Tom's Nov 4 edits of calvo.md lecture

lectures/calvo.md

@@ -64,7 +64,7 @@ We'll use ideas from  papers by Cagan {cite}`Cagan`, Calvo {cite}`Calvo1978`, an
 well as from chapter 19 of {cite}`Ljungqvist2012`.
 
 In addition, we'll use ideas from linear-quadratic dynamic programming
-described in  [Linear Quadratic Control](https://python-intro.quantecon.org/lqcontrol.html) as applied to Ramsey problems in {doc}`Stackelberg problems <dyn_stack>`.
+described in  [Linear Quadratic Control](https://python-intro.quantecon.org/lqcontrol.html) as applied to Ramsey problems in {doc}`Stackelberg plans <dyn_stack>`.
 
 We  specify model fundamentals  in  ways that allow us to use
 linear-quadratic discounted dynamic programming to compute an optimal government
@@ -104,8 +104,7 @@ Let:
 - $\theta_t = p_{t+1} - p_t$ be the net rate of inflation between $t$ and $t+1$
 - $\mu_t = m_{t+1} - m_t$ be the net rate of growth of nominal balances
 
-The demand for real balances is governed by a perfect foresight
-version of a Cagan {cite}`Cagan` demand function for real balances:
+The demand for real balances is governed by a discrete time version of Sargent and Wallace's {cite}`sargent1973stability` perfect foresight version of a Cagan {cite}`Cagan` demand function for real balances:
 
 ```{math}
 :label: eq_old1
@@ -119,9 +118,9 @@ Equation {eq}`eq_old1` asserts that the demand for real balances is inversely
 related to the public's expected rate of inflation, which  equals
 the actual rate of inflation because there is no uncertainty here.
 
-(When there is no uncertainty, an assumption of **rational expectations** that becomes equivalent to  **perfect foresight**).
+(When there is no uncertainty, an assumption of **rational expectations**  becomes equivalent to  **perfect foresight**).
 
-(See {cite}`Sargent77hyper` for a rational expectations version of the model when there is uncertainty.)
+({cite}`Sargent77hyper` presents  a rational expectations version of the model when there is uncertainty.)
 
 Subtracting the demand function {eq}`eq_old1` at time $t$ from the demand
 function at $t+1$ gives:
@@ -204,7 +203,7 @@ as it ordinarily would be in the state-space model described in our lecture on
 
  
 
-We use  form {eq}`eq_old4` because we want to apply an approach described in  our lecture on {doc}`Stackelberg problems <dyn_stack>`.
+We use  form {eq}`eq_old4` because we want to apply an approach described in  our lecture on {doc}`Stackelberg plans <dyn_stack>`.
 
 Notice that $\frac{1+\alpha}{\alpha} > 1$ is an eigenvalue of transition matrix $A$ that threatens to destabilize the state-space system. 
 
@@ -224,14 +223,9 @@ $$
 U(-\alpha \theta_t) = u_0 + u_1 (-\alpha \theta_t) -\frac{u_2}{2}(-\alpha \theta_t)^2 . 
 $$ (eq_old5a)
 
-The ``bliss level`` of real balances is  $\frac{u_1}{u_2}$ and the inflation rate that attains
-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
+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
 
 
 $$
@@ -324,9 +318,10 @@ for all $t \geq 0$.
 Values of $V(\bar \mu)$ computed according to formula {eq}`eq:barvdef` for three different  values of $\bar \mu$ will play important roles below.
 
 * $V(\mu^{MP})$ is the value of attained by the government in a **Markov perfect equilibrium** 
+* $V(\mu^R_\infty)$ is the  value that  a continuation Ramsey planner attains at  $t \rightarrow +\infty$
+  * We shall discover that $V(\mu^R_\infty)$ is the worst continuation value attained along a Ramsey plan 
 * $V(\mu^{CR})$ is the value of attained by the government in a **constrained to constant $\mu$ equilibrium**
-* $V(\mu^R_\infty)$ is the limiting value attained by a continuation Ramsey planner under a Ramsey plan.
-    * We shall see that $V(\mu^R_\infty)$ is a worst continuation value attained along a Ramsey plan  
+     
 
 ## Structure
 
@@ -414,7 +409,7 @@ The models are distinguished by their having  either
   $\{\mu_t\}_{t=0}^\infty$ once and for all at time $0$
   subject to the constraint that $\mu_t = \mu$ for all
   $t \geq 0$; or
-- A sequence indexed by $t =0, 1, 2, \ldots$ of separate policymakers 
+- A sequence of distinct  policymakers indexed by $t =0, 1, 2, \ldots$  
