update typos in calvo and calvo_abreu

Author: HumphreyYang

lectures/calvo.md

@@ -164,7 +164,7 @@ real balances $m_t - p_t = -\alpha \theta_t$.
 
 An equivalence class of continuation money growth sequences $\{\mu_{t+j}\}_{j=0}^\infty$ deliver the same $\theta_t$.
 
-We shall use this insight to help us simplify our  analsis of alternative  government policy problems.
+We shall use this insight to help us simplify our analysis of alternative  government policy problems.
 
 That future rates of money creation influence earlier rates of inflation
 makes  timing protocols matter for modeling optimal government policies.
@@ -221,12 +221,14 @@ U(m_t - p_t) = u_0 + u_1 (m_t - p_t) - \frac{u_2}{2} (m_t - p_t)^2, \quad u_0 >
 The money demand function {eq}`eq_old1` and the utility function {eq}`eq_old5` imply that 
 
 $$
-U(-\alpha \theta_t) = u_1 + u_2 (-\alpha \theta_t) -\frac{u_2}{2}(-\alpha \theta_t)^2 . 
+U(-\alpha \theta_t) = u_0 + u_1 (-\alpha \theta_t) -\frac{u_2}{2}(-\alpha \theta_t)^2 . 
 $$ (eq_old5a)
 
-The``bliss level`` of real balances is  $\frac{u_1}{u_2}$ and the inflation rate that attains
+The ``bliss level`` of real balances is  $\frac{u_1}{u_2}$ and the inflation rate that attains
 it is $-\frac{u_1}{u_2 \alpha}$.
 
+(TO TOM: the first sentece in the next section is very similar to the sentence above.)
+
 ## Friedman's Optimal Rate of Deflation
 
 According to {eq}`eq_old5a`, the "bliss level" of real balances is  $\frac{u_1}{u_2}$ and the inflation rate that attains it is
@@ -447,8 +449,6 @@ Thus, our models have involved two Bellman equations:
 
 A value $\theta$ from one Bellman equation appears as an argument of a second Bellman equation for another value $v$.
 
-.
-
 ## A Ramsey Planner
 
 Here  we consider a Ramsey planner that  chooses
@@ -716,7 +716,7 @@ that, relative to a Ramsey plan,  alter either
 
 We now describe a  model in which we restrict the Ramsey planner's choice set.
 
-Instead of choosing a sequence of money growth rates $\vec \mu \in {\bf R}^2$, we restrict the 
+Instead of choosing a sequence of money growth rates $\vec \mu \in {\bf L}^2$, we restrict the 
 government to choose a time-invariant money growth rate $\bar \mu$. 
 
 We created this version of the model  to highlight an aspect of a Ramsey plan associated with its time inconsistency, namely, the feature that optimal settings of the  policy instrument vary over time.
@@ -1119,7 +1119,7 @@ at time $t=0$.
 
 The figure also plots the limiting value $\theta_\infty^R$ to which  the promised  inflation rate $\theta_t$ converges under the Ramsey plan.
 
-In addition, the figure indicates an MPE inflation rate $\theta^{CR}$ and a bliss inflation $\theta^*$.
+In addition, the figure indicates an MPE inflation rate $\theta^{MPE}$, $\theta^{CR}$, and a bliss inflation $\theta^*$.
 
 ```{code-cell} ipython3
 :tags: [hide-input]
@@ -1194,7 +1194,7 @@ np.allclose(clq.J_θ(θ_inf),
             clq.V_θ(θ_inf))
 ```
 
-So our claim that $J(\theta_\infty^R) = V^{CR}(\theta_\infty^R)$is verified numerically.
+So our claim that $J(\theta_\infty^R) = V^{CR}(\theta_\infty^R)$ is verified numerically.
 
 Since  $J(\theta_\infty^R) = V^{CR}(\theta_\infty^R)$ occurs at a tangency point at which
 $J(\theta)$ is increasing in $\theta$, it follows that
@@ -1337,7 +1337,7 @@ The above graphs and table convey many useful things.
 
 The horizontal dotted lines indicate values 
  $V(\mu_\infty^R), V(\mu^{CR}), V(\mu^{MPE}) $ of time-invariant money
-growth rates $\mu_\infty^R, \mu^{CR}$ and $\mu_{MPE}$, respectfully. 
+growth rates $\mu_\infty^R, \mu^{CR}$ and $\mu^{MPE}$, respectfully. 
 
 Notice how $J(\theta)$ and $V^{CR}(\theta)$ are tangent and increasing at
  $\theta = \theta_\infty^R$, which implies that $\theta^{CR} > \theta_\infty^R$
@@ -1355,6 +1355,8 @@ $$
 \end{aligned}
 $$
 
+(TO TOM: $\theta^{MPE}$ is repeated in the above equations. Should one of them be $\theta^*$?)
+
 
  But let's see what happens when we change $c$.
 
@@ -1531,6 +1533,8 @@ In settings in which governments actually choose sequentially, many economists
 regard a time inconsistent plan as implausible because of the incentives to
 deviate that are presented  along the plan.
 
+(TO TOM: In our meeting, you suggested that we can improve the sentence above.)
+
 A way to state  this reaction   is to say that a Ramsey plan is not credible because there are persistent incentives for policymakers to deviate from it.
 
 For that reason, the Markov perfect equilibrium concept attracts many

lectures/calvo_abreu.md

@@ -32,7 +32,7 @@ kernelspec:
 
 This is a sequel to this  quantecon lecture {doc}`calvo`.
 
-That lecture studied a linear-quadratic version of a model that Guillermo Calvo {cite}`Calvo1978` used to study  the **time inconsistency** of the optimal government plan that emerges when  a ``Stackelberg`` government  (a.k.a.~ a ``Ramsey planner``) at time $0$ once and for all chooses  a sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ of gross rates of growth in the supply of money.
+That lecture studied a linear-quadratic version of a model that Guillermo Calvo {cite}`Calvo1978` used to study  the **time inconsistency** of the optimal government plan that emerges when  a **Stackelberg** government  (a.k.a. a **Ramsey planner**) at time $0$ once and for all chooses  a sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ of gross rates of growth in the supply of money.
 
 A consequence of that choice is a (rational expectations equilibrium)  sequence $\vec \theta = \{\theta_t\}_{t=0}^\infty$  of gross rates of increase in the price level that  we call inflation rates. 
 
@@ -277,8 +277,6 @@ The key is an  object called a **self-enforcing** plan.
 
 We'll proceed to compute one.
 
-
-
 In addition to what's in Anaconda, this lecture will use  the following libraries:
 
 ```{code-cell} ipython3
@@ -293,9 +291,7 @@ We'll start with some imports:
 import numpy as np
 from quantecon import LQ
 import matplotlib.pyplot as plt
-from matplotlib.ticker import FormatStrFormatter
 import pandas as pd
-from IPython.display import display, Math
 ```
 
 ### Abreu's Self-Enforcing Plan