Tom's June 8 afternoon edits of two Cagan lectures plus the table of contents file

Author: thomassargent30

lectures/_toc.yml

@@ -24,6 +24,7 @@ parts:
   - file: cons_smooth
   - file: equalizing_difference
   - file: cagan_ree
+  - file: cagan_adaptive
   - file: geom_series
 - caption: Probability and Distributions
   numbered: true

lectures/cagan_adaptive.md

@@ -0,0 +1,501 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.5
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
++++ {"user_expressions": []}
+
+
+
+# A Fiscal Theory of Price Level with Adaptive Expectations
+
+## Introduction
+
+As usual, we'll start by importing some Python modules.
+
+```{code-cell} ipython3
+import numpy as np
+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:
+
+ * a demand function for real money balances that says 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
+
+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 multplication and matrix inversion.
+
+To facilitate using  linear matrix algebra as our principal mathematical tool, we'll use a finite horizon version of
+the model.
+
+## Structure of 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 $1$.
+  
+  
+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
+  
+$$  
+m_t^d - p_t = -\alpha \pi_t^* \: , \: \alpha > 0 ; \quad t = 0, 1, \ldots, T .
+$$ (eq:caganmd)
+
+
+This equation  asserts that the demand for real balances
+is inversely related to the public's expected rate of inflation.
+
+Equating the logarithm $m_t^d$ of the demand for money  to the logarithm  $m_t$ of the supply of money in equation {eq}`eq:caganmd` and solving for the logarithm $p_t$
+of the price level gives
+
+$$
+p_t = m_t + \alpha \pi_t^*
+$$ (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)
+
+We assume that the expected rate of inflation $\pi_t^*$ is governed
+by the Friedman-Cagan adaptive expectations scheme
+
+$$
+\pi_{t+1}^* = \lambda \pi_t^* + (1 -\lambda) \pi_t 
+$$ (eq:adaptexpn)
+
+As exogenous inputs into the model, we take initial conditions $m_0, \pi_0^*$
+and a money growth sequence $\vec \mu = \{\mu_t\}_{t=0}^T$.  
+
+As endogenous outputs of our model we want to find sequences $\vec \pi = \{\pi_t\}_{t=0}^T, \vec p = \{p_t\}_{t=0}^T$ as functions of the endogenous inputs.
+
+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
+
+
+
+$$
+\begin{bmatrix} 1 & 0 & 0 & \cdots & 0 & 0 \cr
+-\lambda & 1 & 0 & \cdots & 0 & 0 \cr
+0 & - \lambda  & 1  & \cdots & 0 & 0 \cr
+\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \cr
+0 & 0 & 0 & \cdots & -\lambda & 1
+\end{bmatrix}
+\begin{bmatrix} \pi_0^* \cr
+  \pi_1^* \cr
+  \pi_2^* \cr
+  \vdots \cr
+  \pi_{T+1}^* 
+  \end{bmatrix} =
+  (1-\lambda) \begin{bmatrix} 
+  0 & 0 & 0 & \cdots & 0  \cr
+  1 & 0 & 0 & \cdots & 0   \cr
+   0 & 1 & 0 & \cdots & 0  \cr
+    \vdots &\vdots & \vdots & \cdots & \vdots  \cr
+     0 & 0 & 0 & \cdots & 1  \end{bmatrix}
+     \begin{bmatrix}\pi_0 \cr \pi_1 \cr \pi_2 \cr \vdots \cr \pi_T
+  \end{bmatrix} +
+  \begin{bmatrix} \pi_0^* \cr 0 \cr 0 \cr \vdots \cr 0 \end{bmatrix}
+$$
+
+Write this equation as
+
+$$
+A \vec \pi^* = (1-\lambda) B \vec \pi + \vec \pi_0^*
+$$ (eq:eq1)
+
+where the $(T+2) \times (T+2) $matrix $A$, the $(T+2)\times (T+1)$ matrix $B$, and the vectors $\vec \pi^* , \vec \pi_0, \pi_0^*$
+are defined implicitly by aligning these two equations.
+<!-- #endregion -->
+
+<!-- #region -->
+Next we write the key equation {eq}`eq:eqpipi` in matrix notation as
+
+$$ \begin{bmatrix}
+\pi_0 \cr \pi_1 \cr \pi_1 \cr \vdots \cr \pi_T \end{bmatrix}
+= \begin{bmatrix}
+\mu_0 \cr \mu_1 \cr \mu_2 \cr  \vdots \cr \mu_T \end{bmatrix}
++ \begin{bmatrix} - \alpha &  \alpha & 0 & \cdots & 0 & 0 \cr
+0 & -\alpha & \alpha & \cdots & 0 & 0 \cr
+0 & 0 & -\alpha & \cdots & 0 & 0 \cr
+\vdots & \vdots & \vdots & \cdots & \alpha & 0 \cr
+0 & 0 & 0 & \cdots & -\alpha  & \alpha 
+\end{bmatrix}
+\begin{bmatrix} \pi_0^* \cr
+  \pi_1^* \cr
+  \pi_2^* \cr
+  \vdots \cr
+  \pi_{T+1}^* 
+  \end{bmatrix}
+$$
+
+Represent the previous equation system in terms of vectors and matrices as
+
+$$
+\vec \pi = \vec \mu + C \vec \pi^*
+$$ (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 $\vec \pi$ as
+a function of $\vec \mu, \pi_0, \pi_0^*$.  
+
+Combine equations {eq}`eq:eq1`and {eq}`eq:eq2`  to get
+
+$$
+\begin{align*}
+A \vec \pi^* & = (1-\lambda) B \vec \pi + \vec \pi_0^* \cr
+ & = (1-\lambda) B \left[ \vec \mu + C \vec \pi^* \right] + \vec \pi_0^*
+\end{align*}
+$$
+
+which implies that
+
+$$
+\left[ A - (1-\lambda) B C \right] \vec \pi^* = (1-\lambda) B \vec \mu+ \vec \pi_0^*
+$$
+
+Multiplying both sides of the above equation by the inverse of the matrix on the left side gives
+
+$$
+\vec \pi^* = \left[ A - (1-\lambda) B C \right]^{-1} \left[ (1-\lambda) B \vec \mu+ \vec \pi_0^* \right]
+$$ (eq:eq4)
+
+Having solved equation {eq}`eq:eq4` for $\vec \pi^*$, we can use  equation {eq}`eq:eq2`  to solve for $\vec \pi$:
+
+$$
+\vec \pi = \vec \mu + C \vec \pi^*
+$$
+
+
+We have thus solved for two of the key endogenous time series determined by our model, namely, the sequence $\vec \pi^*$
+of expected inflation rates and the sequence $\vec \pi$ of actual inflation rates.  
+
+Knowing these, we can then quickly calculate the associated sequence $\vec p$  of the logarithm of the  price level
+from equation {eq}`eq:eqfiscth1`. 
+
+Let's fill in the details for this step.
+<!-- #endregion -->
+
+Since we now know $\vec \mu$  it is easy to compute $\vec m$.
+
+Thus, notice that we can represent the equations 
+
+$$ 
+m_{t+1} = m_t + \mu_t , \quad t = 0, 1, \ldots, T
+$$
+
+as the matrix equation
+
+$$
+\begin{bmatrix}
+1 & 0 & 0 & \cdots & 0 & 0 \cr
+-1 & 1 & 0 & \cdots & 0 & 0 \cr
+0  & -1 & 1 & \cdots & 0 & 0 \cr
+\vdots  & \vdots & \vdots & \vdots & 0 & 0 \cr
+0  & 0 & 0 & \cdots & 1 & 0 \cr
+0  & 0 & 0 & \cdots & -1 & 1 
+\end{bmatrix}
+\begin{bmatrix}  
+m_1 \cr m_2 \cr m_3 \cr \vdots \cr m_T \cr m_{T+1}
+\end{bmatrix}
+= \begin{bmatrix}  
+\mu_0 \cr \mu_1 \cr \mu_2 \cr \vdots \cr \mu_{T-1} \cr \mu_T
+\end{bmatrix}
++ \begin{bmatrix}  
+m_0 \cr 0 \cr 0 \cr \vdots \cr 0 \cr 0
+\end{bmatrix}
+$$ (eq:eq101)
+
+Multiplying both sides of equation {eq}`eq:eq101`  with the inverse of the matrix on the left will give 
+
+$$
+m_t = m_0 + \sum_{s=0}^{t-1} \mu_s, \quad t =1, \ldots, T+1
+$$ (eq:mcum)
+
+Equation {eq}`eq:mcum` shows that the log of the money supply at $t$ equals the log $m_0$ of the initial money supply 
+plus accumulation of rates of money growth between times $0$ and $t$.
+
+We can then compute $p_t$ for each $t$ from equation {eq}`eq:eqfiscth1`.
+
+We can write a compact formula for $\vec p $ as
+
+$$ 
+\vec p = \vec m + \alpha \hat \pi^*
+$$
+
+where 
+
+$$
+\hat \pi^* = \begin{bmatrix} \pi_0^* \cr
