Tom's first edits of June 18 in main repo

lectures/_static/quant-econ.bib

@@ -3,6 +3,14 @@
 Note: Extended Information (like abstracts, doi, url's etc.) can be found in quant-econ-extendedinfo.bib file in _static/
 ###
 
+@book{cochrane2023fiscal,
+  title={The Fiscal Theory of the Price Level},
+  author={Cochrane, John H},
+  year={2023},
+  publisher={Princeton University Press},
+  address={Princeton, New Jersey}
+}
+
 @incollection{sargent1982ends,
   title={The ends of four big inflations},
   author={Sargent, Thomas J},

lectures/cagan_ree.md

@@ -30,6 +30,11 @@ import matplotlib.pyplot as plt
 <!-- #region -->
 We'll use linear algebra first to explain and then do  some experiments with  a "fiscal theory of the price level".
 
+A fiscal theory of the price level was described by Thomas Sargent and Neil Wallace in chapter 5 of 
+a 1981 article title "Unpleasant Monetarist Arithmetic".  The theory has been extended, criticized, and applied by John Cochrane in {cite}`cochrane2023fiscal`.
+
+In another lecture,  we'll describe some historical episodes in which the elemental forces at work in the fiscal theory help to explain some early twentieth century hyperinflations that occurred in the wake of World War I.
+
 
 According to this model, when the government persistently spends more than it collects in taxes and prints money to finance the shortfall (the "shortfall" is called the "government deficit"), it puts upward pressure on the price level and generates
 persistent inflation.
@@ -368,6 +373,86 @@ This is because the reduction in $\mu$ at $T_1$ has been foreseen from the start
 While the log money supply portrayed in the bottom panel has a kink at $T_1$, the log  price level does not -- it is "smooth" -- once again a consequence of the fact that the
 reduction in $\mu$ has been foreseen.
 
+To set the stage for our next experiment, we want to study the determinants of the price level a little more.
+
+
+### The log price level
+
+We can use equations {eq}`eq:caganmd` and {eq}`eq:ree`
+to discover that the log of the price level satisfies
+
+$$
+p_t = m_t + \alpha \pi_t
+$$ (eq:pformula2)
+
+or, by using equation  {eq}`eq:fisctheory1`,
+
+$$ 
+p_t = m_t + \alpha \left[ (1-\delta) \sum_{s=t}^T \delta^{s-t} \mu_s +  \delta^{T+1-t} \pi_{T+1}^*  \right] 
+$$ (eq:pfiscaltheory2)
+
+In our next experiment, we'll study a "surprise" permanent change in the money growth that beforehand 
+was completely unanticipated.  
+
+At time $T_1$ when the "surprise" money growth rate  change occurs,  to satisfy
+equation {eq}`eq:pformula2`, the log of real balances jumps 
+**upward* as  $\pi_t$ jumps **downward**.
+
+But in order for $m_t - p_t$ to jump, which variable jumps, $m_{T_1}$ or $p_{T_1}$?
+
+
+###  What jumps?
+
+What jumps at $T_1$?
+
+Is it $p_{T_1}$ or  $m_{T_1}$?
+
+
+If we insist that the money supply $m_{T_1}$ is locked at its value $m_{T_1}^1$ inherited from the past, then formula {eq}`eq:pformula2` implies  that the price level jumps downward  at time $T_1$, to coincide with the downward jump in 
+$\pi_{T_1}$ 
+
+An alternative assumption about the money supply level is that as part of the "inflation stabilization",
+the government resets $m_{T_1}$ according to
+
+$$
+ m_{T_1}^2 - m_{T_1}^1 = \alpha (\pi^1 - \pi^2)
+$$ (eq:eqnmoneyjump)
+
+By letting money jump according to equation {eq}`eq:eqnmoneyjump` the monetary authority  prevents  the price level from **falling** at the moment that the unanticipated stabilization arrives.
+
+In various research papers about stabilizations of high inflations, the jump in the money supply described by equation {eq}`eq:eqnmoneyjump` has been called
+"the velocity dividend" that a government reaps from implementing a regime change that sustains a permanently lower inflation rate.
+
+#### Technical details about whether $p$ or $m$ jumps at $T_1$
+
+We have noted that  with a constant expected forward sequence $\mu_s = \bar \mu$ for $s\geq t$, $\pi_{t} =\bar{\mu}$.
+
+A consequence is that at $T_1$, either $m$ or $p$ must "jump" at $T_1$.
+
+We'll study both cases. 
+
+#### $m_{T_{1}}$ does not jump.
+
+$$
+\begin{align*}
+m_{T_{1}}&=m_{T_{1}-1}+\mu_{0}\\\pi_{T_{1}}&=\mu^{*}\\p_{T_{1}}&=m_{T_{1}}+\alpha\pi_{T_{1}}
+\end{align*}
+$$
+Simply glue the sequences $t\leq T_1$ and $t > T_1$.
+
+####  $m_{T_{1}}$ jumps.
+
+We reset $m_{T_{1}}$  so that $p_{T_{1}}=\left(m_{T_{1}-1}+\mu_{0}\right)+\alpha\mu_{0}$, with $\pi_{T_{1}}=\mu^{*}$.
+
+Then, 
+
+$$ 
+m_{T_{1}}=p_{T_{1}}-\alpha\pi_{T_{1}}=\left(m_{T_{1}-1}+\mu_{0}\right)+\alpha\left(\mu_{0}-\mu^{*}\right) 
+$$
+
+We then compute for the remaining $T-T_{1}$ periods with $\mu_{s}=\mu^{*},\forall s\geq T_{1}$ and the initial condition $m_{T_{1}}$ from above.
+
+We are now technically equipped to discuss our next experiment.
 
