Standardize notation: Use capitals for random variables in Job Search I-V

Updated notation in Job Search lectures I-V to use uppercase letters for
random variables and lowercase for realizations:

- Job Search I (mccall_model.md): w_t → W_t in wage offer sequences
- Job Search II (mccall_model_with_separation.md): w_t → W_t for wage offers
- Job Search III (mccall_model_with_sep_markov.md): (s_t, w_t) → (S_t, W_t)
  for state variables
- Job Search IV (mccall_fitted_vfi.md): Already using correct notation
- Job Search V (mccall_persist_trans.md): w_t → W_t, z_t → Z_t, y_t → Y_t
  for wage components

Lowercase notation remains for realizations in sums (e.g., sum_w v(w) q(w))
and for standard normal shocks (zeta_t, epsilon_t).

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <[email protected]>

Author: jstac

lectures/mccall_model.md

@@ -77,19 +77,19 @@ from quantecon.distributions import BetaBinomial
 ```{index} single: Models; McCall
 ```
 
-An unemployed agent receives in each period a job offer at wage $w_t$.
+An unemployed agent receives in each period a job offer at wage $W_t$.
 
 In this lecture, we adopt the following simple environment:
 
-* The offer sequence $\{w_t\}_{t \geq 0}$ is IID, with $q(w)$ being the probability of observing wage $w$ in finite set $\mathbb{W}$.
-* The agent observes $w_t$ at the start of $t$.
-* The agent knows that $\{w_t\}$ is IID with common distribution $q$ and can use this when computing expectations.
+* The offer sequence $\{W_t\}_{t \geq 0}$ is IID, with $q(w)$ being the probability of observing wage $w$ in finite set $\mathbb{W}$.
+* The agent observes $W_t$ at the start of $t$.
+* The agent knows that $\{W_t\}$ is IID with common distribution $q$ and can use this when computing expectations.
 
 (In later lectures, we will relax these assumptions.)
 
 At time $t$, our agent has two choices:
 
-1. Accept the offer and work permanently at constant wage $w_t$.
+1. Accept the offer and work permanently at constant wage $W_t$.
 1. Reject the offer, receive unemployment compensation $c$, and reconsider next period.
 
 The agent is infinitely lived and aims to maximize the expected discounted
@@ -107,7 +107,7 @@ The smaller is $\beta$, the more the agent discounts future earnings relative to
 
 The variable  $y_t$ is income, equal to
 
-* his/her wage $w_t$ when employed
+* his/her wage $W_t$ when employed
 * unemployment compensation $c$ when unemployed
 
 

lectures/mccall_model_with_sep_markov.md

@@ -406,8 +406,8 @@ This is implemented via `jnp.searchsorted` on the precomputed cumulative sum
 
 The function `update_agent` advances the agent's state by one period.
 
-The agent's state is a pair $(s_t, w_t)$, where $s_t$ is employment status (0 if
-unemployed, 1 if employed) and $w_t$ is 
+The agent's state is a pair $(S_t, W_t)$, where $S_t$ is employment status (0 if
+unemployed, 1 if employed) and $W_t$ is 
 
 * their current wage offer, if unemployed, or
 * their current wage, if employed. 
@@ -569,10 +569,10 @@ fraction of time an agent spends unemployed over a long time series.
 We will see that these two values are approximately equal -- if fact they are
 exactly equal in the limit.
 
-The reason is that the process $(s_t, w_t)$, where
+The reason is that the process $(S_t, W_t)$, where
 
-- $s_t$ is the employment status and
-- $w_t$ is the wage 
+- $S_t$ is the employment status and
+- $W_t$ is the wage 
 
 is Markovian, since the next pair depends only on the current pair and iid
 randomness, and ergodic. 
@@ -595,7 +595,7 @@ In particular, the fraction of time a single agent spends unemployed (across all
 wage states) converges to the cross-sectional unemployment rate:
 
 $$
-    \lim_{T \to \infty} \frac{1}{T} \sum_{t=1}^{T} \mathbb{1}\{s_t = \text{unemployed}\} = \sum_{w=1}^{n} \pi(\text{unemployed}, w)
+    \lim_{T \to \infty} \frac{1}{T} \sum_{t=1}^{T} \mathbb{1}\{S_t = \text{unemployed}\} = \sum_{w=1}^{n} \pi(\text{unemployed}, w)
 $$
 
 This holds regardless of initial conditions -- provided that we burn in the

lectures/mccall_model_with_separation.md

@@ -90,7 +90,7 @@ introducing a utility function $u$.
 
 It satisfies $u'> 0$ and $u'' < 0$.
 
-Wage offers $\{ w_t \}$ are IID with common distribution $q$.
+Wage offers $\{ W_t \}$ are IID with common distribution $q$.
 
 The set of possible wage values is denoted by $\mathbb W$.
 
@@ -106,9 +106,9 @@ If currently employed at wage $w$, the worker
 1. receives utility $u(w)$ from their current wage and
 1. is fired with some (small) probability $\alpha$, becoming unemployed next period.
 
-If currently unemployed, the worker receives random wage offer $w_t$ and either accepts or rejects.
+If currently unemployed, the worker receives random wage offer $W_t$ and either accepts or rejects.
 
-If he accepts, then he begins work immediately at wage $w_t$.
+If he accepts, then he begins work immediately at wage $W_t$.
 
 If he rejects, then he receives unemployment compensation $c$.
 

lectures/mccall_persist_trans.md

@@ -68,20 +68,20 @@ from typing import NamedTuple
 Wages at each point in time are given by
 
 $$
-w_t = \exp(z_t) + y_t
+W_t = \exp(Z_t) + Y_t
 $$
 
 where
 
 $$
-y_t \sim \exp(\mu + s \zeta_t)
+Y_t \sim \exp(\mu + s \zeta_t)
 \quad \text{and} \quad
-z_{t+1} = d + \rho z_t + \sigma \epsilon_{t+1}
+Z_{t+1} = d + \rho Z_t + \sigma \epsilon_{t+1}
 $$
 
 Here $\{ \zeta_t \}$ and $\{ \epsilon_t \}$ are both IID and standard normal.
 
-Here $\{y_t\}$ is a transitory component and $\{z_t\}$ is persistent.
+Here $\{Y_t\}$ is a transitory component and $\{Z_t\}$ is persistent.
 
 As before, the worker can either
 
@@ -327,7 +327,7 @@ at all state values.
 
 Next, we study how mean unemployment duration varies with unemployment compensation.
 
-For simplicity, we'll fix the initial state at $z_t = 0$.
+For simplicity, we'll fix the initial state at $Z_0 = 0$.
 
 ```{code-cell} ipython
 @jax.jit