[amss] Update Tom's edits to AMSS and Calvo Lectures

lectures/amss.md

@@ -20,6 +20,28 @@ kernelspec:
 
 # Optimal Taxation without State-Contingent Debt
 
+
+## Overview
+
+In {doc}`another lecture <opt_tax_recur>`, we presented a special case of a model of
+optimal taxation with state-contingent debt due to Robert E. Lucas, Jr.,  and Nancy Stokey  {cite}`LucasStokey1983`.
+
+
+
+In this lecture, we describe how Aiyagari, Marcet, Sargent, and Seppälä {cite}`aiyagari2002optimal`  (hereafter, AMSS) modified that model.
+
+In a nutshell, AMSS studied optimal taxation in a version of Lucas and Stokey's environment  **without** state-contingent debt.
+
+In this lecture, we 
+
+* describe AMSS's assumptions and equilibrium concept
+* describe how to solve the model using a different approach than AMSS did
+* implement the model numerically
+* conduct some policy experiments
+* compare outcomes with those in a corresponding complete-markets model
+
+
+
 In addition to what's in Anaconda, this lecture will need the following libraries:
 
 ```{code-cell} ipython
@@ -29,7 +51,7 @@ tags: [hide-output]
 !pip install --upgrade quantecon
 ```
 
-## Overview
+
 
 Let's start with following imports:
 
@@ -109,28 +131,11 @@ def linear_interp_1d(x_grid, y_values, x_query):
     return linear_interp_1d_scalar(x_min, x_max, x_num, y_values, x_query[0])
 ```
 
-Let's start with following imports:
-
-In {doc}`an earlier lecture <opt_tax_recur>`, we described a model of
-optimal taxation with state-contingent debt due to
-Robert E. Lucas, Jr.,  and Nancy Stokey  {cite}`LucasStokey1983`.
-
-Aiyagari, Marcet, Sargent, and Seppälä {cite}`aiyagari2002optimal`  (hereafter, AMSS)
-studied optimal taxation in a model without state-contingent debt.
-
-In this lecture, we
-
-* describe assumptions and equilibrium concepts
-* solve the model
-* implement the model numerically
-* conduct some policy experiments
-* compare outcomes with those in a corresponding complete-markets model
-
 We begin with an introduction to the model.
 
 ## Competitive equilibrium with distorting taxes
 
-Many but not all features of the economy are identical to those of {doc}`the Lucas-Stokey economy <opt_tax_recur>`.
+Many but not all features of AMSS's economy are identical to those of {doc}`the Lucas-Stokey economy <opt_tax_recur>`.
 
 Let's start with things that are identical.
 
@@ -502,8 +507,11 @@ $$
 where $R_t(s^t)$ is the gross risk-free rate of interest between $t$
 and $t+1$ at history $s^t$ and $T_t(s^t)$ are non-negative transfers.
 
-Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this is  possibly an important  difference from AMSS {cite}`aiyagari2002optimal`, who restricted transfers
-to be non-negative).
+Throughout this lecture, we shall set transfers to zero.
+
+```{note}
+For some issues about the limiting behavior of debt, this is  possibly an important  difference from AMSS {cite}`aiyagari2002optimal`. AMSS  restricted transfers to be non-negative.
+```
 
 In this case, the household faces a sequence of budget constraints
 
@@ -1050,7 +1058,7 @@ the two policies over 200 periods.
 
 This outcome reflects the presence of a force for **precautionary saving** that the incomplete markets structure imparts to the Ramsey plan.
 
-In {doc}`this subsequent lecture <amss2>` and  {doc}`this subsequent lecture <amss3>`, some ultimate consequences of that force are explored.
+In {doc}`this subsequent lecture <amss2>` and  {doc}`this subsequent lecture <amss3>`,  consequences of that force are explored.
 
 ```{code-cell} python3
 T = 200

lectures/amss2.md

@@ -20,19 +20,11 @@ kernelspec:
 
 # Fluctuating Interest Rates Deliver Fiscal Insurance
 
-In addition to what's in Anaconda, this lecture will need the following libraries:
-
-```{code-cell} ipython
----
-tags: [hide-output]
----
-!pip install --upgrade quantecon
-```
 
 ## Overview
 
-This lecture extends our investigations of how optimal policies for levying a flat-rate tax on labor income and  issuing government debt depend
-on whether there are complete  markets for debt.
+This lecture studies  how optimal policies for levying a flat-rate tax on labor income  depend
+on whether a government can buy and sell a complete set of one-period-ahead Arrow securities or whether it is instead able to buy or sell  only one-period risk-free debt.  
 
 A Ramsey allocation and Ramsey policy in the AMSS {cite}`aiyagari2002optimal` model described in {doc}`optimal taxation without state-contingent debt <amss>` generally differs
 from a Ramsey allocation and Ramsey policy in the  Lucas-Stokey {cite}`LucasStokey1983` model described in {doc}`optimal taxation with state-contingent debt <opt_tax_recur>`.
@@ -81,6 +73,16 @@ This lecture studies a special  AMSS model in which
 In a nutshell, the reason for this striking outcome is that at a particular level of risk-free government **assets**, fluctuations in the one-period risk-free interest
 rate provide the government with complete insurance against stochastically varying government expenditures.
 
+
+In addition to what's in Anaconda, this lecture will need the following libraries:
+
+```{code-cell} ipython
+---
+tags: [hide-output]
+---
+!pip install --upgrade quantecon
+```
+
 Let's start with some imports:
 
 ```{code-cell} ipython

lectures/amss3.md

@@ -20,14 +20,6 @@ kernelspec:
 
 # Fiscal Risk and Government Debt
 
-In addition to what's in Anaconda, this lecture will need the following libraries:
-
-```{code-cell} ipython
----
-tags: [hide-output]
----
-!pip install --upgrade quantecon
-```
 
 ## Overview
 
@@ -64,7 +56,19 @@ BEGS that we describe below.
 
 We use code constructed in {doc}`Fluctuating Interest Rates Deliver Fiscal Insurance <amss2>`.
 
+```{note}
 **Warning:** Key equations in  {cite}`BEGS1` section III.D carry  typos  that we correct below.
+```
+
+
+In addition to what's in Anaconda, this lecture will need the following libraries:
+
+```{code-cell} ipython
+---
+tags: [hide-output]
+---
+!pip install --upgrade quantecon
+```
 
 Let's start with some imports:
 

lectures/calvo.md

@@ -138,17 +138,11 @@ or
 
 Because $\alpha > 0$,  $0 < \frac{\alpha}{1+\alpha} < 1$.
 
-**Definition:** For scalar $b_t$, let $L^2$ be the space of sequences
-$\{b_t\}_{t=0}^\infty$ satisfying
 
-$$
-\sum_{t=0}^\infty  b_t^2 < +\infty
-$$
+We assume that the sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ is bounded.
 
-We say that a sequence that belongs to $L^2$   is **square summable**.
 
-When we assume that the sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ is square summable and we require that the sequence $\vec \theta = \{\theta_t\}_{t=0}^\infty$ is square summable,
-the linear difference equation {eq}`eq_old2` can be solved forward to get:
+Consequently the  linear difference equation {eq}`eq_old2` can be solved forward to get:
 
 ```{math}
 :label: eq_old3
@@ -203,10 +197,9 @@ 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 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. 
+
 
-Indeed, for arbitrary, $\vec \mu = \{\mu_t\}_{t=0}^\infty$ sequences, $\vec \theta = \{\theta_t\}_{t=0}^\infty$ will not be  square summable. 
-
-But the  government  planner will design   a decision rule for $\mu_t$ that  stabilizes  the system and renders $\vec \theta$ square summable. 
+But the  government  planner will design   a decision rule for $\mu_t$ that  stabilizes  the system and renders $\vec \theta$ bounded. 
 
 The  government  values  a representative household's utility of real balances at time $t$ according to the utility function
 

lectures/calvo_abreu.md

@@ -103,17 +103,10 @@ or
 
 Because $\alpha > 0$,  $0 < \frac{\alpha}{1+\alpha} < 1$.
 
-**Definition:** For  scalar $b_t$, let $L^2$ be the space of sequences
-$\{b_t\}_{t=0}^\infty$ satisfying
 
-$$
-\sum_{t=0}^\infty  b_t^2 < +\infty
-$$
-
-We say that a sequence that belongs to $L^2$   is **square summable**.
+We assume that the sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ is bounded.
 
-When we assume that the sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ is square summable and we require that the sequence $\vec \theta = \{\theta_t\}_{t=0}^\infty$ is square summable,
-the linear difference equation {eq}`eq_old2_new` can be solved forward to get:
+Consequently, the linear difference equation {eq}`eq_old2_new` can be solved forward to get:
 
 ```{math}
 :label: eq_old3_new

lectures/calvo_machine_learn.md

@@ -139,21 +139,11 @@ or
 
 Because $\alpha > 0$,  $0 < \frac{\alpha}{1+\alpha} < 1$.
 
-```{prf:definition}
-:label: square-summable
 
-For  scalar $b_t$, let $L^2$ be the space of sequences
-$\{b_t\}_{t=0}^\infty$ that satisfy 
+We assume that the sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ is bounded.
 
-$$
-\sum_{t=0}^\infty  b_t^2 < +\infty
-$$
-
-We say that a sequence that belongs to $L^2$ is **square summable**.
-```
-
-When we assume that the sequence $\vec \mu = \{\mu_t\}_{t=0}^\infty$ is square summable and also require that the sequence $\vec \theta = \{\theta_t\}_{t=0}^\infty$ is square summable,
-the linear difference equation {eq}`eq_grad_old2` can be solved forward to get:
+
+Then the linear difference equation {eq}`eq_grad_old2` can be solved forward to get:
 
 ```{math}
 :label: eq_grad_old3

lectures/muth_kalman.md

@@ -23,6 +23,16 @@ kernelspec:
 
 # Reverse Engineering a la Muth
 
+## Overview
+
+This lecture uses the Kalman filter to reformulate John F. Muth’s first
+paper {cite}`Muth1960` about rational expectations.
+
+Muth used *classical* prediction methods to reverse engineer a
+stochastic process that renders optimal Milton Friedman’s {cite}`Friedman1956` “adaptive
+expectations” scheme.
+
+
 In addition to what's in Anaconda, this lecture uses the quantecon library.
 
 ```{code-cell} ipython
@@ -43,13 +53,6 @@ from quantecon import LinearStateSpace
 np.set_printoptions(linewidth=120, precision=4, suppress=True)
 ```
 
-This lecture uses the Kalman filter to reformulate John F. Muth’s first
-paper {cite}`Muth1960` about rational expectations.
-
-Muth used *classical* prediction methods to reverse engineer a
-stochastic process that renders optimal Milton Friedman’s {cite}`Friedman1956` “adaptive
-expectations” scheme.
-
 ## Friedman (1956) and Muth (1960)
 
 Milton Friedman {cite}`Friedman1956` (1956) posited that

lectures/opt_tax_recur.md

@@ -31,32 +31,38 @@ tags: [hide-output]
 
 ## Overview
 
-This lecture describes a celebrated model of optimal fiscal policy by Robert E.
+This lecture describes special case of a celebrated model of optimal fiscal policy by Robert E.
 Lucas, Jr., and Nancy Stokey  {cite}`LucasStokey1983`.
 
-The model revisits classic issues about how to pay for a war.
+
+We use our special case of their  model to  revisit classic issues about how to pay for a war.
 
 Here a *war* means a more  or less temporary surge in an exogenous government expenditure process.
 
