Improve key handling and fix parameter consistency in McCall lectures (#715)

* Improve key handling and fix parameter consistency in McCall lectures

- Refactor random key handling to use fold_in instead of key threading
  - More idiomatic JAX pattern for indexed loops
  - Removes key from loop state for cleaner code
  - Deterministic randomness based on time step

- Fix missing n_agents variable in _simulate_cross_section_compiled
  - Extract from initial_wage_indices using len()

- Standardize separation rate across lectures
  - Set α = 0.05 in mccall_fitted_vfi to match mccall_model_with_sep_markov
  - All economic parameters now consistent between lectures

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

Co-Authored-By: Claude <noreply@anthropic.com>

* fix minor typos and section title capitalization

---------

Co-authored-by: Claude <noreply@anthropic.com>
Co-authored-by: Humphrey Yang <u6474961@anu.edu.au>

Author: 3 people

lectures/mccall_fitted_vfi.md

@@ -268,7 +268,7 @@ class Model(NamedTuple):
 
 def create_mccall_model(
         c: float = 1.0,
-        α: float = 0.1,
+        α: float = 0.05,
         β: float = 0.96,
         ρ: float = 0.9,
         ν: float = 0.2,
@@ -361,14 +361,17 @@ def vfi(
 ```
 
 Here's a function that uses a solution $v_u$ to compute the remaining functions of
-interest: $v_u$, and the continuation value function $h$.
+interest: $v_e$, and the continuation value function $h$.
 
 We use the same expressions as we did in the {doc}`discrete case <mccall_model_with_sep_markov>`, after replacing sums with integrals.
 
 ```{code-cell} ipython3
 def compute_solution_functions(model, v_u):
 
-    # Interpolate v_u 
+    # Unpack model parameters
+    c, α, β, ρ, ν, γ, w_grid, z_draws = model
+
+    # Interpolate v_u on the wage grid
     vf = lambda x: jnp.interp(x, w_grid, v_u)
 
     def compute_expectation(w):
@@ -604,7 +607,7 @@ When unemployed, the agent accepts offers that exceed the reservation wage.
 
 When employed, the agent faces job separation with probability $\alpha$ each period.
 
-### Cross-Sectional Analysis
+### Cross-sectional analysis
 
 Now let's simulate many agents simultaneously to examine the cross-sectional unemployment rate.
 
@@ -633,29 +636,29 @@ def _simulate_cross_section_compiled(
     c, α, β, ρ, ν, γ, w_grid, z_draws = model
 
     # Initialize arrays
-    key, subkey = jax.random.split(key)
+    init_key, subkey = jax.random.split(key)
     wages = jnp.exp(jax.random.normal(subkey, (n_agents,)) * ν)
     status = jnp.zeros(n_agents, dtype=jnp.int32)
 
     def update(t, loop_state):
-        key, status, wages = loop_state
+        status, wages = loop_state
 
         # Shift loop state forwards
-        key, subkey = jax.random.split(key)
-        agent_keys = jax.random.split(subkey, n_agents)
+        step_key = jax.random.fold_in(init_key, t)
+        agent_keys = jax.random.split(step_key, n_agents)
 
         status, wages = update_agents_vmap(
             agent_keys, status, wages, model, w_bar
         )
 
-        return key, status, wages
+        return status, wages
 
     # Run simulation using fori_loop
-    initial_loop_state = (key, status, wages)
+    initial_loop_state = (status, wages)
     final_loop_state = lax.fori_loop(0, T, update, initial_loop_state)
 
     # Return only final employment state
-    _, final_is_employed, _ = final_loop_state
+    final_is_employed, _ = final_loop_state
     return final_is_employed
 
 

lectures/mccall_model_with_sep_markov.md

@@ -49,7 +49,7 @@ libraries
 ```{code-cell} ipython3
 :tags: [hide-output]
 
-!pip install quantecon
+!pip install quantecon jax
 ```
 
