Improve code explanations in ifp_advanced lecture

- Add bridging text connecting mathematical equations to code implementation
- Add detailed code walkthrough for Coleman-Reffett operator
- Add explanation of solver function and convergence
- Add economic interpretation of default parameters
- Expand interpretation of consumption policy results
- Fix grammatical errors (comma splice, missing period)
- Rename variables for clarity: a_in→ae_vals, σ_in→c_vals, a_out→ae_out, σ_out→c_out

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

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

Author: jstac

lectures/ifp_advanced.md

@@ -127,7 +127,7 @@ does not grow too quickly.
 
 When $\{R_t\}$ was constant we required that $\beta R < 1$.
 
-Now it is stochastic, we require that
+Since it is now stochastic, we require that
 
 ```{math}
 :label: fpbc2
@@ -139,7 +139,7 @@ G_R := \lim_{n \to \infty}
 ```
 
 Notice that, when $\{R_t\}$ takes some constant value $R$, this
-reduces to the previous restriction $\beta R < 1$
+reduces to the previous restriction $\beta R < 1$.
 
 The value $G_R$ can be thought of as the long run (geometric) average
 gross rate of return.
@@ -414,16 +414,26 @@ class IFP:
 
 Here's the Coleman-Reffett operator based on EGM:
 
+### Implementation Details
+
+The implementation of operator $K$ maps directly to equation {eq}`k_opr`.
+
+The left side $u'(\xi)$ becomes `u_prime_inv(β * Ez)` after solving for $\xi$.
+
+The expectation term $\mathbb E_z \hat{R} (u' \circ \sigma)[\hat{R}(a - \xi) + \hat{Y}, \hat{Z}]$ is computed via Monte Carlo averaging over future states and shocks.
+
+The max with $u'(a)$ is handled implicitly—the endogenous grid method naturally handles the liquidity constraint since we only solve for interior consumption where $c < a$.
+
 ```{code-cell} ipython
 @jit
-def K(a_in, σ_in, ifp):
+def K(ae_vals, c_vals, ifp):
     """
     The Coleman--Reffett operator for the income fluctuation problem,
     using the endogenous grid method.
 
         * ifp is an instance of IFP
-        * a_in[i, z] is an asset grid
-        * σ_in[i, z] is consumption at a_in[i, z]
+        * ae_vals[i, z] is an asset grid
+        * c_vals[i, z] is consumption at ae_vals[i, z]
     """
 
     # Simplify names
@@ -433,12 +443,12 @@ def K(a_in, σ_in, ifp):
     n = len(P)
 
     # Create consumption function by linear interpolation
-    σ = lambda a, z: np.interp(a, a_in[:, z], σ_in[:, z])
+    σ = lambda a, z: np.interp(a, ae_vals[:, z], c_vals[:, z])
 
     # Allocate memory
-    σ_out = np.empty_like(σ_in)
+    c_out = np.empty_like(c_vals)
 
-    # Obtain c_i at each s_i, z, store in σ_out[i, z], computing
+    # Obtain c_i at each s_i, z, store in c_out[i, z], computing
     # the expectation term by Monte Carlo
     for i, s in enumerate(s_grid):
         for z in range(n):
@@ -452,20 +462,28 @@ def K(a_in, σ_in, ifp):
                         U = u_prime(σ(R_hat * s + Y_hat, z_hat))
                         Ez += R_hat * U * P[z, z_hat]
             Ez = Ez / (len(η_draws) * len(ζ_draws))
-            σ_out[i, z] =  u_prime_inv(β * Ez)
+            c_out[i, z] =  u_prime_inv(β * Ez)
 
     # Calculate endogenous asset grid
-    a_out = np.empty_like(σ_out)
+    ae_out = np.empty_like(c_out)
     for z in range(n):
-        a_out[:, z] = s_grid + σ_out[:, z]
+        ae_out[:, z] = s_grid + c_out[:, z]
 
     # Fixing a consumption-asset pair at (0, 0) improves interpolation
-    σ_out[0, :] = 0
-    a_out[0, :] = 0
+    c_out[0, :] = 0
+    ae_out[0, :] = 0
 
-    return a_out, σ_out
+    return ae_out, c_out
 ```
 
+### Code Walkthrough
+
+The operator creates a consumption function `σ` by interpolating the input policy, then uses triple nested loops to compute the expectation via Monte Carlo averaging over savings grid points, current states, future states, and shock realizations.
+
+After computing optimal consumption $c_i$ at each savings level $s_i$ by inverting marginal utility, we construct the endogenous asset grid using $a_i = s_i + c_i$.
+
+Setting consumption and assets to zero at the origin ensures smooth interpolation near zero assets, where the household consumes everything.
+
 The next function solves for an approximation of the optimal consumption policy via time iteration.
 
 ```{code-cell} ipython
@@ -497,12 +515,24 @@ def solve_model_time_iter(model,        # Class with model information
     return a_new, σ_new
 ```
 