 #### Experiment 2: an unforeseen sudden stabilization
 
@@ -390,7 +475,7 @@ To capture a "completely unanticipated permanent  shock to the $\{\mu\}$ process
 that emerges under path 2 for $t \geq T_1$ to the $\mu_t, \pi_t$ path that had emerged under path 1 for $ t=0, \ldots,
 T_1 -1$.
 
-We can do the MIT shock calculations entirely by hand. 
+We can do the MIT shock calculations mostly by hand. 
 
 Thus, for path 1, $\pi_t = \mu_0 $ for all $t \in  [0, T_1-1]$, while for path 2,
 $\mu_s = \mu^*$ for all $s \geq T_1$.  
@@ -399,7 +484,7 @@ We now move on to experiment 2, our "MIT shock", completely unforeseen
 sudden stabilization.
 
 We set this up so that the $\{\mu_t\}$ sequences that describe the sudden stabilization
-are identical to those for experiment 1, the foreseen suddent stabilization.
+are identical to those for experiment 1, the foreseen sudden stabilization.
 
 The following code does the calculations and plots outcomes.
 <!-- #endregion -->
@@ -523,17 +608,17 @@ ax[2].plot(T_seq, m_seq_1 - p_seq_1,
 ax[2].set_ylabel(r'$m - p$')
 
 ax[3].plot(T_seq, m_seq_2_regime1, 
-             label=r'Unforseen (Insist on $m_{T_1}$)')
+             label=r'Unforeseen (Smooth $m_{T_1}$)')
 ax[3].plot(T_seq, m_seq_2_regime2, 
-             label=r'Unforseen (Reset $m_{T_1}$)')
+             label=r'Unforeseen ($m_{T_1}$ jumps)')
 ax[3].plot(T_seq, m_seq_1, 
              label='Foreseen shock')
 ax[3].set_ylabel(r'$m$')
 
 ax[4].plot(T_seq, p_seq_2_regime1, 
-             label=r'Unforseen (Insist on $m_{T_1}$)')
+             label=r'Unforeseen (Smooth $m_{T_1}$)')
 ax[4].plot(T_seq, p_seq_2_regime2, 
-             label=r'Unforseen (Reset $m_{T_1}$)')
+             label=r'Unforeseen ($m_{T_1}$ jumps)')
 ax[4].plot(T_seq, p_seq_1, 
              label='Foreseen')
 ax[4].set_ylabel(r'$p$')
@@ -550,79 +635,6 @@ plt.show()
 
