Heterogeneous-Agent Macroeconomics and Incomplete Markets

import numpy as np
b, s, r, rho, sig, n = 0.96, 2.0, 0.03, 0.9, 0.2, 7
p = (1 + rho) / 2; P = np.array([[p, 1 - p], [1 - p, p]])
for m_ in range(3, n + 1):
    Pn = np.zeros((m_, m_))
    Pn[:-1, :-1] += p * P; Pn[:-1, 1:] += (1 - p) * P; Pn[1:, :-1] += (1 - p) * P; Pn[1:, 1:] += p * P
    Pn[1:-1] /= 2; P = Pn
psi = sig / np.sqrt(1 - rho**2) * np.sqrt(n - 1)
z = np.exp(np.linspace(-psi, psi, n)); pi = np.ones(n) / n
for _ in range(2000): pi = pi @ P
z = z / (pi @ z)                                          # normalize mean income to 1
a = 50 * np.linspace(0, 1, 500)**2                         # asset grid, dense near the constraint
up = lambda c: c**(-s); upinv = lambda u: u**(-1 / s)
C = (1 + r) * a[:, None] + z[None, :] - a[0]               # initial guess: consume income plus a little
for it in range(5000):
    Emu = (up(C) @ P.T) * b * (1 + r)                      # E[u'(c')] for each (a', z)
    c_endo = upinv(Emu); a_endo = (c_endo + a[:, None] - z[None, :]) / (1 + r)
    Cn = np.empty_like(C)
    for j in range(n):
        Cn[:, j] = np.interp(a, a_endo[:, j], c_endo[:, j])
        bind = a < a_endo[0, j]; Cn[bind, j] = (1 + r) * a[bind] + z[j] - a[0]     # constraint binds
    if np.max(np.abs(Cn - C)) < 1e-9: break
    C = Cn
ap = (1 + r) * a[:, None] + z[None, :] - C
D = np.ones((len(a), n)) / (len(a) * n)
idx = np.clip(np.searchsorted(a, ap) - 1, 0, len(a) - 2); w = np.clip((a[idx + 1] - ap) / (a[idx + 1] - a[idx]), 0, 1)
for _ in range(5000):
    Dn = np.zeros_like(D)
    for j in range(n):
        np.add.at(Dn[:, j], idx[:, j], w[:, j] * D[:, j]); np.add.at(Dn[:, j], idx[:, j] + 1, (1 - w[:, j]) * D[:, j])
    Dn = Dn @ P
    if np.max(np.abs(Dn - D)) < 1e-12: break
    D = Dn
wealth = D.sum(1); order = np.argsort(a); cw = np.cumsum(wealth[order]); cwa = np.cumsum(wealth[order] * a[order])
gini = 1 - np.sum((cwa[1:] + cwa[:-1]) * np.diff(cw)) / cwa[-1] - cwa[0] * cw[0] / cwa[-1]
mpc = (C[1:] - C[:-1]) / ((1 + r) * (a[1:] - a[:-1]))[:, None]
avg_mpc = np.sum(D[:-1] * mpc) / D[:-1].sum()
print(f"EGM converged in {it} iterations; mean assets {np.sum(D.sum(1) * a):.2f} (mean income 1)")
print(f"share at the borrowing constraint: {D[0].sum():.3f};  wealth Gini: {gini:.3f};  average annual MPC: {avg_mpc:.3f}")
np.save("/tmp/aiyagari_D.npy", D); np.save("/tmp/aiyagari_mpc.npy", mpc); np.save("/tmp/aiyagari_a.npy", a)

Output:

EGM converged in 375 iterations; mean assets 7.65 (mean income 1)
share at the borrowing constraint: 0.016;  wealth Gini: 0.487;  average annual MPC: 0.077

EGM converges in a few hundred iterations without any root-finding: inverting the Euler equation on a grid of next-period assets gives consumption directly. The stationary distribution has a wealth Gini of about 0.49 — far below the U.S. value of about 0.8 — and only 1.6% of households at the borrowing constraint. The average annual MPC is 0.08, close to the permanent-income benchmark. One-asset models with these parameters generate too few constrained households and too little wealth inequality; HANK models fix this with two assets (Kaplan, Moll and Violante, 2018) or with a lower discount factor.

import numpy as np, pandas as pd, wooldridge as wd
d = wd.data("401ksubs")
d["htm"] = d["nettfa"] < d["inc"] / 24
x = np.sort(d["nettfa"].to_numpy()); n = len(x); i = np.arange(1, n + 1)
gini = np.sum((2 * i - n - 1) * x) / (n * x.sum())
print(f"share with nettfa <= 0: {np.mean(d['nettfa'] <= 0):.3f};  hand-to-mouth (nettfa < half a month of income): {d['htm'].mean():.3f}")
print(f"Gini of financial wealth: {gini:.3f};  top 10% share: {x[int(0.9*n):].sum()/x.sum():.3f};  median ${np.median(x):.1f}k vs mean ${x.mean():.1f}k")
d["quintile"] = pd.qcut(d["inc"], 5, labels=["Q1 (lowest)", "Q2", "Q3", "Q4", "Q5 (highest)"])
print(d.groupby("quintile", observed=True).agg(income=("inc", "mean"), htm_share=("htm", "mean"), median_nettfa=("nettfa", "median")).round(3).to_string())

Output:

share with nettfa <= 0: 0.350;  hand-to-mouth (nettfa < half a month of income): 0.464
Gini of financial wealth: 0.982;  top 10% share: 0.722;  median $2.0k vs mean $19.1k
              income  htm_share  median_nettfa
quintile                                      
Q1 (lowest)   15.060      0.688          0.000
Q2            23.942      0.590          0.299
Q3            33.410      0.476          1.849
Q4            46.261      0.353          6.375
Q5 (highest)  77.607      0.212         26.299

In these 1991 data, 35% of households have non-positive financial assets, and 46% hold less than half a month of income in financial assets — somewhat above Kaplan, Violante and Weidner's (2014) estimate of about one third, which uses a broader liquid-wealth measure and later survey years. Financial wealth is extremely concentrated (Gini 0.98, top 10% share 72%; the mean is ten times the median). The hand-to-mouth share falls with income but remains above 20% even in the top income quintile. Those are the "wealthy hand-to-mouth" — households with illiquid wealth (housing, pensions) but little cash — which is why transfers raise spending even among middle-income households.