Tom's May 31 edits of two advanced lectures

Author: thomassargent30

lectures/cons_news.md

@@ -137,7 +137,7 @@ $$
 so that the variance of the innovation exceeds the variance of the
 original shock by a multiplicative factor $\beta^{-2}$.
 
-Representation {eq}`eq2` is the **innovations representation** of equation {eq}`eq1` associated with
+Representation {eq}`eqn_2` is the **innovations representation** of equation {eq}`eqn_1` associated with
 Kalman filtering theory.
 
 To see how this works, note that equating representations {eq}`eqn_1`
@@ -387,7 +387,7 @@ $$
 ## Permanent Income Consumption-Smoothing Model
 
 When we computed optimal consumption-saving policies for our  two
-representations {eq}`eq1` and {eq}`eq2` by using formulas obtained with the difference equation
+representations {eq}`eqn_1` and {eq}`eqn_2` by using formulas obtained with the difference equation
 approach described in  [quantecon lecture](https://python-intro.quantecon.org/perm_income_cons.html),
 we obtained:
 
@@ -437,14 +437,15 @@ Ricardian equivalence experiment.
 
 ## State Space Representations
 
-We now cast our  representations {eq}`eq1` and {eq}`eq2`, respectively,  in terms of the following two state
+We now cast our  representations {eq}`eqn_1` and {eq}`eqn_2`, respectively,  in terms of the following two state
 space systems:
+
 $$
 \begin{aligned}  \begin{bmatrix} y_{t+1} \cr \epsilon_{t+1} \end{bmatrix}  &=
       \begin{bmatrix} 1 & - \beta^{-1} \cr 0 & 0 \end{bmatrix} \begin{bmatrix} y_t \cr \epsilon_t \end{bmatrix}
        + \begin{bmatrix} \sigma_\epsilon \cr \sigma_\epsilon \end{bmatrix}
        v_{t+1} \cr
-y_t & =  \begin{bmatrix} 1 & 0 \end{bmatrix}    \begin{bmatrix} y_t \cr \epsilon_t \end{bmatrix} \end{aligned}
+y_t & =  \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} y_t \cr \epsilon_t \end{bmatrix} \end{aligned}
 $$ (eqstsp1)
 
 and
@@ -469,7 +470,9 @@ All the code that we shall use below is presented in that lecture.
 ## Computations
 
 We shall use Python to form  two state-space 
-representations {eq}`eqstsp1` and {eq}`eqstsp2`, using the following parameter values
+representations {eq}`eqstsp1` and {eq}`eqstsp2`.
+
+We set the following parameter values
 $\sigma_\epsilon = 1, \sigma_a = \beta^{-1} \sigma_\epsilon = \beta^{-1}$
 where $\beta$ is the **same** value as the discount factor in the
 household’s problem in the LQ savings problem in the [lecture](https://python-intro.quantecon.org/perm_income_cons.html).
@@ -656,9 +659,9 @@ b_{t+1}^{*} &= \beta^{-1}c_{t}^{*}+\beta^{-1}b_{t}-\beta^{-1}y_{t} \\
 $$
 
 Now we construct two Linear State Space models that emerge from using
-optimal policies $u_t =- F x_t$ for the control variable.
+optimal policies of the form $u_t =- F x_t$.
 
-Take the original representation case as an example,
+Take the original representation {eq}`eqn_1` as an example:
 
 $$
 \left[\begin{array}{c}
@@ -686,8 +689,8 @@ b_{t}
 \end{array}\right]
 $$
 
-To have the Linear State Space model of the innovations representation
-case, we can simply replace the corresponding matrices.
+To have the Linear State Space model be of an innovations representation
+form {eq}`eqn_2`, we can simply replace the corresponding matrices.
 
 ```{code-cell} python3
 # Construct two Linear State Space models
@@ -702,7 +705,7 @@ G2 = np.vstack([-F2, Sb])
 LSS2 = qe.LinearStateSpace(ABF2, CLQ2, G2)
 ```
 
-In the following we compute the impulse response functions of
+The following code computes impulse response functions of
 $c_t$ and $b_t$.
 
 ```{code-cell} python3
@@ -729,7 +732,7 @@ plt.legend()
 ```
 
 The above two impulse response functions show that when the consumer has
-the information assumed in the original representation, his response to
+the information assumed in the original representation {eq}`eqn_1`, his response to
 receiving a positive shock of $\epsilon_t$ is to leave his
 consumption unchanged and to save the entire amount of his extra income
 and then forever roll over the extra bonds that he holds.
@@ -751,20 +754,20 @@ plt.legend()
 
 The above impulse responses show that when the consumer has only the
 information that is assumed to be available under the innovations
-representation for $\{y_t - y_{t-1} \}$, he responds to a positive
-$a_t$ by permanently increasing his consumption.
+representation {eq}`eqn_2` for $\{y_t - y_{t-1} \}$, he responds to a positive
+$a_t$ by **permanently** increasing his consumption.
 
 He accomplishes this by consuming a fraction $(1 - \beta^2)$ of
-the increment $a_t$ to his nonfinancial income and saving the rest
-in order to lower $b_{t+1}$ to finance the permanent increment in
+the increment $a_t$ to his nonfinancial income and saving the rest, thereby
+ lowering $b_{t+1}$ in order  to finance the permanent increment in
 his consumption.
 
 The preceding computations confirm what we had derived earlier using
 paper and pencil.
 