 We use the following imports:
@@ -64,7 +64,7 @@ import matplotlib.pyplot as plt
 from functools import partial
 ```
 
-## Model Setup
+## Model setup
 
 The setting is as follows:
 
@@ -74,7 +74,7 @@ The setting is as follows:
 - Unemployed workers receive compensation $c$ per period
 - Future payoffs are discounted by factor $\beta \in (0,1)$
 
-### Decision Problem
+### Decision problem
 
 When unemployed and receiving wage offer $w$, the agent chooses between:
 
@@ -86,7 +86,7 @@ The wage updates are as follows:
 * If an unemployed agent rejects offer $w$, then their next offer is drawn from $P(w, \cdot)$
 * If an employed agent loses a job in which they were paid wage $w$, then their next offer is drawn from $P(w, \cdot)$
 
-### The Wage Offer Process
+### The wage offer process
 
 To construct the wage offer process we start with an AR1 process.
 
@@ -112,7 +112,7 @@ Actually, in practice, we approximate this wage process as follows:
 
 
 
-### Value Functions
+### Value functions
 
 We let
 
@@ -168,12 +168,12 @@ $$
 
 +++
 
-### Optimal Policy
+### Optimal policy
 
 Once we have the solutions $v_e$ and $v_u$ to these Bellman equations, we can compute the optimal policy: accept at current wage offer $w$ if 
 
 $$
-    v_e(w) ≥ u(c) + β(Pv_u)(w)
+    v_e(w) \geq u(c) + \beta (P v_u)(w)
 $$
 
 The optimal policy turns out to be a reservation wage strategy: accept all wages above some threshold.
@@ -185,7 +185,7 @@ The optimal policy turns out to be a reservation wage strategy: accept all wages
 
 Let's now implement the model.
 
-### Set Up
+### Set up
 
 The default utility function is a CRRA utility function
 
@@ -234,7 +234,7 @@ def create_js_with_sep_model(
 ```
 
 
-### Solution: First Pass
+### Solution: first pass
 
 Let's put together a (not very efficient) routine for calculating the
 reservation wage.
@@ -244,7 +244,7 @@ reservation wage.
 It works by starting with guesses for $v_e$ and $v_u$ and iterating to
 convergence.
 
-Here's are Bellman operators that update $v_u$ and $v_e$ respectively.
+Here are Bellman operators that update $v_u$ and $v_e$ respectively.
 
 
 ```{code-cell} ipython3
@@ -313,7 +313,7 @@ def solve_model_first_pass(
 ```
 
 
-### Road Test
+### Road test
 
 Let's solve the model:
 
@@ -348,9 +348,9 @@ The reservation wage is at the intersection of $v_e$, and the continuation value
 function, which is the value of rejecting.
 
 
-## Improving Efficiency
+## Improving efficiency
 
-The solution method desribed above works fine but we can do much better.
+The solution method described above works fine but we can do much better.
 
 First, we use the employed worker's Bellman equation to express
 $v_e$ in terms of $Pv_u$
@@ -495,7 +495,7 @@ The result is the same as before but we only iterate on one array --- and also
 our JAX code is more efficient.
 
 
-## Sensitivity Analysis
+## Sensitivity analysis
 
 Let's examine how reservation wages change with the separation rate.
 
@@ -523,7 +523,7 @@ Can you provide an intuitive economic story behind the outcome that you see in t
 
 +++
 
-## Employment Simulation
+## Employment simulation
 
 Now let's simulate the employment dynamics of a single agent under the optimal policy.
 
@@ -691,15 +691,15 @@ often leads a high new draw.
 
 +++
 
-## The Ergodic Property
+## Ergodic property
 
 Below we examine cross-sectional unemployment.
 
 In particular, we will look at the unemployment rate in a cross-sectional
 simulation and compare it to the time-average unemployment rate, which is the
 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
+We will see that these two values are approximately equal -- in fact they are
 exactly equal in the limit.
 
 The reason is that the process $(S_t, W_t)$, where
@@ -744,15 +744,15 @@ Often the second approach is better for our purposes, since it's easier to paral
 
 +++
 
-## Cross-Sectional Analysis
+## Cross-sectional analysis
 
 Now let's simulate many agents simultaneously to examine the cross-sectional unemployment rate.
 
 We first create a vectorized version of `update_agent` to efficiently update all agents in parallel:
 
 ```{code-cell} ipython3
-# Create vectorized version of update_agent
-# The last parameter is now w_bar (scalar) instead of σ (array)
+# Create vectorized version of update_agent.
+# Vectorize over key, status, wage_idx
 update_agents_vmap = jax.vmap(
     update_agent, in_axes=(0, 0, 0, None, None)
 )
@@ -761,76 +761,72 @@ update_agents_vmap = jax.vmap(
 Next we define the core simulation function, which uses `lax.fori_loop` to efficiently iterate many agents forward in time:
 