 +++ {"user_expressions": []}
 
-### The log price level
-
-We can use equations {eq}`eq:caganmd` and {eq}`eq:ree`
-to discover that the log of the price level satisfies
-
-$$
-p_t = m_t + \alpha \pi_t
-$$ (eq:pformula2)
-
-or, by using equation  {eq}`eq:fisctheory1`,
-
-$$ 
-p_t = m_t + \alpha \left[ (1-\delta) \sum_{s=t}^T \delta^{s-t} \mu_s +  \delta^{T+1-t} \pi_{T+1}^*  \right] 
-$$ (eq:pfiscaltheory2)
-
-At time $T_1$ when the "surprise" regime change occurs,  to satisfy
-equation {eq}`eq:pformula2`, the log of real balances jumps 
-**upward* as  $\pi_t$ jumps **downward**.
-
-But in order for $m_t - p_t$ to jump, which variable jumps, $m_{T_1}$ or $p_{T_1}$?
-
-
-###  What jumps?
-
-What jumps at $T_1$?
-
-Is it $p_{T_1}$ or  $m_{T_1}$?
-
-
-If we insist that the money supply $m_{T_1}$ is locked at its value $m_{T_1}^1$ inherited from the past, then formula {eq}`eq:pformula2` implies  that the price level jumps downward  at time $T_1$, to coincide with the downward jump in 
-$\pi_{T_1}$ 
-
-An alternative assumption about the money supply level is that as part of the "inflation stabilization",
-the government resets $m_{T_1}$ according to
-
-$$
- m_{T_1}^2 - m_{T_1}^1 = \alpha (\pi^1 - \pi^2)
-$$ (eq:eqnmoneyjump)
-
-By letting money jump according to equation {eq}`eq:eqnmoneyjump` the monetary authority  prevents  the price level
-from **falling** at the moment that the unanticipated stabilization arrives.
-
-In various research papers about stabilizations of high inflations, the jump in the money supply described by equation {eq}`eq:eqnmoneyjump` has been called
-"the velocity dividend" that a government reaps from implementing a regime change that sustains a permanently lower inflation rate.
-
-#### Technical details about whether $p$ or $m$ jumps at $T_1$
-
-We have noted that  with a constant expected forward sequence $\mu_s = \bar \mu$ for $s\geq t$, $\pi_{t} =\bar{\mu}$.
-
-A consequence is that at $T_1$, either $m$ or $p$ must "jump" at $T_1$.
-
-We'll study both cases. 
-
-#### $m_{T_{1}}$ does not jump.
-
-$$
-\begin{align*}
-m_{T_{1}}&=m_{T_{1}-1}+\mu_{0}\\\pi_{T_{1}}&=\mu^{*}\\p_{T_{1}}&=m_{T_{1}}+\alpha\pi_{T_{1}}
-\end{align*}
-$$
-Simply glue the sequences $t\leq T_1$ and $t > T_1$.
-
-####  $m_{T_{1}}$ jumps.
-
-We reset $m_{T_{1}}$  so that $p_{T_{1}}=\left(m_{T_{1}-1}+\mu_{0}\right)+\alpha\mu_{0}$, with $\pi_{T_{1}}=\mu^{*}$.
-
-Then, 
-
-$$ 
-m_{T_{1}}=p_{T_{1}}-\alpha\pi_{T_{1}}=\left(m_{T_{1}-1}+\mu_{0}\right)+\alpha\left(\mu_{0}-\mu^{*}\right) 
-$$
-
-We then compute for the remaining $T-T_{1}$ periods with $\mu_{s}=\mu^{*},\forall s\geq T_{1}$ and the initial condition $m_{T_{1}}$ from above.
 
 
 #### Experiment 3