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[gorman_heterogeneous_households] Update parameters to match the parameters used in paper #303

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81 changes: 57 additions & 24 deletions lectures/gorman_heterogeneous_households.md
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Original file line number Diff line number Diff line change
Expand Up @@ -1203,7 +1203,7 @@ def compute_household_paths(
# Original endowment claim (before trading)
d_j[j] = (U_dj @ z_path)[0, :]

# Dividend income under limited markets: μ_j × aggregate endowment
# Dividend income under limited markets: μ_j aggregate endowment
d_share[j] = (μ[j] * d[0]).squeeze()

# Capital share in mutual fund
Expand All @@ -1217,7 +1217,7 @@ def compute_household_paths(

a_total[j] = k_share[j] + k_hat[j]

# Validate that Gorman weights sum to 1 (required for sharing rule consistency)
# Validate that Gorman weights sum to 1
μ_sum = np.sum(μ)
if abs(μ_sum - 1.0) > 1e-6:
import warnings
Expand Down Expand Up @@ -1569,7 +1569,8 @@ def build_gorman_extended(
n_absorb = max(1, n // 10)
n_absorb = int(n_absorb)
if n_absorb < 1 or n_absorb >= n:
raise ValueError(f"n_absorb must be in [1, n-1], got {n_absorb} with n={n}")
raise ValueError(
f"n_absorb must be in [1, n-1], got {n_absorb} with n={n}")

# Dimensions
n_idio = n - n_absorb # eta_{n_absorb+1}, ..., eta_n
Expand Down Expand Up @@ -1634,12 +1635,17 @@ def build_gorman_extended(
Ud = sum(Ud_per_house)

# Ub_per_house: bliss loading
# b_{jt} = b_bar + xi_{j,t}
# b_{jt} = b_bars[j] + xi_{j,t}
# b_bar can be scalar (same for all) or array (household-specific)
b_bars = np.atleast_1d(b_bar)
if b_bars.size == 1:
b_bars = np.full(n, float(b_bars[0]))

Ub_per_house = []
for j in range(n):
row = np.zeros((1, nz))
row[0, 0] = b_bar # constant bliss
row[0, 3 + n_idio + j] = 1.0 # loading on xi_j
row[0, 0] = b_bars[j] # household-specific bliss
row[0, 3 + n_idio + j] = 1.0 # loading on xi_j (zeroed if gammas=0)
Ub_per_house.append(row)

Ub = sum(Ub_per_house)
Expand All @@ -1649,6 +1655,8 @@ def build_gorman_extended(
x0[0, 0] = 1.0 # constant = 1

return A22, C2, Ub, Ud, Ub_per_house, Ud_per_house, x0


```

### 100-household economy
Expand Down Expand Up @@ -1704,24 +1712,27 @@ n_absorb = 50
# Poorer households face larger, more persistent idiosyncratic shocks
σ_idio_min, σ_idio_max = 0.2, 5.0
σs = σ_idio_min + (σ_idio_max - σ_idio_min) * (poorness ** 2.0)
ρ_idio_min, ρ_idio_max = 0.0, 0.98
ρ_idio_min, ρ_idio_max = 0.0, 0.94
ρ_idio = ρ_idio_min + (ρ_idio_max - ρ_idio_min) * (poorness[n_absorb:] ** 1.0)

# Preference shocks are muted
b_bar = 5.0
# Bliss points scale with endowment so high-α households want more consumption
b_bar_base = 5.0
b_bars = b_bar_base * (αs / αs.mean()) # household-specific bliss points
γs_pref = np.zeros(N)
ρ_pref = 0.0

A22, C2, Ub, Ud, Ub_list, Ud_list, x0 = build_gorman_extended(
n=N,
rho1=ρ1, rho2=ρ2, sigma_a=σ_a,
alphas=αs, phis=φs, sigmas=σs,
b_bar=b_bar, gammas=γs_pref,
b_bar=b_bars, gammas=γs_pref,
rho_idio=ρ_idio, rho_pref=ρ_pref,
n_absorb=n_absorb,
)

print(f"sum(φs) = {np.sum(φs):.6f} (should be 1.0)")

```

```{code-cell} ipython3
Expand Down Expand Up @@ -2069,14 +2080,22 @@ Once we have the redistributed Pareto weights, we can compute the new household
The function below computes household consumption and income under given Pareto weights

