Chapter 33: Income Inequality and Macroeconomic Dynamics

Pareto Laws, Gini Coefficients, and the URE Framework

“The Gini coefficient is a number between 0 and 1. Understanding where it comes from, what drives it, and how policy affects it — that requires a model.”

Cross-reference: Principles Ch. 38 (inequality, Gini coefficient, Piketty’s $r > g$ , distributional consequences of policy); Ch. 25 (HANK, heterogeneous MPC, redistribution channel) [P:Ch.38, P:Ch.25]

33.1 Why Inequality Matters for Macroeconomics¶

Chapter 32 built the tools for solving heterogeneous-agent models. This chapter uses them to analyze inequality — not as a welfare concern in isolation, but as a determinant of aggregate dynamics. Three mechanisms connect inequality to macroeconomic fluctuations:

1. Heterogeneous MPC. Poor households have high MPCs (near 1); wealthy households have low MPCs (near 0). A redistribution from rich to poor raises aggregate consumption. The size of the fiscal multiplier depends on the wealth distribution.

2. Distributional channel of monetary policy. A rate cut benefits net debtors (who spend more) and hurts net savers (who spend less). The aggregate effect depends on the distribution of net financial wealth — the subject of Auclert’s (2019) URE framework developed in Section 33.4.

3. Inequality and savings. Higher inequality concentrates wealth in households with lower MPCs, reducing aggregate demand — a potential explanation for secular stagnation [P:Ch.39.4].

33.2 Pareto Distributions and Power Laws¶

Definition 33.1 (Pareto Distribution). A random variable $X$ has a Pareto distribution with scale $x_m > 0$ and shape $\zeta > 0$ if:

P(X > x) = \left(\frac{x_m}{x}\right)^\zeta \quad \text{for } x \geq x_m.

(1)

The PDF: $f(x) = \zeta x_m^\zeta / x^{\zeta+1}$ . The mean: $\mathbb{E}[X] = \zeta x_m/(\zeta-1)$ (finite iff $\zeta > 1$ ). The variance: finite iff $\zeta > 2$ .

Power law tails in wealth data: Empirically, the upper tail of the wealth distribution follows a Pareto law with $\zeta \approx 1.5$ for the U.S. The top 1% share $s_1$ and the Pareto exponent $\zeta$ are related by $s_1 \approx 0.01^{1-1/\zeta}$ .

Theorem 33.1 (Gini Coefficient for a Pareto Distribution). For $X \sim \text{Pareto}(x_m, \zeta)$ with $\zeta > 1$ :

\text{Gini} = \frac{1}{2\zeta - 1}.

(2)

Proof. The Lorenz curve $L(p)$ gives the share of total income held by the bottom fraction $p$ . For Pareto: $L(p) = 1 - (1-p)^{1-1/\zeta}$ . The Gini coefficient:

G = 1 - 2\int_0^1 L(p)\,dp = 1 - 2\int_0^1[1-(1-p)^{1-1/\zeta}]\,dp.

(3)

Setting $u = 1-p$ : $\int_0^1(1-p)^{1-1/\zeta}dp = \int_0^1 u^{1-1/\zeta}du = 1/(2-1/\zeta) = \zeta/(2\zeta-1)$ .

So $G = 1 - 2[1 - \zeta/(2\zeta-1)] = 1 - 2(\zeta-1)/(2\zeta-1) = 1/(2\zeta-1)$ . $\square$

Applications: For U.S. wealth ( $\zeta \approx 1.5$ ): $G \approx 1/(3-1) = 0.5$ for the Pareto tail alone. The overall Gini (including the non-Pareto middle of the distribution) is higher, approximately 0.87.

33.3 The Gini Coefficient: General Formula¶

For an arbitrary distribution (not necessarily Pareto), the Gini coefficient is computed from the Lorenz curve.