 ```{code-cell} ipython3
-@partial(jax.jit, static_argnums=(3, 4))
+@jax.jit
 def _simulate_cross_section_compiled(
         key: jnp.ndarray,
         model: Model,
         w_bar: float,
-        n_agents: int,
+        initial_wage_indices: jnp.ndarray,
+        initial_status_vec: jnp.ndarray,
         T: int
     ):
-    """JIT-compiled core simulation loop using lax.fori_loop.
-    Returns only the final employment state to save memory."""
+    """
+    JIT-compiled core simulation loop for shifting the cross section
+    using lax.fori_loop. Returns the final employment employment status
+    cross-section.
+
+    """
     n, w_vals, P, P_cumsum, β, c, α, γ = model
+    n_agents = len(initial_wage_indices)
 
-    # Initialize arrays
-    wage_indices = jnp.zeros(n_agents, dtype=jnp.int32)
-    status = jnp.zeros(n_agents, dtype=jnp.int32)
 
     def update(t, loop_state):
-        key, status, wage_indices = loop_state
-
-        # Shift loop state forwards
-        key, subkey = jax.random.split(key)
-        agent_keys = jax.random.split(subkey, n_agents)
-
+        " Shift loop state forwards "
+        status, wage_indices = loop_state
+        step_key = jax.random.fold_in(key, t)
+        agent_keys = jax.random.split(step_key, n_agents)
         status, wage_indices = update_agents_vmap(
             agent_keys, status, wage_indices, model, w_bar
         )
-
-        return key, status, wage_indices
+        return status, wage_indices
 
     # Run simulation using fori_loop
-    initial_loop_state = (key, status, wage_indices)
+    initial_loop_state = (initial_status_vec, initial_wage_indices)
     final_loop_state = lax.fori_loop(0, T, update, initial_loop_state)
 
     # Return only final employment state
-    _, final_is_employed, _ = final_loop_state
+    final_is_employed, _ = final_loop_state
     return final_is_employed
 
 
 def simulate_cross_section(
-        model: Model,
-        n_agents: int = 100_000,
-        T: int = 200,
-        seed: int = 42
+        model: Model,               # Model instance with parameters
+        n_agents: int = 100_000,    # Number of agents to simulate
+        T: int = 200,               # Length of burn-in
+        seed: int = 42              # For reproducibility
     ) -> float:
     """
-    Simulate employment paths for many agents and return final unemployment rate.
+    Wrapper function for _simulate_cross_section_compiled.
 
-    Parameters:
-    - model: Model instance with parameters
-    - n_agents: Number of agents to simulate
-    - T: Number of periods to simulate
-    - seed: Random seed for reproducibility
+    Push forward a cross-section for T periods and return the final
+    cross-sectional unemployment rate.
 
-    Returns:
-    - unemployment_rate: Fraction of agents unemployed at time T
     """
     key = jax.random.PRNGKey(seed)
 
     # Solve for optimal reservation wage
     v_u = vfi(model)
     w_bar = get_reservation_wage(v_u, model)
 
-    # Run JIT-compiled simulation
+    # Initialize arrays
+    initial_wage_indices = jnp.zeros(n_agents, dtype=jnp.int32)
+    initial_status_vec = jnp.zeros(n_agents, dtype=jnp.int32)
+
     final_status = _simulate_cross_section_compiled(
-        key, model, w_bar, n_agents, T
+        key, model, w_bar, initial_wage_indices, initial_status_vec, T
     )
 
-    # Calculate unemployment rate at final period
     unemployment_rate = 1 - jnp.mean(final_status)
-
     return unemployment_rate
 ```
 
@@ -850,8 +846,13 @@ def plot_cross_sectional_unemployment(
     key = jax.random.PRNGKey(42)
     v_u = vfi(model)
     w_bar = get_reservation_wage(v_u, model)
+
+    # Initialize arrays
+    initial_wage_indices = jnp.zeros(n_agents, dtype=jnp.int32)
+    initial_status_vec = jnp.zeros(n_agents, dtype=jnp.int32)
+
     final_status = _simulate_cross_section_compiled(
-        key, model, w_bar, n_agents, t_snapshot
+        key, model, w_bar, initial_wage_indices, initial_status_vec, t_snapshot
     )
 
     # Calculate unemployment rate
@@ -906,7 +907,7 @@ Now let's visualize the cross-sectional distribution:
 plot_cross_sectional_unemployment(model)
 ```
 
-## Lower Unemployment Compensation (c=0.5)
+## Lower unemployment compensation (c=0.5)
 
 What happens to the cross-sectional unemployment rate with lower unemployment compensation?