```{code-cell} ipython3
def allocation_from_weights(paths, econ, U_b_list, weights, γ_1, Λ, h0i=None):
def allocation_from_weights(paths, econ,
U_b_list, weights, γ_1, Λ, h0i=None,
original_weights=None):
"""
Compute household consumption and income under given Pareto weights.
"""

weights = np.asarray(weights).reshape(-1)
N = len(weights)

# For χ computation, use original weights if provided
if original_weights is None:
weights_for_chi = weights
else:
weights_for_chi = np.asarray(original_weights).reshape(-1)

# Extract aggregate paths
x_path = paths["x_path"]
c_agg = paths["c"]
Expand Down Expand Up @@ -2111,7 +2130,8 @@ def allocation_from_weights(paths, econ, U_b_list, weights, γ_1, Λ, h0i=None):
for j in range(N):
U_bj = np.asarray(U_b_list[j], dtype=float)
b_agg = econ.Sb @ x_path
b_tilde = U_bj @ z_path - weights[j] * b_agg
# Use weights_for_chi for b_tilde: keeps χ fixed during redistribution
b_tilde = U_bj @ z_path - weights_for_chi[j] * b_agg

η = np.zeros((n_h, T + 1))
η[:, 0] = np.asarray(h0i).reshape(-1)
Expand Down Expand Up @@ -2163,10 +2183,14 @@ idx_sorted = np.argsort(-μ_values)

h0i_alloc = np.array([[0.0]])

pre = allocation_from_weights(paths, econ,
pre = allocation_from_weights(paths, econ,
Ub_list, μ_values, γ_1, Λ, h0i_alloc)
post = allocation_from_weights(paths, econ,
Ub_list, λ_star, γ_1, Λ, h0i_alloc)

# For post-redistribution: use λ_star for proportional share, but keep χ fixed
# that is to say preferences don't change with taxes
post = allocation_from_weights(paths, econ,
Ub_list, λ_star, γ_1, Λ, h0i_alloc,
original_weights=μ_values)

c_pre = pre["c"][:, t0:]
y_pre = pre["y_net"][:, t0:]
Expand All @@ -2184,31 +2208,40 @@ y_post_pct = _pct(y_post[:, :T_ts])
c_pre_pct = _pct(c_pre[:, :T_ts])
c_post_pct = _pct(c_post[:, :T_ts])

fig, ax = plt.subplots(1, 3, figsize=(14, 4), sharey=True)
fig, ax = plt.subplots(1, 3, figsize=(14, 6), sharey=True)

ax[0].plot(t, y_pre_pct[0], label="p90", lw=2)
ax[0].plot(t, y_pre_pct[1], label="p50", lw=2)
ax[0].plot(t, y_pre_pct[2], label="p10", lw=2)
ax[0].set_xlabel("time")
ax[0].set_ylabel(r"pre-tax income $y^{pre}$")
ax[0].legend()
ax[0].set_xlabel("$t$", fontsize=20)
ax[0].set_ylabel(r"$y^{pre}$", fontsize=20)

ax[1].plot(t, y_post_pct[0], label="p90", lw=2)
ax[1].plot(t, y_post_pct[1], label="p50", lw=2)
ax[1].plot(t, y_post_pct[2], label="p10", lw=2)
ax[1].set_xlabel("time")
ax[1].set_ylabel(r"post-tax income $y^{post}$")
ax[1].legend()
ax[1].set_xlabel("$t$")
ax[1].set_ylabel(r"$y^{post}$", fontsize=20)

ax[2].plot(t, c_post_pct[0], label="p90", lw=2)
ax[2].plot(t, c_post_pct[1], label="p50", lw=2)
ax[2].plot(t, c_post_pct[2], label="p10", lw=2)
ax[2].set_xlabel("time")
ax[2].set_ylabel(r"consumption $c^{post}$")
ax[2].legend()
ax[2].set_xlabel("$t$")
ax[2].set_ylabel(r"$c^{post}$", fontsize=20)

handles, labels = ax[0].get_legend_handles_labels()

fig.legend(
handles,
labels,
loc="upper center",
ncol=3,
frameon=False,
bbox_to_anchor=(0.5, 1.1),
)

plt.tight_layout()
plt.show()

```

The figures reveal a striking reduction in income and consumption inequality after redistribution.
Expand Down
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