8)
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+plt.plot(bal[31:44], a2 + bal[31:44] * b2, color='red', linewidth=0.8)
+plt.plot(a2_rev + b2_rev * infl[31:44], infl[31:44], color='green', linewidth=0.8)
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+plt.plot(bal[44:63], a3 + bal[44:63] * b3, color='orange', linewidth=0.8)
+plt.plot(a3_rev + b3_rev * infl[44:63], infl[44:63], color='green', linewidth=0.8)
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+#plt.savefig('frfinfig7.pdf', dpi=600)
+```
+
+
+
+The graph below is Tom's version of the six lines in one graph. The lines generated by regressing y on x have the same color as the corresponding data points, while the lines generated by regressing x on y are all in green.
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+plt.plot(bal[1:31], a1 + bal[1:31] * b1, color='blue', linewidth=0.8)
+plt.plot(a1_rev + b1_rev * infl[1:31], infl[1:31], color='green', linewidth=0.8)
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+plt.plot(bal[34:44], a2 + bal[34:44] * b2, color='red', linewidth=0.8)
+plt.plot(a2_rev + b2_rev * infl[34:44], infl[34:44], color='green', linewidth=0.8)
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+plt.plot(bal[44:63], a3 + bal[44:63] * b3, color='orange', linewidth=0.8)
+plt.plot(a3_rev + b3_rev * infl[44:63], infl[44:63], color='green', linewidth=0.8)
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7.pdf', dpi=600)
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+plt.plot(bal[1:31], a1 + bal[1:31] * b1, color='blue')
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7_line1.pdf', dpi=600)
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+plt.plot(a1_rev + b1_rev * infl[1:31], infl[1:31], color='blue')
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7_line1_rev.pdf', dpi=600)
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+plt.plot(bal[31:44], a2 + bal[31:44] * b2, color='red')
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7_line2.pdf', dpi=600)
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+plt.plot(a2_rev + b2_rev * infl[31:44], infl[31:44], color='red')
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7_line2_rev.pdf', dpi=600)
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+plt.plot(bal[44:63], a3 + bal[44:63] * b3, color='orange')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7_line3.pdf', dpi=600)
+```
+
+```{code-cell} ipython3
+plt.figure()
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# first subsample
+plt.plot(bal[1:31], infl[1:31], 'o', markerfacecolor='none', color='blue', label='real bills period')
+
+# second subsample
+plt.plot(bal[31:44], infl[31:44], '+', color='red', label='terror')
+
+# third subsample
+plt.plot(bal[44:63], infl[44:63], '*', color='orange', label='classic Cagan hyperinflation')
+plt.plot(a3_rev + b3_rev * infl[44:63], infl[44:63], color='orange')
+
+plt.xlabel('real balances')
+plt.ylabel('inflation')
+plt.legend()
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig7_line3_rev.pdf', dpi=600)
+```
+
+
+## Figure 8
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Real balances of assignats (in gold and goods)"
+ name: fig8
+---
+# Read the data from Excel file
+data7 = pd.read_excel('datasets/assignat.xlsx', sheet_name='Data', usecols='P:Q', skiprows=4, nrows=80, header=None)
+data7a = pd.read_excel('datasets/assignat.xlsx', sheet_name='Data', usecols='L', skiprows=4, nrows=80, header=None)
+
+# Create the figure and plot
+plt.figure()
+h = plt.plot(pd.date_range(start='1789-11-01', periods=len(data7), freq='M'), (data7a.values * [1, 1]) * data7.values, linewidth=1.)
