update sources lecture-python (9ae923d) using tomyst (b06cacb) (#86)

* update sources lecture-python (9ae923d) using tomyst (b06cacb)

* newline bug

Co-authored-by: mmcky <[email protected]>

Author: AakashGfude

lectures/cass_koopmans_2.md

@@ -195,7 +195,7 @@ all other dates $t=1, 2, \ldots, T$.
 There are  sequences of prices
 $\{w_t,\eta_t\}_{t=0}^T= \{\vec{w}, \vec{\eta} \}$
 where $w_t$ is a wage or rental rate for labor at time $t$ and
-$eta_t$ is a rental rate for capital at time $t$.
+$\eta_t$ is a rental rate for capital at time $t$.
 
 In addition there is are intertemporal prices that work as follows.
 
@@ -397,7 +397,7 @@ verify** approach.
 In this lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>`, we  computed an allocation $\{\vec{C}, \vec{K}, \vec{N}\}$
 that solves the planning problem.
 
-(This allocation will constitute the **Big** $K$  to be in the present instance of the *Big** $K$ **, little** $k$ trick
+(This allocation will constitute the **Big** $K$  to be in the present instance of the **Big** $K$ **, little** $k$ trick
 that we'll apply to  a competitive equilibrium in the spirit of [this lecture](https://lectures.quantecon.org/py/rational_expectations.html#)
 and  [this lecture](https://lectures.quantecon.org/py/dyn_stack.html#).)
 
@@ -893,7 +893,7 @@ Vice-versa for lower $\gamma$.
 
 We return to  Hicks-Arrow prices and  calculate how they are related to  **yields**  on loans of alternative maturities.
 
-This will let us plot a **yield curve** that graphs   yields  on bonds of  maturities $j=1, 2, \ldots$ against :math:j=1,2, ldots`.
+This will let us plot a **yield curve** that graphs   yields  on bonds of  maturities $j=1, 2, \ldots$ against $j=1,2, \ldots$.
 
 The formulas we want are:
 

lectures/exchangeable.md

@@ -53,7 +53,7 @@ that are
 Understanding the distinction between these concepts is essential for appreciating how Bayesian updating
 works in our example.
 
-You can read about exchangeability [here](https://en.wikipedia.org/wiki/Exchangeable_random_variables)
+You can read about exchangeability [here](https://en.wikipedia.org/wiki/Exchangeable_random_variables).
 
 Below, we'll often use
 
@@ -116,8 +116,10 @@ $$
 Using the laws of probability, we can always factor such a joint density into a product of conditional densities:
 
 $$
-  p(W_T, W_{T-1}, \ldots, W_1, W_0)    = & p(W_T | W_{t-1}, \ldots, W_0) p(W_{T-1} | W_{T-2}, \ldots, W_0) \cdots  \cr
-  & p(W_1 | W_0) p(W_0)
+\begin{align}
+  p(W_T, W_{T-1}, \ldots, W_1, W_0)    = & p(W_T | W_{T-1}, \ldots, W_0) p(W_{T-1} | W_{T-2}, \ldots, W_0) \cdots  \cr
+  & \quad \quad \cdots p(W_1 | W_0) p(W_0)
+\end{align}
 $$
 
 In general,
@@ -178,7 +180,7 @@ $G$ with probability $1 - \tilde \pi$.
 Thus, we  assume that the decision maker
 
