Tom's Feb 7 edits of 4 advanced lectures

Author: thomassargent30

lectures/markov_jump_lq.md

@@ -56,8 +56,7 @@ the state of an $N$ state Markov chain.
 
 The state of the Markov chain together with   the continuous $n \times 1$ state vector $x_t$ form the state of the system.
 
-For the class of infinite horizon problems being studied in this lecture, we obtain  $N$ interrelated
-matrix Riccati equations that determine $N$ optimal value
+For the class of infinite horizon problems being studied in this lecture, we obtain  $N$ interrelated matrix Riccati equations that determine $N$ optimal value
 functions and $N$ linear decision rules.
 
 One of these value functions and one of these decision rules apply in each of the $N$ Markov states.
@@ -199,20 +198,19 @@ $$
 $$
 
 The optimal value functions $- x' P_i x - \rho_i$ for
-$i = 1, \ldots, n$ satisfy the $N$
-interrelated Bellman equations
+$i = 1, \ldots, n$ satisfy the $N$ interrelated Bellman equations
 
 $$
-\begin{split}
+
+\begin{aligned}
 -x' P_i x - \rho_i & = \max_u -
   \\
-  &
-   \left[
-     x'R_i x + u' Q_i u + 2 u' W_i x +
+    
+     x'R_i x  & + u' Q_i u + 2 u' W_i x +
              \beta \sum_j \Pi_{ij}E ((A_i x + B_i u + C_i w)' P_j
-             (A_i x + B_i u + C_i w) x + \rho_j)
-   \right]
-\end{split}
+             (A_i x + B_i u + C_i w) x + \rho_j)  
+\end{aligned}
+
 $$
 
 The matrices $P_{s_t} = P_i$ and the scalars
@@ -231,7 +229,7 @@ $$
  \sum_j \Pi_{ij} ( \rho_j + {\rm trace}(P_j C_i C_i') )
 $$
 
-and the $F_i$ in the optimal decision rules are
+and the $F_i$ matrices in the optimal decision rules are
 
 $$
 F_i = (Q_i + \beta \sum_j \Pi_{ij} B_i' P_j B_i)^{-1}
@@ -240,7 +238,7 @@ $$
 
 ## Applications
 
-We now describe some Python code and a few examples that put the code to work.
+We now describe Python code and some examples.
 
 To begin, we import these Python modules
 
@@ -297,7 +295,7 @@ $$
 Let $x_{t}=\begin{bmatrix} k_{t}\\ 1 \end{bmatrix}$
 
 We can represent the one-period payoff function
-$r\left(s_{t},k_{t}\right)$ and the state-transition law as
+$r\left(s_{t},k_{t}\right)$  as
 
 $$
 \begin{aligned}
@@ -309,6 +307,8 @@ f_{2,s_t} & -\frac{f_{1,s_t}}{2}\\
  \end{aligned}
 $$
 
+and the state-transition law as
+
 $$
 x_{t+1}=\begin{bmatrix}
 k_{t+1}\\
@@ -804,7 +804,7 @@ $x_{t}=\begin{bmatrix} k_{t}\\ 1\\ w_{t} \end{bmatrix}$
 and continue to set the control $u_{t}=k_{t+1}-k_{t}$.
 
 We can write the one-period payoff function
-$r\left(s_{t},k_{t},w_{t}\right)$ and the state-transition law as
+$r\left(s_{t},k_{t},w_{t}\right)$ as  
 
 $$
 \begin{aligned}
@@ -821,7 +821,8 @@ x_{t}+
 \end{aligned}
 $$
 
-and
+and the state-transition law as
+
 
 $$
 x_{t+1}=\begin{bmatrix}

lectures/tax_smoothing_1.md

@@ -38,13 +38,13 @@ This lecture has two sequels that offer further extensions of the Barro model
 1. {doc}`How to Pay for a War: Part 3 <tax_smoothing_3>`
 
 The extensions are modified versions of
-his 1979 model later suggested by Barro (1999 {cite}`barro1999determinants`, 2003 {cite}`barro2003religion`).
+his 1979 model  suggested by  {cite}`barro1999determinants` and  {cite}`barro2003religion`).
 
