Add labor income volatility analysis to IFP advanced lecture (#749)

Added a new exercise (Exercise 2) that analyzes how wealth inequality varies
with labor income volatility (a_y), complementing the existing analysis of
return volatility (a_r).

Key findings:
- Varying return volatility (a_r: 0.10 to 0.16) increases Gini from 0.19 to 0.79
- Varying labor income volatility (a_y: 0.125 to 0.20) only increases Gini from 0.18 to 0.19
- Demonstrates that capital income risk is a much more powerful driver of wealth inequality than labor income risk

Also cleaned up the wealth distribution plotting code and removed diagnostic output.

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

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

Author: jstac

lectures/ifp_advanced.md

@@ -671,31 +671,24 @@ c_init = a_init
 a_vec, c_vec = solve_model_time_iter(ifp, a_init, c_init)
 assets = compute_asset_stationary(c_vec, a_vec, ifp, num_households=200_000)
 
-# Diagnostic: Check extrapolation issues
-print(f"\n=== Grid and Asset Diagnostics ===")
-print(f"Grid max (s_grid[-1]): {ifp.s_grid[-1]:.2f}")
-print(f"Endogenous grid max (a_vec.max()): {a_vec.max():.2f}")
-print(f"Simulated assets max: {assets.max():.2f}")
-print(f"Simulated assets mean: {assets.mean():.2f}")
-print(f"Simulated assets median: {np.median(assets):.2f}")
-print(f"Fraction of households beyond grid: {(assets > a_vec.max()).mean():.4f}")
-print(f"Fraction beyond 0.9 * grid_max: {(assets > 0.9 * a_vec.max()).mean():.4f}")
-print()
-
 # Compute Gini coefficient for the plot
 gini_plot = gini_coefficient(assets)
 
-# Plot: Histogram with log-scale y-axis
+# Plot histogram of log wealth
 fig, ax = plt.subplots(figsize=(10, 6))
-ax.hist(assets, bins=40, alpha=0.5, density=True)
-ax.set_yscale('log')
-ax.set(xlabel='assets', ylabel='density (log scale)',
-       title="Wealth Distribution")
+ax.hist(jnp.log(assets), bins=40, alpha=0.5, density=True)
+ax.set(xlabel='log assets', ylabel='density', title="Wealth Distribution")
 plt.tight_layout()
 plt.show()
 ```
 
-The histogram shows the wealth distribution with the y-axis on a log scale, allowing us to see both the mass of households at low wealth levels and the long right tail of the distribution.
+The histogram shows the distribution of log wealth. 
+
+Bearing in mind that we are looking at log values, the histogram suggests 
+a long right tail of the distribution.
+
+Below we examine this in more detail.
+
 
 
 ## Wealth Inequality
@@ -751,7 +744,7 @@ We loop over different values of `a_r`, solve the model for each, simulate the w
 
 ```{code-cell} ipython3
 # Range of a_r values to explore
-a_r_vals = np.linspace(0.10, 0.16, 7)
+a_r_vals = np.linspace(0.10, 0.16, 5)
 gini_vals = []
 
 print("Computing Gini coefficients for different return volatilities...\n")
@@ -787,7 +780,7 @@ ax.plot(a_r_vals, gini_vals, 'o-', linewidth=2, markersize=8)
 ax.set(xlabel='Return volatility (a_r)',
        ylabel='Gini coefficient',
        title='Wealth Inequality vs Return Volatility')
-ax.axhline(y=0.8, color='r', linestyle='--', linewidth=1,
+ax.axhline(y=0.8, color='k', linestyle='--', linewidth=1,
            label='Empirical US Gini (~0.8)')
 ax.legend()
 plt.tight_layout()
@@ -803,5 +796,90 @@ high returns accumulate substantially more wealth than unlucky households,
 leading to greater inequality in the wealth distribution.
 
 
+```{solution-end}
+```
+
+```{exercise}
+:label: ifp_advanced_ex2
+
+Plot how the Gini coefficient varies with the volatility of labor income.
+
+Specifically, compute the Gini coefficient for values of `a_y` ranging from
+0.125 to 0.20 and plot the results. Set `a_r=0.10` for this exercise.
+
+What does this tell you about the relationship between labor income risk and
+wealth inequality? Can we achieve the same rise in inequality by varying labor
+income volatility as we can by varying return volatility?
+
+```
+
+```{solution-start} ifp_advanced_ex2
+:class: dropdown
+```
+
+We loop over different values of `a_y`, solve the model for each, simulate the wealth distribution, and compute the Gini coefficient.
+
+```{code-cell} ipython3
+# Range of a_y values to explore
+a_y_vals = np.linspace(0.125, 0.20, 5)
+gini_vals_y = []
+
+print("Computing Gini coefficients for different labor income volatilities...\n")
+
+for a_y in a_y_vals:
+    print(f"a_y = {a_y:.3f}...", end=" ")
+
+    # Create model with this a_y value and a_r=0.10
+    ifp_temp = create_ifp(a_y=a_y, a_r=0.10, grid_max=100)
+
+    # Solve the model
+    s_grid_temp = ifp_temp.s_grid
+    n_z_temp = len(ifp_temp.P)
+    a_init_temp = s_grid_temp[:, None] * jnp.ones(n_z_temp)
+    c_init_temp = a_init_temp
+    a_vec_temp, c_vec_temp = solve_model_time_iter(
+        ifp_temp, a_init_temp, c_init_temp, verbose=False
+    )
+
+    # Simulate households
+    assets_temp = compute_asset_stationary(
+        c_vec_temp, a_vec_temp, ifp_temp, num_households=200_000
+    )
+
+    # Compute Gini coefficient
+    gini_temp = gini_coefficient(assets_temp)
+    gini_vals_y.append(gini_temp)
+    print(f"Gini = {gini_temp:.4f}")
+
+# Plot the results
+fig, ax = plt.subplots(figsize=(10, 6))
+ax.plot(a_y_vals, gini_vals_y, 'o-', linewidth=2, markersize=8, color='green')
+ax.set(xlabel='Labor income volatility (a_y)',
+       ylabel='Gini coefficient',
+       title='Wealth Inequality vs Labor Income Volatility')
+ax.axhline(y=0.8, color='k', linestyle='--', linewidth=1,
+           label='Empirical US Gini (~0.8)')
+ax.legend()
+plt.tight_layout()
+plt.show()
+```
+
+The plot shows that wealth inequality increases with labor income volatility, but the effect is much weaker than the effect of return volatility.
+
+Comparing the two exercises:
+
+- When return volatility (`a_r`) varies from 0.10 to 0.16, the Gini coefficient rises dramatically from around 0.20 to 0.79
+- When labor income volatility (`a_y`) varies from 0.125 to 0.20, a similar amount in percentage terms, the Gini coefficient increases but by a much smaller amount
+
+This suggests that capital income risk is a more important driver of wealth inequality than labor income risk.
+
+The intuition is that wealth accumulation compounds over time: households who
+experience favorable returns on their assets can reinvest those returns, leading
+to exponential growth.
+
+In contrast, labor income shocks, while they affect current consumption and
+savings, do not have the same compounding effect on wealth accumulation.
+
+
 ```{solution-end}
 ```