 The model features
 
-* a government that must finance an exogenous stream of government expenditures with  either
-    * a flat rate tax on labor, or
-    * purchases and sales from a full array of Arrow state-contingent securities
-* a representative household that values consumption and leisure
 * a linear production function mapping labor into a single good
-* a Ramsey planner who at time $t=0$ chooses a plan for taxes and
-  trades of [Arrow securities](https://en.wikipedia.org/wiki/Arrow_security) for all $t \geq 0$
+* a representative household that likes both consumption and leisure
+* an exogenous history-contingent sequence  of government expenditures that a goverment must finance with revenues from a sequence of history-dependent flat rate taxes 
+* an exogenous initial  debt  the government must also finance 
+*  a Ramsey planner who at time $t=0$ chooses a  history contingent plan for flat rate taxes  at all $t \geq 0$
+* a sequence of continuation  governments that at each $t \geq 1$ must pay off the one-period state-contingent debt that a time $t-1$ government has issued
+  * each time $t \geq 1$ government is free to reset the time $t$ flat tax rate 
+  * each time $t \geq 1$ government  can also borrow or lend by selling or buying  
+    one-period-ahead [Arrow securities](https://en.wikipedia.org/wiki/Arrow_security)
+
+```{note}
+Important parts of {cite}`LucasStokey1983` are about how the government should finance a stochastic sequence of debt service payments that at time $0$ the government has inherited from the past. Lucas and Stokey's government must honor this stream of random payments, either by paying promised coupons when they become due at a particular date, state pair or else by issuing new debt, i.e., by somehow rolling it over. We study the special case of Lucas and Stokey's model that assumes that at time $0$ the government  owes only  $b_0(s_0)$ units of the time $0$, state $s_0$ consumption good.     
+```
 
 After first presenting the model in a space of sequences, we shall represent it
 recursively in terms of two Bellman equations formulated along lines that we
 encountered in {doc}`Dynamic Stackelberg models <dyn_stack>`.
 
 As in {doc}`Dynamic Stackelberg models <dyn_stack>`, to apply dynamic programming
-we shall define the state vector artfully.
+we shall define a state vector artfully.
 
-In particular, we shall include forward-looking variables that summarize  optimal
-responses of private agents to a Ramsey plan.
+In particular, we shall include as  components of the state some forward-looking variables that summarize  optimal responses of private agents to a Ramsey tax plan.
 
 See {doc}`Optimal taxation <lqramsey>` for analysis within a linear-quadratic setting.
 
@@ -276,7 +282,9 @@ q_t^0(s^t) = \beta^{t} \pi_{t}(s^{t})
                             {u_c(s^{t})  \over u_c(s^0)}
 ```
 
-(The stochastic process $\{q_t^0(s^t)\}$ is an instance of what finance economists call a *stochastic discount factor* process.)
+```{note}
+The stochastic process $\{\beta^{t}{u_c(s^{t})  \over u_c(s^0)}\}$ is an instance of what finance economists call a *stochastic discount factor* process. The random variable $\beta^{t+1}{u_c(s^{t+1})  \over u_c(s^t)}$ is the time $t+1$ multiplicative increment to the stochastic discount factor process.
+```
 
 Using the first-order conditions {eq}`LSA_taxr` and {eq}`LS101` to eliminate
 taxes and prices from {eq}`TS_bcPV2`, we derive the *implementability condition*
@@ -330,7 +338,7 @@ J  = \sum_{t=0}^\infty
 where  $\{\theta_t(s^t); \forall s^t\}_{t\geq0}$ is a sequence of Lagrange
 multipliers on the feasible conditions {eq}`TSs_techr_opt_tax`.
 
-Given an initial government debt $b_0$,  we want to maximize $J$
+Given an initial government debt  payment $b_0$ due at time $0$,  we want to maximize $J$
 with respect to $\{c_t(s^t), n_t(s^t); \forall s^t \}_{t\geq0}$   and to minimize with respect
 to $\Phi$ and with respect to $\{\theta(s^t); \forall s^t \}_{t\geq0}$.
 
@@ -698,8 +706,8 @@ and
 
 In equation {eq}`TS_barg10`, it is understood that $c$ and $g$ are each functions of the Markov state $s$.
 
-In addition, the time $t=0$ budget constraint is satisfied at $c_0$ and initial government debt
-$b_0$:
+In addition, the time $t=0$ budget constraint is satisfied at $c_0$ and initial  government debt
+$b_0$ due at time $0$:
 
 ```{math}
 :label: opt_tax_eqn_10
@@ -747,7 +755,7 @@ different objective function and faces different constraints and state variables
 Ramsey planner at time $t =0$.
 
 A key step in representing a Ramsey plan recursively is
-to regard the marginal utility scaled government debts
+to regard the marginal utility scaled one-period government debts
 $x_t(s^t) = u_c(s^t) b_t(s_t|s^{t-1})$ as predetermined quantities
 that continuation Ramsey planners at times $t \geq 1$ are
 obligated to attain.