+plt.setp(h[1], linestyle='--', color='red')
+
+plt.vlines([pd.Timestamp('1793-07-15'), pd.Timestamp('1793-07-15')], 0, 3000, linewidth=0.8, color='orange')
+plt.vlines([pd.Timestamp('1794-07-15'), pd.Timestamp('1794-07-15')], 0, 3000, linewidth=0.8, color='purple')
+
+plt.ylim([0, 3000])
+
+# Set properties of the plot
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().set_facecolor('white')
+plt.gca().tick_params(labelsize=12)
+plt.xlim(pd.Timestamp('1789-11-01'), pd.Timestamp('1796-06-01'))
+plt.ylabel('millions of livres', fontsize=12)
+
+# Add text annotations
+plt.text(pd.Timestamp('1793-09-01'), 200, 'Terror', fontsize=12)
+plt.text(pd.Timestamp('1791-05-01'), 750, 'gold value', fontsize=12)
+plt.text(pd.Timestamp('1794-10-01'), 2500, 'real value', fontsize=12)
+
+
+plt.tight_layout()
+plt.show()
+
+# Save the figure as a PDF
+#plt.savefig('frfinfig8.pdf', dpi=600)
+```
+
+TO TEACH TOM: By staring at {numref}`fig8` carefully
+
+
+## Figure 9
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Price Level and Price of Gold (log scale)"
+ name: fig9
+---
+# Create the figure and plot
+plt.figure()
+x = np.arange(1789 + 10/12, 1796 + 5/12, 1/12)
+h, = plt.plot(x, 1. / data7.iloc[:, 0], linestyle='--')
+h, = plt.plot(x, 1. / data7.iloc[:, 1], color='r')
+
+# Set properties of the plot
+plt.gca().tick_params(labelsize=12)
+plt.yscale('log')
+plt.xlim([1789 + 10/12, 1796 + 5/12])
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+# Add vertical lines
+plt.axvline(x=1793 + 6.5/12, linestyle='-', linewidth=0.8, color='orange')
+plt.axvline(x=1794 + 6.5/12, linestyle='-', linewidth=0.8, color='purple')
+
+# Add text
+plt.text(1793.75, 120, 'Terror', fontsize=12)
+plt.text(1795, 2.8, 'price level', fontsize=12)
+plt.text(1794.9, 40, 'gold', fontsize=12)
+
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig9.pdf', dpi=600)
+```
+
+TO TEACH TOM: By staring at {numref}`fig9` carefully
+
+
+## Figure 11
+
+
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: "Spending (blue) and Revenues (orange), (real values)"
+ name: fig11
+---
+# Read data from Excel file
+data11 = pd.read_excel('datasets/assignat.xlsx', sheet_name='Budgets', usecols='J:K', skiprows=22, nrows=52, header=None)
+
+# Prepare the x-axis data
+x_data = np.concatenate([
+ np.arange(1791, 1794 + 8/12, 1/12),
+ np.arange(1794 + 9/12, 1795 + 3/12, 1/12)
+])
+
+# Remove NaN values from the data
+data11_clean = data11.dropna()
+
+# Plot the data
+plt.figure()
+h = plt.plot(x_data, data11_clean.values[:, 0], linewidth=0.8)
+h = plt.plot(x_data, data11_clean.values[:, 1], '--', linewidth=0.8)
+
+
+
+# Set plot properties
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().set_facecolor('white')
+plt.gca().tick_params(axis='both', which='major', labelsize=12)
+plt.xlim([1791, 1795 + 3/12])
+plt.xticks(np.arange(1791, 1796))
+plt.yticks(np.arange(0, 201, 20))
+
+# Set the y-axis label
+plt.ylabel('millions of livres', fontsize=12)
+
+
+
+plt.tight_layout()
+plt.show()
+
+#plt.savefig('frfinfig11.pdf', dpi=600)
+```
+TO TEACH TOM: By staring at {numref}`fig11` carefully
+
+
+## Figure 12
+
+
+```{code-cell} ipython3
+# Read data from Excel file
+data12 = pd.read_excel('datasets/assignat.xlsx', sheet_name='seignor', usecols='F', skiprows=6, nrows=75, header=None).squeeze()
+
+
+# Create a figure and plot the data
+plt.figure()
+plt.plot(pd.date_range(start='1790', periods=len(data12), freq='M'), data12, linewidth=0.8)
+
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+
+plt.axhline(y=472.42/12, color='r', linestyle=':')
+plt.xticks(ticks=pd.date_range(start='1790', end='1796', freq='AS'), labels=range(1790, 1797))
+plt.xlim(pd.Timestamp('1791'), pd.Timestamp('1796-02') + pd.DateOffset(months=2))
+plt.ylabel('millions of livres', fontsize=12)
