Tom's March 20 edits of intro lectures

Author: thomassargent30

lectures/ak2.md

@@ -15,14 +15,13 @@ kernelspec:
 
 
 
-## Introduction
+## Overview
 
 
-This lecture presents a  life-cycle model consisting of overlapping generations of two-period lived people proposed  by Peter Diamond
+This lecture presents a  life-cycle model consisting of overlapping generations of two-period lived people proposed  by 
 {cite}`diamond1965national`.
 
-We'll present the version  that was   analyzed  in chapter 2 of Auerbach and 
-Kotlikoff (1987) {cite}`auerbach1987dynamic`.
+We'll present the version  that was   analyzed  in chapter 2 of {cite}`auerbach1987dynamic`.
 
 Auerbach and Kotlikoff (1987) used their  two period model as a warm-up for their analysis of  overlapping generation models of long-lived people that is the main topic of their book.
 
@@ -1171,234 +1170,3 @@ for i, name in enumerate(['τ', 'D', 'G']):
     ax.legend()
     ax.set_xlabel('t')
 ```
-
-
-## NOTES TO ZEJIN FEB 26
-
-Hi Zejin.
-
-I recommend that we add a few simple experiments in which we do the following.
-
-* take a simple baseline, perhaps on  in which capital starts at or quickly converges to a steady state value.
-
-* start from that steady-state value of capital and add a tax transfer scheme that mimics an unfunded social security system in which young people pay a lump sum tax and old people get a lump sum transfer, starting from the beginning (the initial old will love this!) 
-
-* study how this policy affects the steady state capital stock
-
-* describe in words how this policy experiment compares to one in which at time 0 the government gives government debt to the initial old
-
-* the last bullet point is vague and we'll have to tighten it up
-
-
-## Working when old as well as when young
-
-Now let's assume that each person supplies $1/2$ labor unit when both young and old.
-
-The aggregate labor supply doesn't change.
-
-The lifetime budget constraint becomes
-
-$$
-C_{yt}+\frac{C_{ot+1}}{1+r_{t+1}\left(1-\tau_{t+1}\right)}=\frac{1}{2}W_{t}\left(1-\tau_{t}\right)+\frac{1}{2}\frac{W_{t+1}\left(1-\tau_{t+1}\right)}{1+r_{t+1}\left(1-\tau_{t+1}\right)}
-$$
-
-```{code-cell} ipython3
-@jit
-def Cy_val2(Cy, W, W_next, r_next, τ, τ_next, β):
-
-    # Co given by the budget constraint
-    Co = (W / 2 * (1 - τ) - Cy) * (1 + r_next * (1 - τ_next)) + (W_next / 2) * (1 - τ_next)
-
-    return U(Cy, Co, β)
-```
-
-```{code-cell} ipython3
-W, W_next, r_next, τ, τ_next = W_hat, W_hat, r_hat, τ_hat, τ_hat
-
-# lifetime budget
-C_ub = (W / 2) * (1 - τ) + (W_next / 2) * (1 - τ_next) / (1 + r_next * (1 - τ_next))
-
-Cy_opt, U_opt, _ = brent_max(Cy_val2,      # maximand
-                             1e-3,         # lower bound
-                             C_ub-1e-3,    # upper bound
-                             args=(W, W_next, r_next, τ, τ_next, β))
-
-Cy_opt
-```
-
-
-
-Does the optimal consumption for the young now depend on the future wage $W_{t+1}$?
-
-```{code-cell} ipython3
-W, W_next, r_next, τ, τ_next = W_hat, W_hat, r_hat, τ_hat, τ_hat
-
-W_next = W_hat / 2
-
-# what's the new lifetime income?
-C_ub = (W / 2) * (1 - τ) + (W_next / 2) * (1 - τ_next) / (1 + r_next * (1 - τ_next))
-
-Cy_opt, U_opt, _ = brent_max(Cy_val2,   # maximand
-                             1e-3,      # lower bound
-                             C_ub-1e-3,    # upper bound
-                             args=(W, W_next, r_next, τ, τ_next, β))
-
-Cy_opt
-```
-
-
-