Definition 33.2 (Lorenz Curve). The Lorenz curve $L(p)$ maps the cumulative population share $p$ to the cumulative wealth (or income) share held by the bottom $p$ fraction:

L(p) = \frac{\int_0^p F^{-1}(q)\,dq}{\int_0^1 F^{-1}(q)\,dq} = \frac{\int_0^{F^{-1}(p)}x\,dF(x)}{\mathbb{E}[X]},

(4)

where $F^{-1}(p)$ is the $p$ -th quantile of the wealth distribution.

Definition 33.3 (Gini Coefficient). The Gini coefficient is twice the area between the 45-degree line and the Lorenz curve:

G = 1 - 2\int_0^1 L(p)\,dp = \frac{2\text{Cov}(X, F(X))}{\mathbb{E}[X]}.

(5)

Theorem 33.2 (Gini as Covariance). For a positive random variable $X$ with CDF $F$ :

G = \frac{2}{\mathbb{E}[X]}\text{Cov}(X, F(X)).

(6)

Proof. $\text{Cov}(X, F(X)) = \mathbb{E}[XF(X)] - \mathbb{E}[X]\mathbb{E}[F(X)]$ . Since $\mathbb{E}[F(X)] = 1/2$ (by the probability integral transform), $\text{Cov}(X,F(X)) = \mathbb{E}[XF(X)] - \mathbb{E}[X]/2$ . Rewriting the Gini: $G = (1/\mathbb{E}[X])\mathbb{E}[X(2F(X)-1)] = (2/\mathbb{E}[X])(\mathbb{E}[XF(X)] - \mathbb{E}[X]/2) = (2/\mathbb{E}[X])\text{Cov}(X,F(X))$ . $\square$

Sample formula: For wealth observations $a_1 \leq a_2 \leq \cdots \leq a_n$ (sorted):

G = \frac{2}{n\bar{a}}\sum_{i=1}^n i\cdot a_i - \frac{n+1}{n}.

(7)

In APL: G ← (2÷n×a_bar) × +/ (⍳n) × a_sorted) - (n+1)÷n

33.4 The Auclert (2019) URE Framework¶

Cross-reference: Principles Ch. 25.3 (HANK redistribution channel) [P:Ch.25.3]

Auclert (2019) provides a powerful framework for decomposing the aggregate consumption response to a monetary policy shock into components that depend on the wealth distribution. The key concept is Unhedged Interest Rate Exposure (URE).

Definition 33.4 (Unhedged Interest Rate Exposure). Each household’s URE is:

\text{URE}_i = \underbrace{c_i - y_i}_{\text{net borrower/saver}} + \underbrace{a_i^{maturing} - l_i^{maturing}}_{\text{maturing assets minus liabilities}},

(8)

where $a_i^{maturing}$ and $l_i^{maturing}$ are assets and liabilities maturing in the current period (that will be rolled over at the new interest rate).

Intuition: A household with positive URE (net saver) benefits from a rate increase; a household with negative URE (net borrower) is hurt. A rate cut redistributes purchasing power from high-URE to low-URE households.

Theorem 33.3 (Auclert URE Sufficient Statistic). The aggregate consumption response to a unit change in the real interest rate $dr$ is:

d\bar{C} = \underbrace{-\frac{1}{\sigma}\bar{C}dr}_{\text{intertemporal substitution}} + \underbrace{\text{Cov}\!\left(\text{MPC}_i,\;\text{URE}_i\right)\cdot dr}_{\text{redistribution channel}},

(9)

where $\sigma$ is the EIS, $\bar{C}$ is aggregate consumption, and $\text{MPC}_i$ is household $i$ ’s marginal propensity to consume.

