Tom's final edits of advanced lectures, Jan 23

Author: thomassargent30

lectures/cons_news.md

@@ -37,7 +37,7 @@ tags: [hide-output]
 
 This lecture studies two consumers who have exactly the same
 nonfinancial income process and who both conform to the linear-quadratic
-permanent income of consumption smoothing model described in the
+permanent income of consumption smoothing model described in this
 [quantecon lecture](https://python-intro.quantecon.org/perm_income_cons.html).
 
 The two consumers  have different information about
@@ -87,7 +87,7 @@ We compare behaviors of our two consumers as a way to learn about
 This lecture can be regarded as an introduction to some of the **invertibility** issues that take center stage in
 the analysis of **fiscal foresight** by Eric Leeper, Todd Walker, and Susan Yang {cite}`Leeper_Walker_Yang`.
 
-## Two Representations of the **Same** Nonfinancial Income Process
+## Two Representations of  One Nonfinancial Income Process
 
 Where $\beta \in (0,1)$, we study consequences of endowing a
 consumer with one of the two alternative representations for the change
@@ -201,7 +201,7 @@ $$
 \sigma_a^2 = \sigma_\epsilon^2 + [ 1 + (\beta - \beta^{-1})^2 \sum_{j=0}^\infty \beta^{2j} ] = \beta^{-1} \sigma_\epsilon^2 .
 $$
 
-### Application of Kalman filter
+## Application of Kalman filter
 
 We can also obtain representation {eq}`eqn_2` from representation {eq}`eqn_1` by using
 the **Kalman filter**.
@@ -240,7 +240,7 @@ and $\Sigma = (1-\beta^2) \sigma_\epsilon^2$.
 We can also obtain these formulas via the classical filtering theory
 described in {doc}`this lecture <classical_filtering>`.
 
-### News Shocks and Less Informative Shocks
+## News Shocks and Less Informative Shocks
 
 Representation {eq}`eqn_1` is cast in terms of a **news shock**
 $\epsilon_{t+1}$ that represents a shock to nonfinancial income
@@ -333,7 +333,7 @@ for representation {eq}`eqn_2` is
 $d_a(\beta) = \frac{1 -\beta^2}{1 -\beta } = (1 + \beta)$, another
 fact that will be important below.
 
-### Representation of $\epsilon_t$ in Terms of Future $y$’s
+## Representation of $\epsilon_t$ Shock in Terms of Future $y_t$
 
 Notice that reprentation {eq}`eqn_1`, namely, $y_{t+1} - y_t = -\beta^{-1} \epsilon_t + \epsilon_{t+1}$
 implies the linear difference equation
@@ -360,7 +360,7 @@ $$
 Thus, $\epsilon_t$ contains **exact** information about an
 important linear combination of **future** nonfinancial income.
 
-### Representation in Terms of $a_t$ Shocks
+## Representation in Terms of $a_t$ Shocks
 
 Next notice that representation {eq}`eqn_2`, namely, $y_{t+1} - y_t = -
 \beta a_t + a_{t+1}$ implies the linear difference
@@ -384,7 +384,7 @@ $$
 E [ y_{t+1} | y^t ] = (1-\beta) \sum_{j=0}^\infty \beta^j y_{t-j}
 $$
 
-### Permanent Income Consumption-Smoothing Model
+## Permanent Income Consumption-Smoothing Model
 
 When we computed optimal consumption-saving policies for the two
 representations using formulas obtained with the difference equation
@@ -468,7 +468,7 @@ Equation Approach” in the [quantecon  lecture](https://python-intro.quantecon.
 
 All the code that we shall use below is presented in that lecture.
 