 Now let’s simulate some paths of consumption and debt for our two types
 of consumers while always presenting both types with the same
-$\{y_t\}$ path, constructed as described below.
+$\{y_t\}$ path.
 
 ```{code-cell} python3
 # Set time length for simulation
@@ -791,30 +794,30 @@ plt.legend()
 
 ## 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
+We now  form a **single** $\{y_t\}_{t=0}^T$ realization
+that we will use to simulate decisions associated
 with our two types of consumer.
 
 We accomplish this in the following steps.
 
 1. We form a $\{y_t, \epsilon_t\}$ realization by drawing a long
-   simulation of $\{\epsilon_t\}_{t=0}^T$ where $T$ is a big
-   integer $\epsilon_t = \sigma_\epsilon v_t$, $v_t$ is a
+   simulation of $\{\epsilon_t\}_{t=0}^T$, where $T$ is a big
+   integer, $\epsilon_t = \sigma_\epsilon v_t$, $v_t$ is a
    standard normal scalar, $y_0 =100$, and
    
    $$
    y_{t+1} - y_t =-\beta^{-1} \epsilon_t + \epsilon_{t+1} .
    $$
    
-1. We take the **same** $\{y_t\}$ realization generated in step 1
+1. We take the  $\{y_t\}$ realization generated in step 1
    and form an innovation process $\{a_t\}$ from the formulas
    
    $$
    \begin{aligned} a_0 & = 0 \cr
    a_t & = \sum_{j=0}^{t-1} \beta^j (y_{t-j} - y_{t-j-1}) + \beta^t a_0, \quad t \geq 1 \end{aligned}
    $$
    
-1. We throw away the first $S$ observations and form the sample
+1. We throw away the first $S$ observations and form a sample
    $\{y_t, \epsilon_t, a_t\}_{S+1}^T$ as the realization that
    we’ll use in the following steps.
 1. We use the step 3 realization to **evaluate** and **simulate** the
@@ -838,7 +841,7 @@ $$
 a_{t+1} &=\beta a_{t}+\epsilon_{t+1}-\beta^{-1}\epsilon_{t} \\
     &=\beta\left(\beta a_{t-1}+\epsilon_{t}-\beta^{-1}\epsilon_{t-1}\right)+\epsilon_{t+1}-\beta^{-1}\epsilon_{t} \\
     &=\beta^{2}a_{t-1}+\beta\left(\epsilon_{t}-\beta^{-1}\epsilon_{t-1}\right)+\epsilon_{t+1}-\beta^{-1}\epsilon_{t} \\
-    &=\cdots \\
+    &=\vdots \\
     &=\beta^{t+1}a_{0}+\sum_{j=0}^{t}\beta^{j}\left(\epsilon_{t+1-j}-\beta^{-1}\epsilon_{t-j}\right) \\
     &=\beta^{t+1}a_{0}+\epsilon_{t+1}+\left(\beta-\beta^{-1}\right)\sum_{j=0}^{t-1}\beta^{j}\epsilon_{t-j}-\beta^{t-1}\epsilon_{0}.
 \end{aligned}

lectures/smoothing.md

@@ -43,12 +43,12 @@ This lecture describes two types of consumption-smoothing models.
 * one is in the **complete markets** tradition of [Kenneth Arrow](https://en.wikipedia.org/wiki/Kenneth_Arrow)
 * the other is in the **incomplete markets** tradition  of Hall {cite}`Hall1978`
 
-*Complete markets* allow a consumer  to buy or sell claims contingent on all possible states of the world.
+*Complete markets* allow a consumer  to buy and sell claims contingent on all possible states of the world.
 
-*Incomplete markets* allow a consumer to buy or sell only a limited set of securities, often only a single risk-free security.
+*Incomplete markets* allow a consumer to buy and sell  a limited set of securities, often only a single risk-free security.
 
 Hall {cite}`Hall1978` worked in an incomplete markets tradition by assuming
-that the only asset that can be traded is a risk-free one period bond.
+that the only asset that can be traded is a risk-free one-period bond.
 
 Hall assumed an exogenous stochastic process of nonfinancial income and
 an exogenous and time-invariant gross interest rate on one period risk-free debt that equals
@@ -60,10 +60,10 @@ consumption at time $t+1$ for sure.
 
 So $\beta$ is the price of a one-period risk-free claim to consumption next period.
 
-We maintain Hall's assumption about the interest rate when we describe an
+We preserve Hall's assumption about the interest rate when we describe an
 incomplete markets version of our model.
 
-In addition, we extend Hall's assumption about the risk-free interest rate to its appropriate counterpart when we create another model in which there are markets
+In addition, we extend Hall's assumption about the risk-free interest rate to an appropriate counterpart when we create another model in which there are markets
 in a complete array of one-period Arrow state-contingent securities.
 
 We'll consider two closely related but distinct alternative assumptions about the consumer's
@@ -113,7 +113,7 @@ payoffs depend on next period's realization of the Markov state.
 
 * In the two-state Markov chain case,  two such securities are traded  each period.
 * In an $N$ state Markov state version,  $N$ such securities are traded each period.
-* In a continuous state Markov state version, a continuum of such securities are traded each period.
+* In a continuous state Markov state version, a continuum of such securities is traded each period.
 
 These state-contingent securities are commonly called Arrow securities, after [Kenneth Arrow](https://en.wikipedia.org/wiki/Kenneth_Arrow).