+plt.text(pd.Timestamp('1793-11'), 39.5, 'revenues in 1788', verticalalignment='top', fontsize=12)
+
+
+plt.tight_layout()
+plt.show()
+
+#plt.savefig('frfinfig12.pdf', dpi=600)
+```
+
+
+## Figure 13
+
+
+```{code-cell} ipython3
+# Read data from Excel file
+data13 = pd.read_excel('datasets/assignat.xlsx', sheet_name='Exchge', usecols='P:T', skiprows=3, nrows=502, header=None)
+
+# Plot the last column of the data
+plt.figure()
+plt.plot(data13.iloc[:, -1], linewidth=0.8)
+
+# Set properties of the plot
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().set_xlim([1, len(data13)])
+
+# Set x-ticks and x-tick labels
+ttt = np.arange(1, len(data13) + 1)
+plt.xticks(ttt[~np.isnan(data13.iloc[:, 0])],
+ ['Mar', 'Apr', 'May', 'Jun', 'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec', 'Jan', 'Feb',
+ 'Mar', 'Apr', 'May', 'Jun', 'Jul', 'Aug', 'Sep'])
+
+# Add text to the plot
+plt.text(1, 120, '1795', fontsize=12, ha='center')
+plt.text(262, 120, '1796', fontsize=12, ha='center')
+
+# Draw a horizontal line and add text
+plt.axhline(y=186.7, color='red', linestyle='-', linewidth=0.8)
+plt.text(150, 190, 'silver parity', fontsize=12)
+
+# Add an annotation with an arrow
+plt.annotate('end of the assignat', xy=(340, 172), xytext=(380, 160),
+ arrowprops=dict(facecolor='black', arrowstyle='->'), fontsize=12)
+
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig13.pdf', dpi=600)
+```
+
+
+## Figure 14
+
+
+```{code-cell} ipython3
+# figure 14
+data14 = pd.read_excel('datasets/assignat.xlsx', sheet_name='Post-95', usecols='I', skiprows=9, nrows=91, header=None).squeeze()
+data14a = pd.read_excel('datasets/assignat.xlsx', sheet_name='Post-95', usecols='F', skiprows=100, nrows=151, header=None).squeeze()
+
+plt.figure()
+h = plt.plot(data14, '*-', markersize=2, linewidth=0.8)
+plt.plot(np.concatenate([np.full(data14.shape, np.nan), data14a]), linewidth=0.8)
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.gca().set_xticks(range(20, 237, 36))
+plt.gca().set_xticklabels(range(1796, 1803))
+plt.xlabel('*: Before the 2/3 bankruptcy')
+plt.ylabel('Francs')
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig14.pdf', dpi=600)
+```
+
+
+## Figure 15
+
+
+```{code-cell} ipython3
+# figure 15
+data15 = pd.read_excel('datasets/assignat.xlsx', sheet_name='Post-95', usecols='N', skiprows=4, nrows=88, header=None).squeeze()
+
+plt.figure()
+h = plt.plot(range(2, 90), data15, '*-', linewidth=0.8)
+plt.setp(h, markersize=2)
+plt.gca().spines['top'].set_visible(False)
+plt.gca().spines['right'].set_visible(False)
+plt.text(47.5, 11.4, '17 brumaire', horizontalalignment='left', fontsize=12)
+plt.text(49.5, 14.75, '19 brumaire', horizontalalignment='left', fontsize=12)
+plt.text(15, -1, 'Vendémiaire 8', fontsize=12, horizontalalignment='center')
+plt.text(45, -1, 'Brumaire', fontsize=12, horizontalalignment='center')
+plt.text(75, -1, 'Frimaire', fontsize=12, horizontalalignment='center')
+plt.ylim([0, 25])
+plt.xticks([], [])
+plt.ylabel('Francs')
+
+plt.tight_layout()
+plt.show()
+#plt.savefig('frfinfig15.pdf', dpi=600)
+```
+
+```{code-cell} ipython3
+
+```
diff --git a/lectures/greek_square.md b/lectures/greek_square.md
index 80664a2c..db14f12b 100644
--- a/lectures/greek_square.md
+++ b/lectures/greek_square.md
@@ -48,13 +48,95 @@ In this lecture, we'll describe this method.
We'll also use invariant subspaces to describe variations on this method that are faster.
-In this lecture, we use the following import:
+## Primer on second order linear difference equation
-```{code-cell} ipython3
-:tags: []
+Consider the following second-order linear difference equation
+
+$$
+y_t = a_1 y_{t-1} + a_2 y_{t-2}, \quad t \geq 0
+$$ (eq:2diff1)
+
+where $(y_{-1}, y_{-2})$ is a pair of given initial conditions.