-### Computations
+## Computations
 
 We shall use Python to form **both** of the above two state-space
 representations, using the following parameter values
@@ -792,7 +792,7 @@ plt.title("innovations representation")
 plt.legend()
 ```
 
-### Simulating the Income Process and Two Associated Shock Processes
+## Simulating  Income Process and Two Associated Shock Processes
 
 We now describe how we form a **single** $\{y_t\}_{t=0}^T$ realization
 that we will use to simulate the two different decision rules associated
@@ -828,7 +828,7 @@ The above steps implement the experiment of comparing decisions made by
 two consumers having **identical** incomes at each date but at each date
 having **different** information about their future incomes.
 
-### Calculating Innovations in Another Way
+## Calculating Innovations in Another Way
 
 Here we use formula {eq}`eqn_3` above to compute $a_{t+1}$ as a function
 of the history
@@ -850,7 +850,7 @@ $$
 We can verify that we recover the same $\{a_t\}$ sequence
 computed earlier.
 
-### Another Invertibility Issue
+## Another Invertibility Issue
 
 This {doc}`quantecon lecture <hs_invertibility_example>` contains another example of a shock-invertibility issue that is endemic
 to the LQ permanent income or consumption smoothing model.

lectures/dyn_stack.md

@@ -124,7 +124,7 @@ In choosing $\vec q_2$, firm 2 takes into account that firm 1 will
 base its choice of $\vec q_1$ on firm 2's choice of
 $\vec q_2$.
 
-### Abstract Statement of the Leader's and Follower's Problems
+### Statement of  Leader's and Follower's Problems
 
 We can express firm 1's problem as
 
@@ -320,7 +320,7 @@ recursively.
 We'll put our little duopoly model into a broader class of models with
 the same conceptual structure.
 
-## The Stackelberg Problem
+## Stackelberg Problem
 
 We formulate a class of linear-quadratic Stackelberg leader-follower
 problems of which our duopoly model is an instance.
@@ -401,7 +401,7 @@ or
 y_{t+1} = A y_t + B u_t
 ```
 
-### Interpretation of the Second Block of Equations
+### Interpretation of  Second Block of Equations
 
 The Stackelberg follower's best response mapping is summarized by the
 second block of equations of {eq}`new3`.
@@ -494,7 +494,7 @@ The value function $w(z_0)$ tells the value of the Stackelberg plan
 as a function of the vector of natural state variables at time $0$,
 $z_0$.
 
-### Two Bellman Equations
+## Two Bellman Equations
 
 We now describe Bellman equations for $v(y)$ and $w(z_0)$.
 
@@ -773,7 +773,7 @@ sequence $\vec q_2$, we must use representation
 $\check z^t$ and **not** a corresponding representation cast in
 terms of $z^t$.
 
-### Dynamic Programming and Time Consistency of Follower's Problem
+## Dynamic Programming and Time Consistency of Follower's Problem
 
 Given the sequence $\vec q_2$ chosen by the Stackelberg leader in
 our duopoly model, it turns out that the Stackelberg **follower's**
@@ -914,7 +914,7 @@ $$
 
 It follows that the follower's plan is time consistent.
 
-## Computing the Stackelberg Plan
+## Computing  Stackelberg Plan
 
 Here is our code to compute a Stackelberg plan via a linear-quadratic
 dynamic program as outlined above
@@ -988,7 +988,7 @@ print("Computed policy for Stackelberg leader\n")
 print(f"F = {F}")
 ```
 
-### Implied Time Series for Price and Quantities
+## Time Series for Price and Quantities
 
 The following code plots the price and quantities
 
@@ -1039,7 +1039,7 @@ v_expanded = -((y0.T @ R @ y0 + ut[:, 0].T @ Q @ ut[:, 0] +
 (v_leader_direct - v_expanded < tol0)[0, 0]
 ```
 
-## Exhibiting Time Inconsistency of Stackelberg Plan
+## Time Inconsistency of Stackelberg Plan
 
 In the code below we compare two values
 
@@ -1086,7 +1086,7 @@ plt.tight_layout()
 plt.show()
 ```
 
-## Recursive Formulation of the Follower's Problem
+## Recursive Formulation of Follower's Problem
 
 We now formulate and compute the recursive version of the follower's
 problem.

lectures/lu_tricks.md

@@ -549,7 +549,7 @@ For now, we note that by creating the matrix $W$ for large
 $N$ and factoring it into the $LU$ form, good approximations
 to $c(L)$ and $c(\beta L^{-1})^{-1}$ can be obtained.
 
-## The Infinite Horizon Limit
+## Infinite Horizon Limit
 
 For the infinite horizon problem, we propose to discover first-order
 necessary conditions by taking the limits of {eq}`onefour` and {eq}`onefive` as