 - **knows** both $F$ and $G$
-- **doesnt't know** which of these two distributions that nature has drawn
+- **doesn't know** which of these two distributions that nature has drawn
 - summarizing his ignorance by acting  as if or **thinking** that nature chose distribution $F$ with probability $\tilde \pi \in (0,1)$ and distribution
   $G$ with probability $1 - \tilde \pi$
 - at date $t \geq 0$ has observed  the partial history $w_t, w_{t-1}, \ldots, w_0$ of draws from the appropriate joint
@@ -480,9 +482,9 @@ learning_example()
 ```
 
 Please look at the three graphs above created for an instance in which $f$ is a uniform distribution on $[0,1]$
-(i.e., a Beta distribution with parameters $F_a=1, F_b=1$, while  $g$ is a Beta distribution with the default parameter values $G_a=3, G_b=1.2$.
+(i.e., a Beta distribution with parameters $F_a=1, F_b=1$), while  $g$ is a Beta distribution with the default parameter values $G_a=3, G_b=1.2$.
 
-The graph on the left  plots the likehood ratio $l(w)$ on the coordinate axis against $w$ on the ordinate axis.
+The graph on the left  plots the likelihood ratio $l(w)$ on the coordinate axis against $w$ on the ordinate axis.
 
 The middle graph plots both $f(w)$ and $g(w)$  against $w$, with the horizontal dotted lines showing values
 of $w$ at which the likelihood ratio equals $1$.
@@ -491,7 +493,7 @@ The graph on the right plots arrows to the right that show when Bayes' Law  make
 to the left that show when Bayes' Law make $\pi$ decrease.
 
 Notice how the length of the arrows, which show the magnitude of the force from Bayes' Law impelling $\pi$ to change,
-depend on both the prior probability $\pi$ on the ordinate axis and the evidence in the form of the current draw of
+depends on both the prior probability $\pi$ on the ordinate axis and the evidence in the form of the current draw of
 $w$ on the coordinate axis.
 
 The fractions in the colored areas of the middle graphs are probabilities under $F$ and $G$, respectively,
@@ -528,7 +530,7 @@ assumptions about nature's choice of distribution:
 - that nature permanently draws from $G$
 
 Outcomes depend on a peculiar property of likelihood ratio processes that are discussed in
-[this lecture](https://python-advanced.quantecon.org/additive_functionals.html)
+[this lecture](https://python-advanced.quantecon.org/additive_functionals.html).
 
 To do this, we create some Python code.
 

lectures/finite_markov.md

@@ -1218,22 +1218,23 @@ Let $F$ be the cumulative distribution function of the normal distribution $N(0,
 The values $P(x_i, x_j)$ are computed to approximate the AR(1) process --- omitting the derivation, the rules are as follows:
 
 1. If $j = 0$, then set
-
-$$
-P(x_i, x_j) = P(x_i, x_0) = F(x_0-\rho x_i + s/2)
-$$
-
+   
+   $$
+   P(x_i, x_j) = P(x_i, x_0) = F(x_0-\rho x_i + s/2)
+   $$
+   
 1. If $j = n-1$, then set
-
-$$
-P(x_i, x_j) = P(x_i, x_{n-1}) = 1 - F(x_{n-1} - \rho x_i - s/2)
-$$
-
+   
+   $$
+   P(x_i, x_j) = P(x_i, x_{n-1}) = 1 - F(x_{n-1} - \rho x_i - s/2)
+   $$
+   
 1. Otherwise, set
-
-$$
-P(x_i, x_j) = F(x_j - \rho x_i + s/2) - F(x_j - \rho x_i - s/2)
-$$
+   
+   $$
+   P(x_i, x_j) = F(x_j - \rho x_i + s/2) - F(x_j - \rho x_i - s/2)
+   $$
+   
 
 The exercise is to write a function `approx_markov(rho, sigma_u, m=3, n=7)` that returns
 $\{x_0, \ldots, x_{n-1}\} \subset \mathbb R$ and $n \times n$ matrix

lectures/harrison_kreps.md

@@ -175,7 +175,7 @@ Remember that state $1$ is the high dividend state.
 * In state $0$, a type $a$ agent is more optimistic about next period's dividend than a type $b$ agent.
 * In state $1$, a type $b$ agent is more optimistic about next period's dividend.
 
-However, the stationary distributions $\pi_A = \begin{bmatrix} .57 & .43 \end{bmatrix}$ and $\pi_B = \begin{bmatrix} .43 & .57 \end{bmatrix}$ tell us that a type $B$ person is more optimistic about the dividend process in the long run than is a type A person.
+However, the stationary distributions $\pi_A = \begin{bmatrix} .57 & .43 \end{bmatrix}$ and $\pi_B = \begin{bmatrix} .43 & .57 \end{bmatrix}$ tell us that a type $B$ person is more optimistic about the dividend process in the long run than is a type $A$ person.
 
 Transition matrices for the temporarily optimistic and pessimistic investors are constructed as follows.
 

lectures/lake_model.md

@@ -378,7 +378,7 @@ there exists an $\bar x$  such that
 
 This equation tells us that a steady state level $\bar x$ is an  eigenvector of $\hat A$ associated with a unit eigenvalue.
 
-We also have $x_t \to \bar x$ as $t \to \infty$ provided that the remaining eigenvalue of $\hat A$ has modulus less that 1.
+We also have $x_t \to \bar x$ as $t \to \infty$ provided that the remaining eigenvalue of $\hat A$ has modulus less than 1.
 
 This is the case for our default parameters:
 

lectures/likelihood_bayes.md

@@ -50,7 +50,7 @@ We'll study how, at least  in our setting, a Bayesian eventually learns the prob
 rests on the asymptotic behavior of likelihood ratio processes studied in {doc}`this lecture <likelihood_ratio_process>`.
 
 This lecture provides technical results that underly outcomes to be studied in {doc}`this lecture <odu>`
-and {doc}`this lecture <wald_friedman>` and {doc}`this lecture <navy_captain>`
+and {doc}`this lecture <wald_friedman>` and {doc}`this lecture <navy_captain>`.
 
 ## The Setting
 
@@ -262,7 +262,7 @@ and the initial prior $\pi_{0}$
 \pi_{t+1}=\frac{\pi_{0}L\left(w^{t+1}\right)}{\pi_{0}L\left(w^{t+1}\right)+1-\pi_{0}} .
 ```
 
-Formula {eq}`eq_Bayeslaw103` generalizes generalizes formula {eq}`eq_recur1`.
+Formula {eq}`eq_Bayeslaw103` generalizes formula {eq}`eq_recur1`.
 