Proof sketch. Each household’s consumption responds to a rate change through: (1) the Euler equation effect ( $-\dot{c}/c = (r-\rho)/\sigma$ , so $dc_i/dr \propto -c_i/\sigma$ ); and (2) the income effect from interest rate changes on assets and liabilities at maturity ( $dc_i/dr \propto \text{MPC}_i \cdot \text{URE}_i$ ). Aggregating: $d\bar{C} = -\bar{C}/\sigma \cdot dr + \mathbb{E}[\text{MPC}_i\cdot\text{URE}_i] \cdot dr$ . Since $\mathbb{E}[\text{MPC}_i]=\bar{\text{MPC}}$ and $\mathbb{E}[\text{URE}_i]=0$ (economy-wide, interest payments are transfers between agents), $\mathbb{E}[\text{MPC}_i\cdot\text{URE}_i] = \text{Cov}(\text{MPC}_i, \text{URE}_i)$ . $\square$

Key insight: The redistribution channel $\text{Cov}(\text{MPC},\text{URE})$ is positive if poor households (high MPC) are net borrowers (negative URE) and wealthy households (low MPC) are net savers (positive URE). In this case, a rate cut redistributes to high-MPC households, amplifying the aggregate consumption response beyond what the representative-agent Euler equation would predict.

Calibration of the redistribution channel: For the U.S.:

$\text{Cov}(\text{MPC}, \text{URE}) \approx -0.07$ (Auclert 2019 estimate).
Average MPC $\approx 0.25$ .
So the redistribution channel contributes $-0.07 \cdot dr$ to aggregate consumption.
For a 1pp rate cut ( $dr = -0.01$ ): redistribution channel contributes $+0.07\%$ of aggregate consumption — substantial relative to the total effect.

33.5 Decomposing the Gini by Income Source¶

For understanding the contribution of different income sources (labor, capital, transfers) to total inequality, the Gini decomposition by source is useful.

Theorem 33.4 (Gini Decomposition). If total income $Y_i = Y_i^L + Y_i^K + Y_i^T$ (labor + capital + transfers), the Gini of total income:

G_Y = \sum_k \frac{\mathbb{E}[Y_i^k]}{\mathbb{E}[Y_i]} \cdot G_{Y^k} \cdot \rho_k,

(10)

where $G_{Y^k}$ is the concentration coefficient of source $k$ and $\rho_k$ is the rank correlation between source $k$ and total income.

33.6 Worked Example: Gini and URE in a Calibrated HANK Model¶

Cross-reference: Principles Ch. 25.3, Ch. 38 [P:Ch.25.3, P:Ch.38]

Python¶

import numpy as np
from scipy.interpolate import interp1d

# Using the Aiyagari wealth distribution from Chapter 32
# Here: simplified version with known distribution
np.random.seed(42)
N = 50000

# Generate a wealth distribution with Pareto upper tail
# Mix: 60% near zero (hand-to-mouth), 40% Pareto
frac_htm = 0.45  # hand-to-mouth fraction
n_htm = int(frac_htm * N)
n_pareto = N - n_htm

wealth_htm = np.random.exponential(0.5, n_htm)  # near-zero wealth
zeta_pareto = 1.5  # Pareto exponent
wealth_pareto = np.random.pareto(zeta_pareto, n_pareto) * 10.0 + 5.0

wealth = np.concatenate([wealth_htm, wealth_pareto])
wealth = np.sort(wealth)

# Gini coefficient
n = len(wealth); mean_w = np.mean(wealth)
G_sample = (2/(n*mean_w)) * np.sum((np.arange(1, n+1))*wealth) - (n+1)/n
print(f"Wealth Gini: {G_sample:.3f}  (data: ~0.87)")

# Pareto tail fit
threshold = np.percentile(wealth, 90)  # fit Pareto to top 10%
tail = wealth[wealth > threshold]
# MLE for Pareto: ζ_hat = n / sum(log(x/x_m))
zeta_hat = len(tail) / np.sum(np.log(tail/threshold))
print(f"Pareto exponent (top 10%): ζ = {zeta_hat:.3f}")
print(f"Implied Gini for Pareto tail: {1/(2*zeta_hat-1):.3f}")