-Barro’s original 1979 {cite}`Barro1979` model is about a government that borrows and lends
+{cite}`Barro1979` m is about a government that borrows and lends
 in order to minimize an intertemporal measure of distortions
 caused by taxes.
 
-Technical tractability induced Barro {cite}`Barro1979` to assume that
+Technical tractability induced  {cite}`Barro1979` to assume that
 
 - the government trades only one-period risk-free debt, and
 - the one-period risk-free interest rate is constant
@@ -57,17 +57,17 @@ Also, by expanding the dimension of the
 state, we can add a maturity composition decision to the government’s
 problem.
 
-It is by doing these two things that we extend Barro’s 1979 {cite}`Barro1979`
-model along lines he suggested in Barro (1999 {cite}`barro1999determinants`, 2003 {cite}`barro2003religion`).
+By doing these two things we extend  {cite}`Barro1979`
+along lines he suggested in  {cite}`barro1999determinants` and  {cite}`barro2003religion`).
 
-Barro (1979) {cite}`Barro1979` assumed
+{cite}`Barro1979` assumed
 
 - that a government faces an **exogenous sequence** of expenditures
   that it must finance by a tax collection sequence whose expected
   present value equals the initial debt it owes plus the expected
   present value of those expenditures.
-- that the government wants to minimize the following measure of tax
-  distortions: $E_0 \sum_{t=0}^{\infty} \beta^t T_t^2$, where $T_t$ are total tax collections and $E_0$
+- that the government wants to minimize a measure of tax
+  distortions that is proportional to $E_0 \sum_{t=0}^{\infty} \beta^t T_t^2$, where $T_t$ are total tax collections and $E_0$
   is a mathematical expectation conditioned on time $0$
   information.
 - that the government trades only one asset, a risk-free one-period
@@ -79,9 +79,8 @@ Barro (1979) {cite}`Barro1979` assumed
 Barro’s model can be mapped into a discounted linear quadratic dynamic
 programming problem.
 
-Partly inspired by Barro
-(1999) {cite}`barro1999determinants` and Barro (2003) {cite}`barro2003religion`,
-our generalizations of Barro’s (1979) {cite}`Barro1979` model assume
+Partly inspired by  {cite}`barro1999determinants` and  {cite}`barro2003religion`,
+our generalizations of  {cite}`Barro1979`,  assume
 
 - that the government borrows or saves in the form of risk-free bonds
   of maturities $1, 2, \ldots , H$.
@@ -121,7 +120,7 @@ states.
 
 ## Public Finance Questions
 
-Barro’s 1979 {cite}`Barro1979` model is designed to answer questions such as
+{cite}`Barro1979`  is designed to answer questions such as
 
 - Should a government finance an exogenous surge in government
   expenditures by raising taxes or borrowing?
@@ -130,8 +129,7 @@ Barro’s 1979 {cite}`Barro1979` model is designed to answer questions such as
   whether the surge in government expenditures can be expected to be
   temporary or permanent?
 
-Barro’s 1999 {cite}`barro1999determinants` and 2003 {cite}`barro2003religion`
-models are designed to answer more fine-grained
+{cite}`barro1999determinants` and {cite}`barro2003religion` are designed to answer more fine-grained
 questions such as
 
 - What determines whether a government wants to issue short-term or
@@ -174,23 +172,27 @@ import matplotlib.pyplot as plt
 
 ## Barro (1979) Model
 
-We begin by solving a version of the Barro (1979) {cite}`Barro1979` model by mapping it
+We begin by solving a version of  {cite}`Barro1979`  by mapping it
 into the original LQ framework.
 
 As mentioned [in this lecture](https://python-intro.quantecon.org/perm_income_cons.html), the
 Barro model is mathematically isomorphic with the LQ permanent income
 model.
 
-Let $T_t$ denote tax collections, $\beta$ a discount factor,
-$b_{t,t+1}$ time $t+1$ goods that the government promises to
-pay at $t$, $G_t$ government purchases, $p_{t,t+1}$
-the number of time $t$ goods received per time $t+1$ goods
-promised.
+Let 
+ 
+ * $T_t$ denote tax collections
+ *  $\beta$ be a discount factor
+ * $b_{t,t+1}$ be time $t+1$ goods that at $t$ the government promises to
+deliver to time $t$ buyers of  one-period   government bonds 
+ * $G_t$ be government purchases
+ *  $p_{t,t+1}$ the number of time $t$ goods received per time $t+1$ goods
+promised to one-period bond purchasers.
 