 Formula {eq}`eq_Bayeslaw103`  can be regarded as a one step  revision of prior probability $\pi_0$ after seeing
 the batch of data $\left\{ w_{i}\right\} _{i=1}^{t+1}$.
@@ -276,8 +276,7 @@ limiting behavior of $\pi_t$.
 
 To illustrate this insight, below we will plot  graphs showing **one** simulated
 path of the  likelihood ratio process $L_t$ along with two paths of
-$\pi_t$ that are associated with the *same* realization of the likelihood ratio process but *different* initial prior probabilities
-probabilities $\pi_{0}$.
+$\pi_t$ that are associated with the *same* realization of the likelihood ratio process but *different* initial prior probabilities $\pi_{0}$.
 
 First, we tell Python two values of $\pi_0$.
 
@@ -374,5 +373,5 @@ $g$.
 This lecture has been devoted to building some useful infrastructure.
 
 We'll build on results highlighted in this lectures to understand inferences that are the foundations of
-results described  in {doc}`this lecture <odu>` and {doc}`this lecture <wald_friedman>` and {doc}`this lecture <navy_captain>`
+results described  in {doc}`this lecture <odu>` and {doc}`this lecture <wald_friedman>` and {doc}`this lecture <navy_captain>`.
 

lectures/lq_inventories.md

@@ -35,7 +35,8 @@ tags: [hide-output]
 
 ## Overview
 
-This lecture can be viewed as an application of the {doc}`quantecon lecture <lqcontrol>`.
+This lecture can be viewed as an application of this {doc}`quantecon lecture <lqcontrol>` about linear quadratic control
+theory.
 
 It formulates a discounted dynamic program for a firm that
 chooses a production schedule to balance
@@ -46,19 +47,19 @@ chooses a production schedule to balance
 In the tradition of a classic book by Holt, Modigliani, Muth, and
 Simon {cite}`Holt_Modigliani_Muth_Simon`, we simplify the
 firm’s problem by formulating it as a linear quadratic discounted
-dynamic programming problem of the type studied in this {doc}`quantecon <lqcontrol>`.
+dynamic programming problem of the type studied in this {doc}`quantecon lecture <lqcontrol>`.
 
 Because its costs of production are increasing and quadratic in
-production, the firm wants to smooth production across time provided
+production, the firm holds inventories as a buffer stock in order to smooth production across time, provided
 that holding inventories is not too costly.
 
-But the firm also prefers to sell out of existing inventories, a
+But the firm also wants to make its sales  out of existing inventories, a
 preference that we represent by a cost that is quadratic in the
 difference between sales in a period and the firm’s beginning of period
 inventories.
 
-We compute examples designed to indicate how the firm optimally chooses
-to smooth production and manage inventories while keeping inventories
+We compute examples designed to indicate how the firm optimally
+smooths production  while keeping inventories
 close to sales.
 
 To introduce components of the model, let
@@ -72,7 +73,7 @@ To introduce components of the model, let
 - $d(I_t, S_t) = d_1 I_t + d_2 (S_t - I_t)^2$, where
   $d_1>0, d_2 >0$, be a cost-of-holding-inventories function,
   consisting of two components:
-    - a cost $d_1 t$ of carrying inventories, and
+    - a cost $d_1 I_t$ of carrying inventories, and
     - a cost $d_2 (S_t - I_t)^2$ of having inventories deviate
       from sales
 - $p_t = a_0 - a_1 S_t + v_t$ be an inverse demand function for a
@@ -84,7 +85,7 @@ To introduce components of the model, let
   be the present value of the firm’s profits at
   time $0$
 - $I_{t+1} = I_t + Q_t - S_t$ be the law of motion of inventories
-- $z_{t+1} = A_{22} z_t + C_2 \epsilon_{t+1}$ be the law
+- $z_{t+1} = A_{22} z_t + C_2 \epsilon_{t+1}$ be a law
   of motion for an exogenous state vector $z_t$ that contains
   time $t$ information useful for predicting the demand shock
   $v_t$
@@ -133,17 +134,20 @@ appears in the firm’s one-period profit function)
 We can express the firm’s profit as a function of states and controls as
 
 $$
-\pi_t =  - (x_t' R x_t + u_t' Q u_t + 2 u_t' H x_t )
+\pi_t =  - (x_t' R x_t + u_t' Q u_t + 2 u_t' N x_t )
 $$
 