# Lorenz curve
cumshare_pop = np.linspace(0, 1, n)
cumshare_wealth = np.cumsum(wealth) / np.sum(wealth)

# Top shares
for pct in [1, 5, 10, 50]:
    share = np.sum(wealth[int((1-pct/100)*n):]) / np.sum(wealth)
    print(f"Top {pct}% wealth share: {share*100:.1f}%")

# URE computation
# Assign MPC inversely proportional to wealth (hand-to-mouth MPC=1, wealthy MPC→0)
mpc = np.maximum(0.05, 1 - 0.9*cumshare_wealth)  # declining in wealth rank

# URE: net borrowers (bottom 60%) negative; net savers (top 40%) positive
ure_median = 0.0
ure = (cumshare_pop - 0.6) * 2  # roughly: bottom 60% negative URE

# Auclert redistribution channel
cov_mpc_ure = np.cov(mpc, ure)[0,1]
sigma_eis = 1.0; C_bar = np.mean(wealth) * 0.1  # consumption ≈ 10% of wealth
dr = -0.01  # 100bp rate cut
dC_euler = -C_bar/sigma_eis * dr
dC_redist = cov_mpc_ure * dr
print(f"\nResponse to 100bp rate cut:")
print(f"  Intertemporal substitution: {dC_euler*100:.3f}% of C")
print(f"  Redistribution (URE) channel: {dC_redist*100:.3f}% of C")
print(f"  Total dC: {(dC_euler+dC_redist)*100:.3f}% of C")
print(f"  Cov(MPC, URE): {cov_mpc_ure:.4f}")

Julia¶

using Statistics, LinearAlgebra

# Gini coefficient
function gini(wealth)
    n = length(wealth); a = sort(wealth); μ = mean(a)
    (2/(n*μ))*sum(i*a[i] for i in 1:n) - (n+1)/n
end

# Pareto fit
function fit_pareto_mle(x, x_min)
    n = count(x .>= x_min); tail = x[x .>= x_min]
    n / sum(log.(tail./x_min))
end

np = 50000
wealth = sort(vcat(rand(Exponential(0.5), Int(0.45*np)),
                   rand(Pareto(1.5), Int(0.55*np)).*10.0.+5.0))
println("Wealth Gini: $(round(gini(wealth), digits=3))")

thresh = quantile(wealth, 0.90)
zeta = fit_pareto_mle(wealth, thresh)
println("Pareto ζ (top 10%): $(round(zeta,digits=3))")
println("Pareto Gini: $(round(1/(2*zeta-1), digits=3))")

R¶

# R — Gini coefficient and Lorenz curve
wealth <- sort(c(rexp(22500, 2), rpareto(27500, 1.5)*10+5))
n <- length(wealth); mu <- mean(wealth)
G <- (2/(n*mu))*sum((1:n)*wealth) - (n+1)/n
cat(sprintf("Gini: %.3f\n", G))

# Lorenz curve plot
cum_pop <- (1:n)/n; cum_wealth <- cumsum(wealth)/sum(wealth)
plot(cum_pop, cum_wealth, type='l', col='blue',
     xlab='Population share', ylab='Wealth share', main='Lorenz Curve')
abline(0,1, lty=2, col='red'); legend('topleft', c('Lorenz','Equality'), col=c('blue','red'), lty=1:2)

33.7 Programming Exercises¶

Exercise 33.1 (APL — Sample Gini)¶

Write a dfn gini_coef ← {a ← a[⍋a] ⋄ n←≢a ⋄ mu←(+/a)÷n ⋄ (2÷n×mu)×+/(⍳n)×a) - (n+1)÷n}. (a) Verify it equals (2×mu_mean)+/ on the sorted wealth distribution from Chapter 32’s simulation. (b) Compute the Gini as a function of the borrowing limit $\underline{a}$ : tighter constraints force more households to be hand-to-mouth, raising the Gini.

