Tom's July 4 edits of two lectures

Author: thomassargent30

lectures/career.md

@@ -275,7 +275,7 @@ def operator_factory(cw, parallel_flag=True):
 ```
 
 Lastly, `solve_model` will  take an instance of `CareerWorkerProblem` and
-iterate using the Bellman operator to find the fixed point of the value function.
+iterate using the Bellman operator to find the fixed point of the Bellman equation.
 
 ```{code-cell} python3
 def solve_model(cw,

lectures/odu.md

@@ -97,7 +97,12 @@ At time $t$, our worker either
 The wage sequence ${W_t}$ is IID and generated from known density $q$.
 
 The worker aims to maximize the expected discounted sum of earnings
-$\mathbb{E} \sum_{t=0}^{\infty}\beta^t y_t$ The function $V$ satisfies the recursion
+$\mathbb{E} \sum_{t=0}^{\infty}\beta^t y_t$.
+
+Let $v(w)$ be the optimal value of the problem for a previously unemployed worker who has just received offer $w$ and is
+yet to decide whether to accept or reject the offer.
+
+The value  function $v$ satisfies the recursion
 
 ```{math}
 :label: odu_odu_pv
@@ -122,16 +127,34 @@ The model is as above, apart from the fact that
 - the worker learns about $q$ by starting with a prior and updating
   based on wage offers that he/she observes
 
-The worker knows there are two possible distributions $F$ and $G$ —
-with densities $f$ and $g$.
+The worker knows there are two possible distributions $F$ and $G$.
+
+These two distributions have  densities $f$ and $g$, repectively.
+
+Just before time starts,  “nature” selects $q$ to be either $f$ or $g$.
 
-At the start of time, “nature” selects $q$ to be either $f$ or $g$
-— the wage distribution from which the entire sequence ${W_t}$ will be
+This is then the wage distribution from which the entire sequence ${W_t}$ will be
 drawn.
 
-This choice is not observed by the worker, who puts prior probability $\pi_0$ on $f$ being chosen.
+The worker does not know which distribution nature has drawn, but the worker does know
+the two possible distributions $f$ and $g$.
+
+The worker  puts a (subjective) prior probability $\pi_0$ on $f$ having been  chosen.
+
+The worker's time $0$ subjective distribution for the distribution of $W_0$ is
+
+$$
+\pi_0 f + (1 - \pi_0) g
+$$
+
+
+
+The  worker's time $t$ subjective belief about the  the distribution of $W_t$  is 
+
+$$
+\pi_t f + (1 - \pi_t) g, 
+$$
 
-Update rule: worker's time $t$ estimate of the distribution is $\pi_t f + (1 - \pi_t) g$,
 where $\pi_t$ updates via
 
 ```{math}
@@ -228,8 +251,8 @@ fact, it should be decreasing in $\pi$ because
 - larger $\pi$ means more weight on $f$ and less on
   $g$
 
-Thus larger $\pi$ depresses the worker’s assessment of
-her future prospects, and relatively low current offers become more
+Thus, s larger $\pi$ depresses the worker’s assessment of
+her future prospects, so relatively low current offers become more
 attractive.
 
 **Summary:** We conjecture that the optimal policy is of the form