Tom's edits of three more lectures in intro repo

lectures/cagan_ree.md

@@ -13,23 +13,26 @@ kernelspec:
 
 # A Monetarist Theory of  Price Levels
 
-## Introduction
+## Overview
 
 
 We'll use linear algebra first to explain and then do  some experiments with  a "monetarist theory of  price levels".
 
-Sometimes this theory is also called a "fiscal theory of  price levels".
 
-Such a theory of  price levels was described by Thomas Sargent and Neil Wallace in chapter 5 of 
-{cite}`sargent2013rational`, which reprints a  1981 Federal Reserve Bank of Minneapolis article entitled "Unpleasant Monetarist Arithmetic".  
+
 
-Sometimes people call it a "monetary" or "monetarist" theory of  price levels because fiscal effects on  price levels occur through the effects of government fiscal policy decisions on the path of the money supply.  
+Economist  call it a "monetary" or "monetarist" theory of  price levels because  effects on  price levels occur via a central banks's  decisions to print  money supply.  
 
   * a goverment's fiscal policies determine whether it **expenditures** exceed its **tax collections**
-  * if its expenditures exceeds it tax collections, it can cover the difference by **printing money**
+  * if its expenditures exceeds it tax collections, the government can instruct the central bank to  cover the difference by **printing money**
   * that leads to effects on the price level as price level path adjusts to equate the supply of money to the demand for money
 
-The theory has been extended, criticized, and applied by John Cochrane in {cite}`cochrane2023fiscal`.
+Such a theory of  price levels was described by Thomas Sargent and Neil Wallace in chapter 5 of 
+{cite}`sargent2013rational`, which reprints a  1981 Federal Reserve Bank of Minneapolis article entitled "Unpleasant Monetarist Arithmetic". 
+
+Sometimes this theory is also called a "fiscal theory of  price levels" to emphasize the importance of fisal deficits in shaping changes in the money supply.  
+
+The theory has been extended, criticized, and applied by John Cochrane  {cite}`cochrane2023fiscal`.
 
 In another lecture {doc}`price level histories <inflation_history>`,  we described some  European hyperinflations that occurred in the wake of World War I.
 
@@ -39,29 +42,29 @@ Elemental forces at work in the fiscal theory of the price level help to underst
 According to this theory, when the government persistently spends more than it collects in taxes and prints money to finance the shortfall (the "shortfall" is called the "government deficit"), it puts upward pressure on the price level and generates
 persistent inflation.
 
-The "monetarist or fiscal theory of  price levels" asserts that  
+The ''monetarist'' or ''fiscal theory of  price levels" asserts that  
 
-* to **start** a persistent inflation the government simply persistently runs a money-financed government deficit
+* to **start** a persistent inflation the government beings persistently to run a money-financed government deficit
 
-* to **stop** a persistent inflation the government simply  stops persistently running  a money-financed government deficit
+* to **stop** a persistent inflation the government   stops persistently running  a money-financed government deficit
 
-Our model is a "rational expectations" (or "perfect foresight") version of a model that Philip Cagan {cite}`Cagan`  used to study the monetary dynamics of hyperinflations.  
+The  model in this lecture is a "rational expectations" (or "perfect foresight") version of a model that Philip Cagan {cite}`Cagan`  used to study the monetary dynamics of hyperinflations.  
 
 While Cagan didn't use that  "rational expectations" version of the model, Thomas Sargent {cite}`sargent1982ends` did when he studied the Ends of Four Big Inflations in Europe after World War I.
 
 * this lecture {doc}`fiscal theory of the price level with adaptive expectations <cagan_adaptive>` describes  a version  of the model that does not impose "rational expectations" but instead uses 
    what Cagan and his teacher Milton Friedman called "adaptive expectations"
 
-   * a reader of both lectures will notice that the algebra is easier and more streamlined in the present rational expectations version of the model
-   * this can be traced to the following source: the adaptive expectations version of the model has more endogenous variables and more free parameters 
+   * a reader of both lectures will notice that the algebra is less complicated in the present rational expectations version of the model
+   * the difference in algebra complications  can be traced to the following source: the adaptive expectations version of the model has more endogenous variables and more free parameters 
 
-Some of our quantitative experiments with our rational expectations version of the  model are designed to illustrate how the fiscal theory explains the abrupt end of those big inflations.
+Some of our quantitative experiments with the  rational expectations version of the  model are designed to illustrate how the fiscal theory explains the abrupt end of those big inflations.
 
