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

```{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")

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()

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}

```