+  \pi_1^* \cr
+  \pi_2^* \cr
+  \vdots \cr
+  \pi_{T}^* 
+  \end{bmatrix},
+ $$
+
+which is just $\vec \pi^*$ with the last element dropped.
+ 
+
+
+## Forecast  Errors  
+
+Our computations will verify that 
+
+$$
+\hat \pi^* \neq \vec  \pi,
+$$
+
+so that in general
+
+$$ 
+\pi_t^* \neq \pi_t, \quad t = 0, 1, \ldots , T
+$$ (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}`fiscal theory of the price level <cagan_ree>`, we studied a version of the model that replaces hypothesis {eq}`eq:adaptexpn` with
+a "perfect foresight" or "rational expectations" hypothesis.
+
+```{code-cell} ipython3
+Cagan_Adaptive = namedtuple("Cagan_Adaptive", 
+                        ["α", "m0", "Eπ0", "T", "λ"])
+
+def create_cagan_model(α, m0, Eπ0, T, λ):
+    return Cagan_Adaptive(α, m0, Eπ0, T, λ)
+```
++++ {"user_expressions": []}
+
+Here we define the parameters.
+
+```{code-cell} ipython3
+# parameters
+T = 80
+T1 = 60
+α = 5
+λ = 0.9   # 0.7
+m0 = 1
+
+μ0 = 0.5
+μ_star = 0
+
+md = create_cagan_model(α=α, m0=m0, Eπ0=μ0, T=T, λ=λ)
+```
++++ {"user_expressions": []}
+
+We solve the model and plot variables of interests using the following functions.
+
+```{code-cell} ipython3
+def solve(model, μ_seq):
+    " Solve the Cagan model in finite time. "
+    
+    model_params = model.α, model.m0, model.Eπ0, model.T, model.λ
+    α, m0, Eπ0, T, λ = model_params
+    
+    A = np.eye(T+2, T+2) - λ*np.eye(T+2, T+2, k=-1)
+    B = np.eye(T+2, T+1, k=-1)
+    C = -α*np.eye(T+1, T+2) + α*np.eye(T+1, T+2, k=1)
+    Eπ0_seq = np.append(Eπ0, np.zeros(T+1))
+
+    # Eπ_seq is of length T+2
+    Eπ_seq = np.linalg.inv(A - (1-λ)*B @ C) @ ((1-λ) * B @ μ_seq + Eπ0_seq)
+
+    # π_seq is of length T+1
+    π_seq = μ_seq + C @ Eπ_seq
+
+    D = np.eye(T+1, T+1) - np.eye(T+1, T+1, k=-1)
+    m0_seq = np.append(m0, np.zeros(T))
+
+    # m_seq is of length T+2
+    m_seq = np.linalg.inv(D) @ (μ_seq + m0_seq)
+    m_seq = np.append(m0, m_seq)
+
+    # p_seq is of length T+2
+    p_seq = m_seq + α * Eπ_seq
+
+    return π_seq, Eπ_seq, m_seq, p_seq
+```
+
++++ {"user_expressions": []}
+
+```{code-cell} ipython3
+def solve_and_plot(model, μ_seq):
+    
+    π_seq, Eπ_seq, m_seq, p_seq = solve(model, μ_seq)
+    
+    T_seq = range(model.T+2)
+    
+    fig, ax = plt.subplots(2, 3, figsize=[10,5], dpi=200)
+    ax[0,0].plot(T_seq[:-1], μ_seq)
+    ax[0,1].plot(T_seq[:-1], π_seq, label=r'$\pi_t$')
+    ax[0,1].plot(T_seq, Eπ_seq, label=r'$\pi^{*}_{t}$')
+    ax[0,2].plot(T_seq, m_seq - p_seq)
+    ax[1,0].plot(T_seq, m_seq)
+    ax[1,1].plot(T_seq, p_seq)
+    
+    x_lab = r'$t$'
+    y_labs = [r'$\mu$', r'$\pi$', r'$m - p$', r'$m$', r'$p$']
+
+    k = 0
+    for i,j in zip([0, 0, 0, 1, 1], [0, 1, 2, 0, 1]):
+        ax[i,j].set_xlabel(x_lab)
+        ax[i,j].set_ylabel(y_labs[k])
+        k = k + 1
+
+    ax[0,1].legend()
+    ax[1,2].set_axis_off()
+    plt.tight_layout()
+    plt.show()
+    
+    return π_seq, Eπ_seq, m_seq, p_seq
+```
+
++++ {"user_expressions": []}
+
+
+
+## Technical condition for stability
+
+In constructing our examples, we shall assume that $(\lambda, \alpha)$ satisfy
+
+$$
+\Bigl| \frac{\lambda-\alpha(1-\lambda)}{1-\alpha(1-\lambda)} \Bigr| < 1
+$$ (eq:suffcond)
+
+The  source of this condition is the following string of deductions:
+
+$$
+\begin{align*}