 Evidently, $p_{t, t+1}$ is inversely related to
 appropriate corresponding gross interest rates on government debt.
 
-In the spirit of Barro (1979) {cite}`Barro1979`, the stochastic process of government
+In the spirit of  {cite}`Barro1979`, the stochastic process of government
 expenditures is exogenous.
 
 The government’s problem is to choose a plan

lectures/tax_smoothing_2.md

@@ -29,20 +29,19 @@ kernelspec:
 This lecture presents another application of Markov jump linear quadratic dynamic programming and constitutes a  {doc}`sequel to an earlier lecture <tax_smoothing_1>`.
 
 
-We use a method introduced in lecture {doc}`Markov Jump LQ dynamic programming <markov_jump_lq>` to
-implement suggestions by Barro (1999 {cite}`barro1999determinants`, 2003 {cite}`barro2003religion`) for extending his
+We use a method introduced in lecture {doc}`Markov Jump LQ dynamic programming <markov_jump_lq>` toimplement suggestions by  {cite}`barro1999determinants` and  {cite}`barro2003religion`) for extending his
 classic 1979 model of tax smoothing.
 
-Barro’s 1979 {cite}`Barro1979` model is about a government that borrows and lends in order
+ {cite}`Barro1979` model is about a government that borrows and lends in order
 to help it minimize an intertemporal measure of distortions caused by
 taxes.
 
-Technically, Barro’s 1979 {cite}`Barro1979` model looks a lot like a consumption-smoothing model.
+Technically, {cite}`Barro1979` model looks a lot like a consumption-smoothing model.
 
-Our generalizations of his 1979 {cite}`Barro1979` model will also look
+Our generalizations of  {cite}`Barro1979`  will also look
 like souped-up consumption-smoothing models.
 
-Wanting tractability induced Barro in 1979 {cite}`Barro1979` to assume that
+Wanting tractability induced  {cite}`Barro1979` to assume that
 
 - the government trades only one-period risk-free debt, and
 - the one-period risk-free interest rate is constant
@@ -111,7 +110,7 @@ time $t$ goods received per time $t+2$ goods promised.
 Evidently, $p_{t, t+1}, p_{t,t+2}$ are inversely related to
 appropriate corresponding gross interest rates on government debt.
 
-In the spirit of Barro (1979) {cite}`Barro1979`, government
+In the spirit of {cite}`Barro1979`, government
 expenditures are governed by an exogenous stochastic process.
 
 Given initial conditions $b_{-2,0}, b_{-1,0}, z_0, i_0$, where

lectures/tax_smoothing_3.md

@@ -29,17 +29,17 @@ kernelspec:
 This lecture presents another application of Markov jump linear quadratic dynamic programming and constitutes a  {doc}`sequel to an earlier lecture <tax_smoothing_1>`.
 
 We again use a method introduced in lecture {doc}`Markov Jump LQ dynamic programming <markov_jump_lq>`
-to implement some ideas Barro (1999 {cite}`barro1999determinants`, 2003 {cite}`barro2003religion`) that
-extend his classic 1979 {cite}`Barro1979` model of tax smoothing.
+to implement some ideas of  {cite}`barro1999determinants` and  {cite}`barro2003religion`) that
+extend the  classic  {cite}`Barro1979` model of tax smoothing.
 
-Barro’s 1979 {cite}`Barro1979` model is about a government that borrows and lends in order
+{cite}`Barro1979`  is about a government that borrows and lends in order
 to help it minimize an intertemporal measure of distortions caused by
 taxes.
 
-Technically, Barro’s 1979 {cite}`Barro1979` model looks a lot like a consumption-smoothing model.
+Technically,  {cite}`Barro1979`  looks a lot like a consumption-smoothing model.
 
-Our generalizations of his 1979 model will also look
-like souped-up consumption-smoothing models.
+Our generalization  will also look
+like a souped-up consumption-smoothing model.
 
 In this lecture, we describe a  tax-smoothing problem of a
 government that faces **roll-over risk**.