-Does it depend on the future interest rate $r_{t+1}$?
-
-```{code-cell} ipython3
-W, W_next, r_next, τ, τ_next = W_hat, W_hat, r_hat, τ_hat, τ_hat
-
-r_next = r_hat / 2
-
-# what's the new lifetime income?
-C_ub = (W / 2) * (1 - τ) + (W_next / 2) * (1 - τ_next) / (1 + r_next * (1 - τ_next))
-
-Cy_opt, U_opt, _ = brent_max(Cy_val2,   # maximand
-                             1e-3,      # lower bound
-                             C_ub-1e-3,    # upper bound
-                             args=(W, W_next, r_next, τ, τ_next, β))
-
-Cy_opt
-```
-
-
-
-Does it depend on the future tax rate $\tau_{t+1}$?
-
-```{code-cell} ipython3
-W, W_next, r_next, τ, τ_next = W_hat, W_hat, r_hat, τ_hat, τ_hat
-
-τ_next = τ_hat / 2
-
-# what's the new lifetime income?
-C_ub = (W / 2) * (1 - τ) + (W_next / 2) * (1 - τ_next) / (1 + r_next * (1 - τ_next))
-
-Cy_opt, U_opt, _ = brent_max(Cy_val2,   # maximand
-                             1e-3,      # lower bound
-                             C_ub-1e-3,    # upper bound
-                             args=(W, W_next, r_next, τ, τ_next, β))
-
-Cy_opt
-```
-
-```{code-cell} ipython3
-T = 20
-tax_cut = 1 / 3
-
-K_hat, Y_hat, Cy_hat, Co_hat = init_ss[:4]
-W_hat, r_hat = init_ss[4:6]
-τ_hat, Ag_hat, G_hat = init_ss[6:9]
-
-# initial guesses of prices
-W_seq = np.ones(T+2) * W_hat
-r_seq = np.ones(T+2) * r_hat
-
-# initial guesses of policies
-τ_seq = np.ones(T+2) * τ_hat
-
-Ag_seq = np.zeros(T+1)
-G_seq = np.ones(T+1) * G_hat
-
-# containers
-K_seq = np.empty(T+2)
-Y_seq = np.empty(T+2)
-C_seq = np.empty((T+1, 2))
-
-# t=0, starting from steady state
-K_seq[0], Y_seq[0] = K_hat, Y_hat
-W_seq[0], r_seq[0] = W_hat, r_hat
-
-# tax cut
-τ_seq[0] = τ_hat * (1 - tax_cut)
-Ag1 = Ag_hat * (1 + r_seq[0] * (1 - τ_seq[0])) + τ_seq[0] * Y_hat - G_hat
-Ag_seq[1:] = Ag1
-
-# immediate effect on consumption
-# only know about Co,0 but not Cy,0
-C_seq[0, 1] = (W_hat / 2) + K_seq[0] * (1 + r_hat * (1 - τ_hat))
-
-# prepare to plot iterations until convergence
-fig, axs = plt.subplots(1, 3, figsize=(14, 4))
-
-# containers for checking convergence (Don't use np.copy)
-W_seq_old = np.empty_like(W_seq)
-r_seq_old = np.empty_like(r_seq)
-τ_seq_old = np.empty_like(τ_seq)
-
-max_iter = 500
-i_iter = 0
-tol = 1e-5 # tolerance for convergence
-
-# start iteration
-while True:
-
-    # plot current prices at ith iteration
-    for i, seq in enumerate([W_seq, r_seq, τ_seq]):
-            axs[i].plot(range(T+2), seq)
-
-    # store old prices from last iteration
-    W_seq_old[:] = W_seq
-    r_seq_old[:] = r_seq
-    τ_seq_old[:] = τ_seq
-
-    # start update quantities and prices
-    for t in range(T+1):
-
-        # note that r_seq[t+1] and τ_seq[t+1] are guesses!
-        W, W_next, r_next, τ, τ_next = W_seq[t], W_seq[t+1], r_seq[t+1], τ_seq[t], τ_seq[t+1]
-        
-        C_ub = (W / 2) * (1 - τ) + (W_next / 2) * (1 - τ_next) / (1 + r_next * (1 - τ_next))
-
-        out = brent_max(Cy_val2,   # maximand
-                        1e-3,      # lower bound
-                        C_ub-1e-3,    # upper bound
-                        args=(W, W_next, r_next, τ, τ_next, β))
-        Cy = out[0]
-
-        # private saving, Ap[t+1]
-        Ap_next = W * (1 - τ) - Cy
-
-        # asset next period
-        K_next = Ap_next + Ag1
-        W_next, r_next, Y_next = K_to_W(K_next, α), K_to_r(K_next, α), K_to_Y(K_next, α)
-
-        K_seq[t+1] = K_next
-        W_seq[t+1] = W_next
-        r_seq[t+1] = r_next
