Restore Tom's edits (#128)

Author: mmcky

lectures/calvo.md

@@ -42,7 +42,7 @@ This lecture describes a linear-quadratic version of a model that Guillermo Calv
 used to illustrate the **time inconsistency** of optimal government
 plans.
 
-Like Chang {cite}`chang1998credible`, we use the model as a laboratory in which to explore consequences of
+Like Chang {cite}`chang1998credible`, we use the model as a laboratory in which to explore the consequences of
 different timing protocols for government decision making.
 
 The model focuses attention on intertemporal tradeoffs between
@@ -53,9 +53,9 @@ The model focuses attention on intertemporal tradeoffs between
 The model features
 
 - rational expectations
-- costly government actions at  dates $t \geq 1$ that increase household
-  utilities at all dates before $t$
-- two Bellman equations, one that expresses the private sector's  forecast of future inflation
+- costly government actions at all dates $t \geq 1$ that increase household
+  utilities at dates before $t$
+- two Bellman equations, one that expresses the private sector's  expectation of future inflation
   as a function of current and future government actions, another that
   describes  the value function of a  Ramsey planner
 
@@ -112,7 +112,7 @@ the actual rate of inflation.
 
 (When there is no uncertainty, an assumption of **rational expectations** simplifies to **perfect foresight**).
 
-(See {cite}`Sargent77hyper` for a rational expectations version of the model when there is uncertainty)
+(See {cite}`Sargent77hyper` for a rational expectations version of the model when there is uncertainty.)
 
 Subtracting the demand function at time $t$ from the demand
 function at $t+1$ gives:
@@ -190,12 +190,12 @@ or
 x_{t+1} = A x_t + B \mu_t
 ```
 
-We write the model in the state-space form {eq}`eq_old4` even though $\theta_0$ is to be determined and so is not an initial condition
+We write the model in the state-space form {eq}`eq_old4` even though $\theta_0$ is to be determined by our  model and so is not an initial condition
 as it ordinarily would be in the state-space model described in [Linear Quadratic Control](https://python-intro.quantecon.org/lqcontrol.html).
 
 We write the model in the form {eq}`eq_old4` because we want to apply an approach described in  {doc}`Stackelberg problems <dyn_stack>`.
 
-Assume that a representative household's utility of real balances at
+We assume that a benevolent government  believes that a representative household's utility of real balances at
 time $t$ is:
 
 ```{math}
@@ -218,7 +218,7 @@ both use to discount future utilities.
 
 (If we set parameters so that $\theta^* = \log(\beta)$, then we can
 regard a recommendation to set $\theta_t = \theta^*$ as a "poor
-man's Friedman rule" that attains Milton Friedman's **optimal quantity of money**)
+man's Friedman rule" that attains Milton Friedman's **optimal quantity of money**.)
 
 Via equation {eq}`eq_old3`, a government plan
 $\vec \mu = \{\mu_t \}_{t=0}^\infty$ leads to an equilibrium
@@ -257,8 +257,9 @@ v_t = - s(\theta_t, \mu_t) + \beta v_{t+1}
 ## Structure
 
 The following structure is induced by private agents'
-behavior as summarized by the demand function for money {eq}`eq_old1` that leads to equation {eq}`eq_old3` that tells how future
-settings of $\mu$ affect the current value of $\theta$.
+behavior as summarized by the demand function for money {eq}`eq_old1` that leads to equation {eq}`eq_old3`.
+
+It  tells how future settings of $\mu$ affect the current value of $\theta$.
 
 Equation {eq}`eq_old3` maps a **policy** sequence of money growth rates
 $\vec \mu =\{\mu_t\}_{t=0}^\infty \in L^2$  into an inflation sequence
@@ -284,7 +285,7 @@ At this point $\vec \mu \in L^2$ is an arbitrary exogenous policy.
 To make $\vec \mu$ endogenous, we require a theory of government
 decisions.
 
-## Intertemporal Influences
+## Intertemporal Structure 
 
 Criterion function {eq}`eq_old7` and the constraint system {eq}`eq_old4` exhibit the following
 structure:
@@ -296,8 +297,8 @@ structure:
   household's one-period utilities at all dates
   $s = 0, 1, \ldots, t$.
 
-That settings of $\mu$ at one date affect household utilities at
-earlier dates sets the stage for the emergence of a time-inconsistent
+
+This structure  sets the stage for the emergence of a time-inconsistent
 optimal government plan  under a Ramsey (also called a Stackelberg)  timing protocol.
 
 We'll study outcomes under a Ramsey timing protocol below.
@@ -308,9 +309,9 @@ But we'll also study the consequences of other timing protocols.
 
 We consider four models of policymakers that  differ in
 
-- what a policymaker is allowed to choose, either a sequence
+- what a  policymaker is allowed to choose, either a sequence
   $\vec \mu$ or just a single period  $\mu_t$.
-- when a policymaker chooses, either at time $0$ or at times
+- when a  policymaker chooses, either at time $0$ or at times
   $t \geq 0$.
 - what a policymaker assumes about how its choice of $\mu_t$
   affects private agents' expectations about earlier and later
@@ -455,7 +456,9 @@ $\vec \mu$ recursively with the following system created in the spirit of Chang
 
 To interpret this system, think of the  sequence
 $\{\theta_t\}_{t=0}^\infty$ as a sequence of
-synthetic **promised inflation rates** that are just computational devices for
+synthetic **promised inflation rates**.
+
+These  are just computational devices for
 generating a sequence $\vec\mu$ of money growth rates that are to
 be substituted into equation {eq}`eq_old3` to form actual rates of inflation.
 
@@ -464,15 +467,14 @@ $\vec \mu = \{\mu_t\}_{t=0}^\infty$ that satisfies these equations
 into equation {eq}`eq_old3`, we obtain the same sequence $\vec \theta$
 generated by the system {eq}`eq_old9`.
 
-(Here an application of the Big $K$, little $k$ trick could once again be enlightening)
+(Here an application of the Big $K$, little $k$ trick could once again be enlightening.)
 
 Thus, our construction of a Ramsey plan guarantees that **promised
 inflation** equals **actual inflation**.
 
 ### Multiple roles of $\theta_t$
 
-The inflation rate $\theta_t$ that appears in the system {eq}`eq_old9` and
-equation {eq}`eq_old3` plays three roles simultaneously:
+The inflation rate $\theta_t$ plays three roles simultaneously:
 
 - In equation {eq}`eq_old3`, $\theta_t$ is the actual rate of inflation
   between $t$ and $t+1$.
@@ -495,7 +497,7 @@ that alter either
 
 ## A Constrained-to-a-Constant-Growth-Rate Ramsey Government
 
-We now consider the following peculiar model of optimal government behavior.
+We now consider a peculiar model of optimal government behavior.
 
 We have created this model in order to highlight an aspect of an optimal government policy associated with its time inconsistency,
 namely, the feature that optimal settings of the  policy instrument vary over time.
@@ -525,14 +527,14 @@ $\mu_t$ as a telltale sign of  time inconsistency of a Ramsey plan.
 
 ## Markov Perfect Governments
 
-We now  change the timing protocol by considering a sequence of
+We now  alter the timing protocol by considering a sequence of
 government policymakers, the time $t$ representative of which
 chooses $\mu_t$ and expects all future governments to set
 $\mu_{t+j} = \bar \mu$.
 
 This assumption mirrors an assumption made in a different setting [Markov Perfect Equilibrium](https://python-intro.quantecon.org/markov_perf.html).
 
-Further, a government  policymaker at $t$ believes that $\bar \mu$ is
+A government  policymaker at $t$ believes that $\bar \mu$ is
 unaffected by its choice of $\mu_t$.
 
 The time $t$ rate of inflation is then:
@@ -584,7 +586,7 @@ $$
 \bar \mu = - \frac{\alpha a_1}{\alpha^2 a_2 + (1+\alpha)c}
 $$
 
-## Equilibrium Outcomes for Three Models of Government Policy Making
+## Equilibrium Outcomes for Three Models
 
 Below we compute sequences $\{ \theta_t,\mu_t \}$ under a Ramsey
 plan and compare these with the constant levels of $\theta$ and
@@ -903,17 +905,15 @@ is a symptom of time inconsistency.
   past utilities and to reoptimize at date $t \geq 1$ would, if allowed, want
   to deviate from a Ramsey plan.
 
-```{note}
-A modified Ramsey plan constructed under the restriction that
+**Note:** A modified Ramsey plan constructed under the restriction that
 $\mu_t$ must be constant over time is time consistent (see
 $\check \mu$ and $\check \theta$ in the above graphs).
-```
 
 ### Meaning of Time Inconsistency
 
 In settings in which governments actually choose sequentially, many economists
-regard a time inconsistent plan implausible because of the incentives to
-deviate that occur along the plan.
+regard a time inconsistent plan as controversial because of the incentives to
+deviate that are presented  along the plan.
 
 A way to summarize this *defect* in a Ramsey plan is to say that it
 is not credible because there  endure incentives for policymakers
@@ -941,7 +941,7 @@ We turn to such theories of **sustainable plans** next.
 
 This is a model in which
 
-- The government chooses $\{\mu_t\}_{t=0}^\infty$ not once and
+- the government chooses $\{\mu_t\}_{t=0}^\infty$ not once and
   for all at $t=0$ but chooses to set $\mu_t$ at time $t$, not before.
 - private agents' forecasts of
   $\{\mu_{t+j+1}, \theta_{t+j+1}\}_{j=0}^\infty$ respond to