+\pi_{t}&=\mu_{t}+\alpha\pi_{t+1}^{*}-\alpha\pi_{t}^{*}\\\pi_{t+1}^{*}&=\lambda\pi_{t}^{*}+(1-\lambda)\pi_{t}\\\pi_{t}&=\frac{\mu_{t}}{1-\alpha(1-\lambda)}-\frac{\alpha(1-\lambda)}{1-\alpha(1-\lambda)}\pi_{t}^{*}\\\implies\pi_{t}^{*}&=\frac{1}{\alpha(1-\lambda)}\mu_{t}-\frac{1-\alpha(1-\lambda)}{\alpha(1-\lambda)}\pi_{t}\\\pi_{t+1}&=\frac{\mu_{t+1}}{1-\alpha(1-\lambda)}-\frac{\alpha(1-\lambda)}{1-\alpha(1-\lambda)}\left(\lambda\pi_{t}^{*}+(1-\lambda)\pi_{t}\right)\\&=\frac{\mu_{t+1}}{1-\alpha(1-\lambda)}-\frac{\lambda}{1-\alpha(1-\lambda)}\mu_{t}+\frac{\lambda-\alpha(1-\lambda)}{1-\alpha(1-\lambda)}\pi_{t}
+\end{align*}
+$$
+
+By assuring that the coefficient on $\pi_t$ is less than one in absolulte 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 $\mu_0$
+from $t=0$ to $t= T_1$ and then permanently falls to $\mu^*$ at $t=T_1$.
+
+Thus, let $T_1 \in (0, T)$. 
+
+So where $\mu_0 > \mu^*$, we assume that
+
+$$
+\mu_{t+1} = \begin{cases}
+    \mu_0  , & t = 0, \ldots, T_1 -1 \\
+     \mu^* , & t \geq T_1
+     \end{cases}
+$$
+
+Notice that  we studied exactly this experiment  in a rational expectations version of the model in this lecture {doc}`fiscal theory of the price level <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.
+
+
+```{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>`.
+
+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 $T_1$.
+
+We invite you to explain to  yourself the source of this overshooting and why it does not occur in  the rational expectations version of the model.
+
+### Experiment 2
+
+Now we'll do a different experiment, namely, a gradual stabilization in which the rate of growth of the money supply smoothly 
+decline from a high value to a persistently low value.  
+
+While price level inflation eventually falls, it falls more slowly than the driving  force that ultimately causes it to fall, namely, the falling rate of growth of the money supply.
+
+The sluggish fall in inflation is explained by how anticipated  inflation $\pi_t^*$ persistently exceeds actual inflation $\pi_t$ during the transition from a high inflation to a low inflation situation. 
+
+```{code-cell} ipython3
+# parameters
+ϕ = 0.9
+μ_seq_2 = np.array([ϕ**t * μ0 + (1-ϕ**t)*μ_star for t in range(T)])
+μ_seq_2 = np.append(μ_seq_2, μ_star)
+
+
+# solve and plot
+π_seq_2, Eπ_seq_2, m_seq_2, p_seq_2 = solve_and_plot(md, μ_seq_2)
+```
++++ {"user_expressions": []}

lectures/cagan_ree.md

@@ -50,8 +50,7 @@ In those experiments, we'll encounter an instance of a ''velocity dividend'' tha
 
 To facilitate using  linear matrix algebra as our main mathematical tool, we'll use a finite horizon version of the model.
 
-As in the {doc}`present values <pv>` and {doc}`consumption smoothing<cons_smooth>` lectures, the only linear algebra that we'll be  using
-are matrix multplication and matrix inversion.
+As in the {doc}`present values <pv>` and {doc}`consumption smoothing<cons_smooth>` lectures, the only linear algebra that we'll be  using are matrix multplication and matrix inversion.
 
 
 ## Structure of the Model
@@ -293,7 +292,7 @@ $$
 
 so that, in terms of our notation and formula for $\theta_{T+1}^*$ above, $\tilde \gamma = 1$. 
 
-#### Experiment 1: foreseen sudden stabilization
+#### Experiment 1: Foreseen sudden stabilization
 
 In this experiment, we'll study how, when $\alpha >0$, a foreseen inflation stabilization has effects on inflation that proceed it.