+This function implements fixed-point iteration by repeatedly applying the operator $K$ until the policy converges.
+
+Convergence is measured by the maximum absolute change in consumption across all states.
+
+The operator is guaranteed to converge due to the contraction property discussed earlier.
+
 Now we are ready to create an instance at the default parameters.
 
 ```{code-cell} ipython
 ifp = IFP()
 ```
 
+The default parameters represent a calibration with moderate risk aversion ($\gamma = 1.5$, CRRA utility) and a quarterly discount factor ($\beta = 0.96$, corresponding to roughly 4% annual discounting).
+
+The Markov chain has high persistence (90% probability of staying in the current state), while returns have 10% volatility around a zero mean log return ($a_r = 0.1$, $b_r = 0.0$).
+
+Labor income is state-dependent: $Y_t = \exp(0.2 \eta_t + 0.5 Z_t)$ implies higher expected income in the good state ($Z_t = 1$) compared to the bad state ($Z_t = 0$).
+
 Next we set up an initial condition, which corresponds to consuming all
 assets.
 
@@ -537,8 +567,11 @@ Notice that we consume all assets in the lower range of the asset space.
 
 This is because we anticipate income $Y_{t+1}$ tomorrow, which makes the need to save less urgent.
 
-Can you explain why consuming all assets ends earlier (for lower values of
-assets) when $z=0$?
+Observe that consuming all assets ends earlier (at lower asset levels) when $z=0$ compared to $z=1$.
+
+This occurs because expected future income is lower in the bad state ($z=0$), so the household begins precautionary saving at lower wealth levels.
+
+In contrast, when $z=1$ (good state), higher expected future income allows the household to consume all assets up to a higher threshold before savings become optimal.
 
 ### Law of Motion
 
@@ -684,14 +717,14 @@ Here's the Coleman-Reffett operator using JAX:
 
 ```{code-cell} ipython
 @jax_jit
-def K_jax(a_in, σ_in, ifp):
+def K_jax(ae_vals, c_vals, ifp):
     """
     The Coleman--Reffett operator for the income fluctuation problem,
     using the endogenous grid method with JAX.
 
         * ifp is an instance of IFP_JAX
-        * a_in[i, z] is an asset grid
-        * σ_in[i, z] is consumption at a_in[i, z]
+        * ae_vals[i, z] is an asset grid
+        * c_vals[i, z] is consumption at ae_vals[i, z]
     """
 
     # Extract parameters from ifp
@@ -701,9 +734,9 @@ def K_jax(a_in, σ_in, ifp):
     n = len(P)
 
     # Allocate memory
-    σ_out = jnp.empty_like(σ_in)
+    c_out = jnp.empty_like(c_vals)
 
-    # Obtain c_i at each s_i, z, store in σ_out[i, z], computing
+    # Obtain c_i at each s_i, z, store in c_out[i, z], computing
     # the expectation term by Monte Carlo
     def compute_expectation(s, z):
         """Compute expectation for given s and z"""
@@ -714,7 +747,7 @@ def K_jax(a_in, σ_in, ifp):
                 Y_hat = Y(z_hat, η, a_y, b_y)
                 a_val = R_hat * s + Y_hat
                 # Interpolate consumption
-                c_interp = jnp.interp(a_val, a_in[:, z_hat], σ_in[:, z_hat])
+                c_interp = jnp.interp(a_val, ae_vals[:, z_hat], c_vals[:, z_hat])
                 U = u_prime(c_interp, γ)
                 return R_hat * U
 
@@ -728,17 +761,17 @@ def K_jax(a_in, σ_in, ifp):
         return u_prime_inv(β * Ez, γ)
 
     # Vectorize over s_grid and z
-    σ_out = vmap(vmap(compute_expectation, in_axes=(None, 0)),
+    c_out = vmap(vmap(compute_expectation, in_axes=(None, 0)),
                   in_axes=(0, None))(s_grid, jnp.arange(n))
 
     # Calculate endogenous asset grid
-    a_out = s_grid[:, None] + σ_out
+    ae_out = s_grid[:, None] + c_out
 
     # Fixing a consumption-asset pair at (0, 0) improves interpolation
-    σ_out = σ_out.at[0, :].set(0)
-    a_out = a_out.at[0, :].set(0)
+    c_out = c_out.at[0, :].set(0)
+    ae_out = ae_out.at[0, :].set(0)
 
-    return a_out, σ_out
+    return ae_out, c_out
 ```
 
 The next function solves for an approximation of the optimal consumption policy via time iteration using JAX: