Improve McCall model lecture: Fix beta unpacking and enhance readability (#694)

Made several improvements to the mccall_model.md lecture to enhance code clarity and pedagogical value:

**Key changes:**
1. Fixed beta unpacking in Comparative statics section - now unpacks model parameters before use
2. Improved derivation of nonlinear equation in h under "Take 2" - added step-by-step algebraic transformations
3. Enhanced JAX solution in Exercise 1 - extracted wage drawing logic into documented draw_wage() function
4. Broke vectorization into clear steps - separated vmap calls with explanatory comments
5. Simplified Exercise 2 solution - removed redundant keyword arguments and meshgrid, unpacked model directly
6. Made while_loop pattern clearer - explicitly created final_state before unpacking

All changes tested and verified to run correctly.

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

Co-authored-by: Claude <noreply@anthropic.com>

Author: jstac

lectures/mccall_model.md

@@ -498,27 +498,29 @@ colors = prop_cycle.by_key()['color']
 
 # Plot the wage offer distribution
 ax.plot(w, q, '-', alpha=0.6, lw=2,
-        label='wage offer distribution', 
+        label='wage offer distribution',
         color=colors[0])
 
 # Compute reservation wage with default beta
 model_default = McCallModel()
-v_init = model_default.w / (1 - model_default.β)
+c, β, w, q = model_default
+v_init = w / (1 - β)
 v_default, res_wage_default = compute_reservation_wage(
     model_default, v_init
 )
 
 # Compute reservation wage with lower beta
 β_new = 0.96
 model_low_beta = McCallModel(β=β_new)
-v_init_low = model_low_beta.w / (1 - model_low_beta.β)
+c, β_low, w, q = model_low_beta
+v_init_low = w / (1 - β_low)
 v_low, res_wage_low = compute_reservation_wage(
     model_low_beta, v_init_low
 )
 
 # Plot vertical lines for reservation wages
 ax.axvline(x=res_wage_default, color=colors[1], lw=2,
-           label=f'reservation wage (β={model_default.β})')
+           label=f'reservation wage (β={β})')
 ax.axvline(x=res_wage_low, color=colors[2], lw=2,
            label=f'reservation wage (β={β_new})')
 
@@ -621,12 +623,43 @@ Let $h$ denote the continuation value:
 
 The Bellman equation can now be written as
 
-$$
+```{math}
+:label: j1b
+
     v^*(w')
     = \max \left\{ \frac{w'}{1 - \beta}, \, h \right\}
+```
+
+Now let's derive a nonlinear equation for $h$ alone.
+
+Starting from {eq}`j1b`, we multiply both sides by $q(w')$ to get
+
+$$
+    v^*(w') q(w') = \max \left\{ \frac{w'}{1 - \beta}, h \right\} q(w')
+$$
+
+Next, we sum both sides over $w' \in \mathbb{W}$:
+
+$$
+    \sum_{w' \in \mathbb W} v^*(w') q(w')
+    = \sum_{w' \in \mathbb W} \max \left\{ \frac{w'}{1 - \beta}, h \right\} q(w')
+$$
+
+Now multiply both sides by $\beta$:
+
+$$
+    \beta \sum_{w' \in \mathbb W} v^*(w') q(w')
+    = \beta \sum_{w' \in \mathbb W} \max \left\{ \frac{w'}{1 - \beta}, h \right\} q(w')
 $$
 
-Substituting this last equation into {eq}`j1` gives
+Add $c$ to both sides:
+
+$$
+    c + \beta \sum_{w' \in \mathbb W} v^*(w') q(w')
+    = c + \beta \sum_{w' \in \mathbb W} \max \left\{ \frac{w'}{1 - \beta}, h \right\} q(w')
+$$
+
+Finally, using the definition of $h$ from {eq}`j1`, the left-hand side is just $h$, giving us
 
 ```{math}
 :label: j2
@@ -638,7 +671,7 @@ Substituting this last equation into {eq}`j1` gives
         \right\}  q (w')
 ```
 
-This is a nonlinear equation that we can solve for $h$.
+This is a nonlinear equation in the single scalar $h$ that we can solve for $h$.
 
 As before, we will use successive approximations:
 
@@ -781,8 +814,28 @@ plt.show()
 And here's a solution using JAX.
 
 ```{code-cell} ipython3
+# First, we set up a function to draw random wage offers from the distribution.
+# We use the inverse transform method: draw a uniform random variable u,
+# then find the smallest wage w such that the CDF at w is >= u.
 cdf = jnp.cumsum(q_default)
 
+def draw_wage(uniform_rv):
+    """
+    Draw a wage from the distribution q_default using the inverse transform method.
+
+    Parameters:
+    -----------
+    uniform_rv : float
+        A uniform random variable on [0, 1]
+
+    Returns:
+    --------
+    wage : float
+        A wage drawn from w_default with probabilities given by q_default
+    """
+    return w_default[jnp.searchsorted(cdf, uniform_rv)]
+
+
 def compute_stopping_time(w_bar, key):
     """
     Compute stopping time by drawing wages until one exceeds `w_bar`.
@@ -791,7 +844,7 @@ def compute_stopping_time(w_bar, key):
         t, key, accept = loop_state
         key, subkey = jax.random.split(key)
         u = jax.random.uniform(subkey)
-        w = w_default[jnp.searchsorted(cdf, u)]
+        w = draw_wage(u)
         accept = w >= w_bar
         t = t + 1
         return t, key, accept
@@ -831,7 +884,8 @@ def compute_stop_time_for_c(c):
     return compute_mean_stopping_time(w_bar)
 
 # Vectorize across all c values
-stop_times = jax.vmap(compute_stop_time_for_c)(c_vals)
+compute_stop_time_vectorized = jax.vmap(compute_stop_time_for_c)
+stop_times = compute_stop_time_vectorized(c_vals)
 
 fig, ax = plt.subplots()
 
@@ -928,12 +982,12 @@ def create_mccall_continuous(
     key = jax.random.PRNGKey(seed)
     s = jax.random.normal(key, (mc_size,))
     w_draws = jnp.exp(μ + σ * s)
-    return McCallModelContinuous(c=c, β=β, σ=σ, μ=μ, w_draws=w_draws)
+    return McCallModelContinuous(c, β, σ, μ, w_draws)
 
 
 @jax.jit
 def compute_reservation_wage_continuous(model, max_iter=500, tol=1e-5):
-    c, β, σ, μ, w_draws = model.c, model.β, model.σ, model.μ, model.w_draws
+    c, β, σ, μ, w_draws = model
     
     h = jnp.mean(w_draws) / (1 - β)  # initial guess
     
@@ -949,8 +1003,9 @@ def compute_reservation_wage_continuous(model, max_iter=500, tol=1e-5):
         return jnp.logical_and(i < max_iter, error > tol)
     
     initial_state = (h, 0, tol + 1)
-    h_final, _, _ = jax.lax.while_loop(cond, update, initial_state)
-    
+    final_state = jax.lax.while_loop(cond, update, initial_state)
+    h_final, _, _ = final_state
+
     # Now compute the reservation wage
     return (1 - β) * h_final
 ```
@@ -969,12 +1024,13 @@ def compute_R_element(c, β):
     model = create_mccall_continuous(c=c, β=β)
     return compute_reservation_wage_continuous(model)
 
-# Create meshgrid and vectorize computation
-c_grid, β_grid = jnp.meshgrid(c_vals, β_vals, indexing='ij')
-compute_R_vectorized = jax.vmap(
-    jax.vmap(compute_R_element, 
-    in_axes=(None, 0)), 
-    in_axes=(0, None))
+# First, vectorize over β (holding c fixed)
+compute_R_over_β = jax.vmap(compute_R_element, in_axes=(None, 0))
+
+# Next, vectorize over c (applying the above function to each c)
+compute_R_vectorized = jax.vmap(compute_R_over_β, in_axes=(0, None))
+
+# Apply to compute the full grid
 R = compute_R_vectorized(c_vals, β_vals)
 ```