-        τ_seq[t+1] = (G_hat - r_next * Ag1) / (Y_next - r_next * Ag1)
-
-        # record consumption
-        C_seq[t, 0] = Cy
-        if t < T:
-            C_seq[t+1, 1] = (W / 2 * (1 - τ) - Cy) * (1 + r_next * (1 - τ_next)) + W_next / 2
-
-    # one iteration finishes
-    i_iter += 1
-
-    # check convergence
-    if (np.max(np.abs(W_seq_old - W_seq)) < tol) & \
-       (np.max(np.abs(r_seq_old - r_seq)) < tol) & \
-       (np.max(np.abs(τ_seq_old - τ_seq)) < tol):
-        print(f"Converge using {i_iter} iterations")
-        break
-
-    if i_iter > max_iter:
-        print(f"Fail to converge using {i_iter} iterations")
-        break
-
-axs[0].set_title('W')
-axs[1].set_title('r')
-axs[2].set_title('τ')
-
-plt.show();
-```
-
-```{code-cell} ipython3
-plt.plot(K_seq)
-plt.title('K')
-```

lectures/cagan_adaptive.md

@@ -79,7 +79,7 @@ $$
 $$ (eq:eqpipi)
 
 We assume that the expected rate of inflation $\pi_t^*$ is governed
-by the Friedman-Cagan adaptive expectations scheme
+by the following adaptive expectations scheme proposed by {cite}`Friedman1956` and {citet}`Cagan`:
 
 $$
 \pi_{t+1}^* = \lambda \pi_t^* + (1 -\lambda) \pi_t 

lectures/french_rev.md

@@ -18,7 +18,7 @@ kernelspec:
 ## Overview 
 
 This lecture describes some monetary and fiscal  features of the French Revolution
-described by Sargent and Velde {cite}`sargent_velde1995`.
+described by {cite}`sargent_velde1995`.
 
 We use matplotlib to replicate several of the graphs that they used to present salient patterns.
 

lectures/laffer_adaptive.md

@@ -13,26 +13,37 @@ kernelspec:
 
 +++ {"user_expressions": []}
 
-# Inflation Rate Laffer Curves  with Adaptive Expectations 
+# Laffer Curves  with Adaptive Expectations 
 
 ## Overview
 
 This lecture studies stationary and dynamic **Laffer curves** in the inflation tax rate in a non-linear version of the model studied in this  lecture {doc}`money_inflation`.
 
-As in the lecture {doc}`money_inflation`, this lecture uses the log-linear version of the demand function for money that Cagan {cite}`Cagan` used in his classic paper in place of the linear demand function used in this  lecture {doc}`money_inflation`.
+As in the lecture {doc}`money_inflation`, this lecture uses the log-linear version of the demand function for money that  {cite}`Cagan` used in his classic paper in place of the linear demand function used in this  lecture {doc}`money_inflation`.
 
 But now, instead of assuming  ''rational expectations'' in the form of ''perfect foresight'',
-we'll adopt the ''adaptive expectations'' assumption used by Cagan {cite}`Cagan`.
+we'll adopt the ''adaptive expectations'' assumption used by  {cite}`Cagan` and {cite}`Friedman1956`.
 
 
 
-This means that instead of assuming that expected inflation $\pi_t^*$ is described by
+This means that instead of assuming that expected inflation $\pi_t^*$ is described by the "perfect foresight" or "rational expectations" hypothesis 
 
 $$
 \pi_t^* = p_{t+1} - p_t
 $$ 
 
-as we assumed in lecture XXXX, we'll now assume that $\pi_t^*$ is determined by the adaptive expectations scheme described in equation {eq}`eq:adaptex`  reported below. 
+that we adopted in lectures {doc}`money_inflation` and lectures {doc}`money_inflation_nonlinear`, we'll now assume that $\pi_t^*$ is determined by the adaptive expectations hypothesis described in equation {eq}`eq:adaptex`  reported below. 
+
+We shall discover that changing our hypothesis about expectations formation in this way will change some our findings and leave others intact.  In particular, we shall discover that