 In those experiments, we'll encounter an instance of a ''velocity dividend'' that has sometimes accompanied successful inflation stabilization programs. 
 
 To facilitate using  linear matrix algebra as our main mathematical tool, we'll use a finite horizon version of the model.
 
-As in the {doc}`present values <pv>` and {doc}`consumption smoothing<cons_smooth>` lectures, the only linear algebra that we'll be  using are matrix multiplication and matrix inversion.
+As in the {doc}`present values <pv>` and {doc}`consumption smoothing<cons_smooth>` lectures, our mathematical tools  are matrix multiplication and matrix inversion.
 
 
 ## Structure of the model
@@ -71,7 +74,7 @@ The model consists of
 
 * a function that expresses the demand for real balances of government printed money as an inverse function of the public's expected rate of inflation
 
-* an exogenous sequence of rates of growth of the money supply.  The money supply grows because the government is printing it to finance some of its expenditures
+* an exogenous sequence of rates of growth of the money supply.  The money supply grows because the government  prints  it to pay for goods and services
 
 * an equilibrium condition that equates the demand for money to the supply
 

lectures/equalizing_difference.md

@@ -18,27 +18,30 @@ kernelspec:
 This lecture presents a model of the college-high-school wage gap in which the
 "time to build" a college graduate plays a key role.
 
-```{note}
-Milton Friedman used our   model  to study whether  differences in the earnings of US dentists and doctors were justified by competitive labor markets or whether
-they reflected entry barriers imposed by US governments working in conjunction with doctors' lobbies.  Chapter 4 of Jennifer Burns {cite}`Burns_2023` presents an 
-interesting account of Milton Friedman's joint work with Simon Kuznets that eventually  led to the publication of {cite}`kuznets1939incomes` and {cite}`friedman1954incomes`. To map  Friedman's application to our model, think of our high school students as Friedman's dentists and our college graduates as Friedman's doctors.  
-```
 
-The model is "incomplete" in the sense that it is just one "condition" in the form of a single equation that would be part of set equations comprising all  "equilibrium conditions" of   a more fully articulated model.
+Milton Friedman invented the   model  to study whether  differences in  earnings of US dentists and doctors were outcomes of  competitive labor markets or whether
+they reflected entry barriers imposed by governments working in conjunction with doctors' professional organizations. 
+
+Chapter 4 of Jennifer Burns {cite}`Burns_2023` describes  Milton Friedman's joint work with Simon Kuznets that eventually  led to the publication of {cite}`kuznets1939incomes` and {cite}`friedman1954incomes`.
+
+To map  Friedman's application into our model, think of our high school students as Friedman's dentists and our college graduates as Friedman's doctors.  
+
 
-The condition featured in our model determines  a college, high-school wage ratio that equalizes the present values of a high school worker and a college educated worker.
+Our presentation is "incomplete" in the sense that it is based on  a single equation that would be part of set equilibrium conditions of a more fully articulated model.
 
-The idea behind this condition is that lifetime earnings have to adjust to make someone indifferent between going to college and not going to college.
+This ''equalizing difference'' equation  determines  a college, high-school wage ratio that equalizes present values of a high school educated  worker and a college educated worker.
 
-(The job of the "other equations" in a more complete model would be to fill in details about what adjusts to bring about this outcome.)
+The idea  is that lifetime earnings somehow adjust to make a new high school worker indifferent between going to college and not going to college but instead going to work immmediately.
 
-It is just one instance of an  "equalizing difference" theory of relative wage rates, a class of theories dating back at least to Adam Smith's **Wealth of Nations** {cite}`smith2010wealth`.  
+(The job of the "other equations" in a more complete model would be to describe what adjusts to bring about this outcome.)
+
+Our model is just one example  of an  "equalizing difference" theory of relative wage rates, a class of theories dating back at least to Adam Smith's **Wealth of Nations** {cite}`smith2010wealth`.  
 