+
+We want to find expressions for $y_t, t \geq 0$ as functions of the initial conditions $(y_{-1}, y_{-2})$:
+
+$$
+y_t = g((y_{-1}, y_{-2});t), \quad t \geq 0
+$$ (eq:2diff2)
+
+We call such a function $g$ a **solution** of the difference equation {eq}`eq:2diff1`.
+
+One way to discover a solution is to use a guess and verify method.
+
+We shall begin by considering a special initial pair of initial conditions
+that satisfy
+
+$$
+y_{-1} = \delta y_{-2}
+$$ (eq:2diff3)
+
+where $\delta$ is a scalar to be determined.
+
+For initial condition that satisfy {eq}`eq:2diff3`
+equation {eq}`eq:2diff1` impllies that
+
+$$
+y_0 = \left(a_1 + \frac{a_2}{\delta}\right) y_{-1}
+$$ (eq:2diff4)
+
+We want
+
+$$
+\left(a_1 + \frac{a_2}{\delta}\right) = \delta
+$$ (eq:2diff5)
+
+which we can rewrite as the **characteristic equation**
+
+$$
+\delta^2 - a_1 \delta - a_2 = 0
+$$ (eq:2diff6)
+
+Applying the quadratic formula to solve for the roots of {eq}`eq:2diff6` we find that
+
+$$
+\delta = \frac{ a_1 \pm \sqrt{a_1^2 + 4 a_2}}{2}
+$$ (eq:2diff7)
+
+For either of the two $\delta$'s that satisfy equation {eq}`eq:2diff7`,
+a solution of difference equation {eq}`eq:2diff1` is
+
+$$
+y_t = \delta^t y_0 , \forall t \geq 0
+$$ (eq:2diff8)
+
+and $y_0 = a_1 y_{-1} + a_2 y_{-2}$
+
+The **general** solution of difference equation {eq}`eq:2diff1` takes the form
+
+$$
+y_t = \eta_1 \delta_1^t + \eta_2 \delta_2^t
+$$ (eq:2diff9)
+
+where $\delta_1, \delta_2$ are the two solutions {eq}`eq:2diff7` of the characteristic equation {eq}`eq:2diff6`, and $\eta_1, \eta_2$ are two constants chosen to satisfy
+
+$$
+ \begin{bmatrix} y_{-1} \cr y_{-2} \end{bmatrix} = \begin{bmatrix} \delta_1^{-1} & \delta_2^{-1} \cr \delta_1^{-2} & \delta_2^{-2} \end{bmatrix} \begin{bmatrix} \eta_1 \cr \eta_2 \end{bmatrix}
+$$ (eq:2diff10)
+
+or
+
+$$
+\begin{bmatrix} \eta_1 \cr \eta_2 \end{bmatrix} = \begin{bmatrix} \delta_1^{-1} & \delta_2^{-1} \cr \delta_1^{-2} & \delta_2^{-2} \end{bmatrix}^{-1} \begin{bmatrix} y_{-1} \cr y_{-2} \end{bmatrix}
+$$ (eq:2diff11)
+
+Sometimes we are free to choose the initial conditions $(y_{-1}, y_{-2})$, in which case we
+use system {eq}`eq:2diff10` to find the associated $(\eta_1, \eta_2)$.
+
+If we choose $(y_{-1}, y_{-2})$ to set $(\eta_1, \eta_2) = (1, 0)$, then $y_t = \delta_1^t$ for all $t \geq 0$.
+
+
+If we choose $(y_{-1}, y_{-2})$ to set $(\eta_1, \eta_2) = (0, 1)$, then $y_t = \delta_2^t$ for all $t \geq 0$.
-import numpy as np
-```
## Setup
@@ -91,6 +173,8 @@ $$ (eq:cha_eq0)
+++
+(This is an instance of equation {eq}`eq:2diff6` above.)
+
If we factor the right side of the equation {eq}`eq:cha_eq0`, we obtain
$$
@@ -206,7 +290,16 @@ We'll study these special initial conditions soon. But first let's write some P
## Implementation
-We now implement the above algorithm to compute the square root of $\sigma$
+We now implement the above algorithm to compute the square root of $\sigma$.
+
+
+In this lecture, we use the following import:
+
+```{code-cell} ipython3
+:tags: []
+
+import numpy as np
+```
```{code-cell} ipython3
:tags: []