-To form the matrices $R, Q, H$, we note that the firm’s profits at
+To form the matrices $R, Q, N$ in an LQ dynamic programming problem, we note that the firm’s profits at
 time $t$ function can be expressed
 
 $$
+\begin{equation}
+\begin{split}
 \pi_{t} =&p_{t}S_{t}-c\left(Q_{t}\right)-d\left(I_{t},S_{t}\right)  \\
     =&\left(a_{0}-a_{1}S_{t}+v_{t}\right)S_{t}-c_{1}Q_{t}-c_{2}Q_{t}^{2}-d_{1}I_{t}-d_{2}\left(S_{t}-I_{t}\right)^{2}  \\
     =&a_{0}S_{t}-a_{1}S_{t}^{2}+Gz_{t}S_{t}-c_{1}Q_{t}-c_{2}Q_{t}^{2}-d_{1}I_{t}-d_{2}S_{t}^{2}-d_{2}I_{t}^{2}+2d_{2}S_{t}I_{t}  \\
-    =&-\left(\underset{x_{t}^{\prime}Rx_{t}}{\underbrace{d_{1}I_{t}+d_{2}I_{t}^{2}}}\underset{u_{t}^{\prime}Qu_{t}}{\underbrace{+a_{1}S_{t}^{2}+d_{2}S_{t}^{2}+c_{2}Q_{t}^{2}}}\underset{2u_{t}^{\prime}Hx_{t}}{\underbrace{-a_{0}S_{t}-Gz_{t}S_{t}+c_{1}Q_{t}-2d_{2}S_{t}I_{t}}}\right) \\
+    =&-\left(\underset{x_{t}^{\prime}Rx_{t}}{\underbrace{d_{1}I_{t}+d_{2}I_{t}^{2}}}\underset{u_{t}^{\prime}Qu_{t}}{\underbrace{+a_{1}S_{t}^{2}+d_{2}S_{t}^{2}+c_{2}Q_{t}^{2}}}
+    \underset{2u_{t}^{\prime}N x_{t}}{\underbrace{-a_{0}S_{t}-Gz_{t}S_{t}+c_{1}Q_{t}-2d_{2}S_{t}I_{t}}}\right) \\
     =&-\left(\left[\begin{array}{cc}
 I_{t} & z_{t}^{\prime}\end{array}\right]\underset{\equiv R}{\underbrace{\left[\begin{array}{cc}
 d_{2} & \frac{d_{1}}{2}S_{c}\\
@@ -166,12 +170,14 @@ Q_{t} & S_{t}\end{array}\right]\underset{\equiv N}{\underbrace{\left[\begin{arra
 I_{t}\\
 z_{t}
 \end{array}\right]\right)
+\end{split}
+\end{equation}
 $$
 
 where $S_{c}=\left[1,0\right]$.
 
 **Remark on notation:** The notation for cross product term in the
-QuantEcon library is $N$ instead of $H$.
+QuantEcon library is $N$.
 
 The firms’ optimum decision rule takes the form
 
@@ -185,6 +191,16 @@ $$
 x_{t+1} = (A - BF ) x_t + C \epsilon_{t+1}
 $$
 
+The firm chooses a decision rule for $u_t$ that maximizes
+
+$$
+E_0 \sum_{t=0}^\infty \beta^t \pi_t
+$$
+
+subject to a given $x_0$.
+
+This is a stochastic discounted LQ dynamic program.
+
 Here is code for computing an optimal decision rule and for analyzing
 its consequences.
 
@@ -330,7 +346,7 @@ class SmoothingExample:
 Notice that the above code sets parameters at the following default
 values
 
-- discount factor β=0.96,
+- discount factor $\beta=0.96$,
 - inverse demand function: $a0=10, a1=1$
 - cost of production $c1=1, c2=1$
 - costs of holding inventories $d1=1, d2=1$
@@ -465,7 +481,7 @@ We introduce this $I_t$ **is hardwired to zero** specification in
 order to shed light on the role that inventories play by comparing outcomes
 with those under our two other versions of the problem.
 
-The bottom right panel displays an production path for the original
+The bottom right panel displays a production path for the original
 problem that we are interested in (the blue line) as well with an
 optimal production path for the model in which inventories are not
 useful (the green path) and also for the model in which, although