 For most of this lecture, the only mathematical tools that we'll use are from linear algebra, in particular, matrix multiplication and matrix inversion.
 
-However, at the very end of the lecture, we'll use calculus just in case readers want to see how computing partial derivatives could let us present some findings more concisely.  
+However, near the  end of the lecture, we'll use calculus just in case readers want to see how computing partial derivatives could let us present some findings more concisely.  
 
-(And doing that will let us show off how good Python is at doing calculus!)
+And doing that will let illustrate how good Python is at doing calculus!
 
 But if you don't know calculus, our tools from linear algebra are certainly enough.
 
@@ -51,15 +54,15 @@ import matplotlib.pyplot as plt
 
 ## The indifference condition
 
-The key idea is that the initial college wage premium has to adjust to make a representative worker indifferent between going to college and not going to college.
+The key idea is that the entry level college wage premium has to adjust to make a representative worker indifferent between going to college and not going to college.
 
 Let
 
  * $R > 1$ be the gross rate of return on a one-period bond
 
  * $t = 0, 1, 2, \ldots T$ denote the years that a person either works or attends college
 
- * $0$ denote the first period after high school that a person can go to work
+ * $0$ denote the first period after high school that a person can work if he does not go to college
 
  * $T$ denote the last period  that a person  works
 
@@ -75,7 +78,12 @@ Let
 
  * $D$ be the upfront monetary costs of going to college
 
+We now compute present values that a new high school graduate earns if
+
+  * he goes to work immediately and earns wages paid to someone without a college education
+  * he goes to college for four years and after graduating earns wages paid to a college graduate
 
+### Present value of a high school educated worker
 
 If someone goes to work immediately after high school  and  works for the  $T+1$ years $t=0, 1, 2, \ldots, T$, she earns present value
 
@@ -91,6 +99,8 @@ $$
 
 The present value $h_0$ is the "human wealth" at the beginning of time $0$ of someone who chooses not to attend college but instead to go to work immediately at the wage of a high school graduate.
 
+### Present value of a college-bound new high school graduate
+
 
 If someone goes to college for the four years $t=0, 1, 2, 3$ during which she earns $0$, but then goes to work  immediately after college   and  works for the $T-3$ years $t=4, 5, \ldots ,T$, she earns present value
 
@@ -101,31 +111,29 @@ $$
 where
 
 $$
-A_c = (R^{-1} \gamma_c)^4  \left[ \frac{1 - (R^{-1} \gamma_c)^{T-3} }{1 - R^{-1} \gamma_c } \right] 
+A_c = (R^{-1} \gamma_c)^4  \left[ \frac{1 - (R^{-1} \gamma_c)^{T-3} }{1 - R^{-1} \gamma_c } \right] .
 $$ 
 
 The present value $c_0$  is the "human wealth" at the beginning of time $0$ of someone who chooses to attend college for four years and then start to work at time $t=4$ at the wage of a college graduate.
 
 
-Assume that college tuition plus four years of room and board paid for up front costs $D$.
+Assume that college tuition plus four years of room and board amount to  $D$ and must be paid at time $0$.
 
 So net of monetary cost of college, the present value of attending college as of the first period after high school is
 
 $$ 
 c_0 - D
 $$
 
-We now formulate a pure **equalizing difference** model of the initial college-high school wage gap $\phi$ defined by 
-
-Let
+We now formulate a pure **equalizing difference** model of the initial college-high school wage gap $\phi$ that verifies 
 
 $$
 w_0^c = \phi w_0^h 
 $$
 
 We suppose that $R, \gamma_h, \gamma_c, T$ and also $w_0^h$  are fixed parameters. 
 
-We start by noting that the pure equalizing difference model asserts that the college-high-school wage gap $\phi$ solves 
+We start by noting that the pure equalizing difference model asserts that the college-high-school wage gap $\phi$ solves an 
 "equalizing" equation that sets the present value not going to college equal to the present value of going go college:
 
 
@@ -139,25 +147,27 @@ $$
 w_0^h A_h  = \phi w_0^h A_c - D .
 $$ (eq:equalize)
 
-This is the "indifference condition" that is at the heart of the model.
+This "indifference condition"  is the heart of the model.
 