+
+
+* replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$ 
+* it reverse the pervese dynamics by making the **lower** stationary inflation rate the one to which the system typically converges
+* a more plausible comparative dynamic outcome emerges in which now inflation can be **reduced** by running **lower**  government deficits
+
+These more plausible comparative dynamics underly the "old time religion" that states that 
+"inflation is always and everwhere caused by government deficits".
+
 
 ## The Model
 

lectures/money_inflation.md

@@ -54,7 +54,7 @@ In this lecture we will encounter these concepts
 
 The same qualitive outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture.  
 
-These outcomes will set the stage for the analysis of this lecture {doc}`laffer_adaptive` that studies a version of the present model that  uses a version of "adaptive expectations" instead of rational expectations.
+These outcomes will set the stage for the analysis of this lecture {doc}`laffer_adaptive` that studies a nonlinear version of the present model that  uses a version of "adaptive expectations" instead of rational expectations.
 
 That lecture will show that 
 

lectures/money_inflation_nonlinear.md

@@ -19,7 +19,7 @@ kernelspec:
 
 This lecture studies stationary and dynamic **Laffer curves** in the inflation tax rate in a non-linear version of the model studied in this lecture {doc}`money_inflation`.
 
-This lecture uses the log-linear version of the demand function for money that Cagan {cite}`Cagan`
+This lecture uses the log-linear version of the demand function for money that  {cite}`Cagan`
 used in his classic paper in place of the linear demand function used in this lecture {doc}`money_inflation`. 
 
 That change requires that we modify parts of our analysis.
@@ -29,6 +29,32 @@ In particular, our dynamic system is no longer linear in state variables.
 Nevertheless, the economic logic underlying an  analysis based on what we called ''method 2''  remains unchanged.  
 
 
+in this lecture we shall discover   qualitatively similar outcomnes to those that we studied  in the lecture  {doc}`money_inflation`.
+
+That lecture presented a linear version of the model in this lecture.  
+
+As in that  lecture,  we discussed these topics:
+
+* an **inflation tax** that a government gathers by printing paper or electronic money
+* a dynamic **Laffer curve** in the inflation tax rate that has two stationary equilibria
+* perverse dynamics under rational expectations in which the system converges to the higher stationary inflation tax rate
+* a peculiar comparative stationary-state analysis connected with that stationary inflation rate that assert that inflation can be **reduced** by running **higher**  government deficits 
+
+
+
+These outcomes will set the stage for the analysis of this lecture {doc}`laffer_adaptive` that studies a version of the present model that  uses a version of "adaptive expectations" instead of rational expectations.
+
+That lecture will show that 
+
+* replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$ 
+* it reverse the pervese dynamics by making the **lower** stationary inflation rate the one to which the system typically converges
+* a more plausible comparative dynamic outcome emerges in which now inflation can be **reduced** by running **lower**  government deficits
+
+
+GGHH
+
+
+
 
 ## The Model