     - a time $t$ policymaker chooses $\mu_t$ only and forecasts that future government decisions are unaffected by its choice.
 
 
@@ -438,7 +433,7 @@ The relationship between  outcomes in  the first (Ramsey) timing protocol and th
 We'll begin with the timing protocol associated with a Ramsey plan and deploy 
 an application of what we  nickname **dynamic programming squared**.
 
-The nickname refers to the feature that a value satisfying one Bellman equation appears as an argument in a second Bellman equation.
+The nickname refers to the feature that a value satisfying one Bellman equation appears as an argument in a  value function associated with a  second Bellman equation.
 
 Thus, our models have involved two Bellman equations:
 
@@ -455,7 +450,7 @@ Here  we consider a Ramsey planner that  chooses
 $\{\mu_t, \theta_t\}_{t=0}^\infty$ to maximize {eq}`eq_old7`
 subject to the law of motion {eq}`eq_old4`.
 
-We can split this problem into two stages, as in {doc}`Stackelberg problems <dyn_stack>` and  {cite}`Ljungqvist2012` Chapter 19.
+We can split this problem into two stages, as in the lecture  {doc}`Stackelberg plans <dyn_stack>` and  {cite}`Ljungqvist2012` Chapter 19.
 
 In the first stage, we take the initial inflation rate $\theta_0$ as given
 and solve what looks like an ordinary  LQ discounted dynamic programming problem.
@@ -491,7 +486,7 @@ $$
 x' = Ax + B\mu
 $$
 
-As in {doc}`Stackelberg problems <dyn_stack>`, we can map this problem into a linear-quadratic control problem and deduce an optimal value function $J(x)$.
+As in the lecture {doc}`Stackelberg plans <dyn_stack>`, we can map this problem into a linear-quadratic control problem and deduce an optimal value function $J(x)$.
 
 Guessing that $J(x) = - x'Px$ and substituting into the Bellman
 equation gives rise to the algebraic matrix Riccati equation:
@@ -698,7 +693,7 @@ about dynamic or time inconsistency.
 
 ## Time inconsistency
 
-As discussed in {doc}`Stackelberg problems <dyn_stack>` and {doc}`Optimal taxation with state-contingent debt <opt_tax_recur>`, a continuation Ramsey plan is not a Ramsey plan.
+As discussed in {doc}`Stackelberg plans <dyn_stack>` and {doc}`Optimal taxation with state-contingent debt <opt_tax_recur>`, a continuation Ramsey plan is not a Ramsey plan.
 
 This is a concise way of characterizing the time inconsistency of a Ramsey plan.
 
@@ -1351,11 +1346,10 @@ $$
 \begin{aligned}
 \theta^{CR} & = - \frac{\alpha u_1}{\alpha^2 u_2 + c } \\
 \theta^{MPE} &  = - \frac{\alpha u_1}{\alpha^2 u_2 + (1+\alpha)c} \\
-\theta^{MPE} & = - \frac{\alpha u_1}{\alpha^2 u_2 + (1+\alpha)c}
+\theta^{*} &  = -\frac{u_1}{u_2 \alpha}
 \end{aligned}
 $$
 
-(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 $c$.
@@ -1376,7 +1370,7 @@ generate_table(clqs, dig=4)
 The above table and figures show how 
   changes in $c$ alter $\theta_\infty^R$
  and $\theta_0^R$ as well as  $\theta^{CR}$ and  $\theta^{MPE}$, but not
- $\theta^*$, again  in accord with formulas
+ $\theta^*,$ again  in accord with formulas
  {eq}`eq:Friedmantheta`,  {eq}`eq:muRamseyconstrained`, and {eq}`eq:Markovperfectmu`. 
 
 Notice that as $c $ gets larger and larger,   $\theta_\infty^R, \theta_0^R$ 
@@ -1529,20 +1523,16 @@ A constrained-to-constant-$\mu$  Ramsey plan  is  time consistent by constructio
 
 ### Implausibility of Ramsey Plan 
 
-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.)
+Many economists regard a time inconsistent plan as implausible because they question the plausibility of  timing protocol in 
+which a plan for setting a sequence of policy variables is chosen once-and-for-all at time $0$.
 
-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
 economists.
 
-* A Markov perfect equilibrium plan is constructed to insure that government policymakers who choose sequentially do not want to deviate from it.
+* A Markov perfect equilibrium plan is constructed to insure that a sequence of  government policymakers who choose sequentially do not want to deviate from it.
 
-The *no incentive to deviate from the plan* property is what makes the Markov perfect equilibrium concept attractive.
+The  property of a Markov perfect equilibrium that there is *no incentive to deviate from the plan*   makes it  attractive.
 
 
 ## Comparison of Equilibrium Values