 Solving equation {eq}`eq:equalize` for the college wage premium $\phi$ we obtain
 
 $$
 \phi  = \frac{A_h}{A_c} + \frac{D}{w_0^h A_c} .
 $$ (eq:wagepremium)
 
-In a **free college** special case $D =0$ so that the only cost of going to college is the forgone earnings from not working as a high school worker.  
+In a **free college** special case $D =0$.
+
+Here  the only cost of going to college is the forgone earnings from being  a high school educated worker.  
 
 In that case,
 
 $$
 \phi  = \frac{A_h}{A_c} . 
 $$
 
-Soon we'll write Python code to compute the gap and plot it as a function of its determinants.
+Soon we'll write Python code to compute $\phi$  and plot it as a function of its determinants.
 
-But first we'll describe a possible alternative interpretation of our model.
+But first we'll describe an  alternative interpretation of our model that mostly just relabels variables.
 
 
 
@@ -185,13 +195,18 @@ This cost might include costs of hiring workers, office space, and lawyers.
 What we used to call the college, high school wage gap $\phi$ now becomes the ratio
 of a successful entrepreneur's earnings to a worker's earnings.  
 
-We'll find that as $\pi$ decreases, $\phi$ increases.  
+We'll find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to
+be an entrepreuner, the higher must be the reward for a successful project. 
+
+## Computations
 
-Now let's write some Python code to compute $\phi$ and plot it as a function of some of its determinants.
 
-We can have some fun providing some example calculations that tweak various parameters,
+We can have some fun with examples that tweak various parameters,
 prominently including $\gamma_h, \gamma_c, R$.
 
+Now let's write some Python code to compute $\phi$ and plot it as a function of some of its determinants.
+
+
 ```{code-cell} ipython3
 class equalizing_diff:
     """
@@ -219,10 +234,10 @@ class equalizing_diff:
 ```
 
 
+Using vectorization instead of loops,
+we  build some functions to help do comparative statics .
 
-We can build some functions to help do comparative statics using vectorization instead of loops.
-
-For a given instance of the class, we want to compute $\phi$ when one parameter changes and others remain unchanged.
+For a given instance of the class, we want to recompute $\phi$ when one parameter changes and others remain fixed.
 
 Let's do an example.
 
@@ -315,7 +330,7 @@ plt.show()
 ```
 Notice how  the intitial wage gap falls when the rate of growth $\gamma_c$ of college wages rises.  
 
-It falls to "equalize" the present values of the two types of career, one as a high school worker, the other as a college worker.
+The wage gap falls to "equalize" the present values of the two types of career, one as a high school worker, the other as a college worker.
 
 Can you guess what happens to the initial wage ratio $\phi$ when next we vary the rate of growth of high school wages, holding all other determinants of $\phi$ constant?  
 
@@ -363,9 +378,9 @@ Does the graph make sense to you?
 
 So far, we have used only linear algebra and it has been a good enough tool for us to  figure out how our model works.
 
-However, someone who knows calculus might ask "Instead of plotting those graphs, why didn't you just take partial derivatives?"
+However, someone who knows calculus might want us  just to  take partial derivatives.
 
-We'll briefly do just that,  yes, the questioner is correct and that partial derivatives are indeed a good tool for discovering the "comparative statics" properities of our model.
+We'll do that now.
 
 A reader who doesn't know calculus could read no further and feel confident that applying linear algebra has taught us the main properties of the model.
 
@@ -433,7 +448,7 @@ Now let's compute $\frac{\partial \phi}{\partial D}$ and then evaluate it at the
 ϕ_D_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
 ```
 
-Thus, as with our graph above, we find that raising $R$ increases the initial college wage premium $\phi$.
+Thus, as with our earlier graph, we find that raising $R$ increases the initial college wage premium $\phi$.
 
 +++
 
@@ -469,7 +484,7 @@ Let's compute $\frac{\partial \phi}{\partial γ_h}$ and evaluate it at default p
 ϕ_γ_h_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value)
 ```
 
-We find that raising $\gamma_h$ increases the initial college wage premium $\phi$, as we did with our graphical analysis earlier
+We find that raising $\gamma_h$ increases the initial college wage premium $\phi$, as we did with our  earlier graphical analysis.
 
 +++