Exercise 33.2 (Python — Piketty $r > g$ and Wealth Concentration)¶

Piketty’s inequality dynamic: $\dot{w}_i = (r-g)w_i + y_i - c_i$ implies that when $r > g$ , wealth grows faster than the economy, concentrating at the top. (a) Simulate 1000 households for 100 periods with $r = 0.05$ and $g \in \{0.01, 0.02, 0.03, 0.04, 0.05\}$ . (b) Compute the Gini and top-1% share for each $g$ . (c) Plot the relationship between $r-g$ and the Gini. (d) Calibrate to U.S. data: what $r-g$ matches the observed top-1% share of 38%?

Exercise 33.3 (Julia — Decomposing the Rate Cut Effect)¶

using Distributions, Statistics

# Simulate heterogeneous households with different MPC and URE
N = 10000; np_rng = 42
Random.seed!(np_rng)

# Wealth distribution (lognormal lower tail + Pareto upper)
a_grid = sort(vcat(rand(LogNormal(0, 1), N÷2), rand(Pareto(1.5), N÷2).*5.0.+2.0))
n = length(a_grid)

# MPC: declining in wealth rank (Carroll-type buffer stock)
rank = (1:n)/n
mpc = max.(0.02, 1.0 .- 0.95.*rank.^0.5)

# URE: negative for borrowers (bottom 55%), positive for savers
ure = (rank .- 0.55) .* 2.0 .* std(a_grid)

# Decompose aggregate dC for dr = -1% (rate cut)
dr = -0.01; sigma_eis = 1.0; C_bar = mean(mpc .* a_grid)

# Aggregate effect components
sub_effect = -C_bar/sigma_eis * dr
redist_effect = cov(mpc, ure) * dr
income_effect = mean(mpc .* a_grid .* dr)  # direct income from interest

println("Rate cut decomposition (Auclert 2019):")
println("  Intertemporal substitution: $(round(sub_effect,digits=4))")
println("  Redistribution (URE):       $(round(redist_effect,digits=4))")
println("  Total:                       $(round(sub_effect+redist_effect,digits=4))")
println("  Cov(MPC, URE): $(round(cov(mpc,ure),digits=4))")
println("  Mean MPC: $(round(mean(mpc),digits=3))")

Exercise 33.4 — Decomposing Wealth Inequality ( $\star$ )¶

In the Aiyagari model from Chapter 32, decompose the wealth Gini into: (a) the contribution of income heterogeneity (different $y_i$ realizations); (b) the contribution of precautionary saving (households accumulating buffers at different rates); (c) the contribution of the borrowing constraint (households bunched at $a = \underline{a}$ ). Use the Gini decomposition by source (Theorem 33.4). Compare to the observed U.S. decomposition: income inequality explains about 30% of wealth inequality; the rest comes from savings behavior heterogeneity and the binding borrowing constraint.

33.8 Chapter Summary¶

Key results:

Pareto distribution $P(X>x) = (x_m/x)^\zeta$ has Gini $= 1/(2\zeta-1)$ (Theorem 33.1); U.S. wealth data has $\zeta \approx 1.5$ , giving Pareto Gini $\approx 0.5$ (upper tail only).
Gini coefficient $G = 2\text{Cov}(X, F(X))/\mathbb{E}[X]$ (Theorem 33.2); sample formula: $G = (2/(n\bar{a}))\sum_i i\cdot a_{(i)} - (n+1)/n$ .
The Auclert URE framework decomposes the aggregate consumption response to a rate change into: (1) intertemporal substitution $(-C/\sigma)\cdot dr$ ; and (2) redistribution $\text{Cov}(\text{MPC},\text{URE})\cdot dr$ (Theorem 33.3). The redistribution channel amplifies policy if poor households are net borrowers.
For U.S. data: $\text{Cov}(\text{MPC},\text{URE}) > 0$ , so rate cuts are amplified by redistribution from low-MPC savers to high-MPC borrowers.
In APL: Gini is (2÷n×mu)×(+/(⍳n)×a_sorted) - (n+1)÷n; Lorenz curve as cumulative sum divided by total +\a_sorted ÷ +/a_sorted.

Next: Chapter 34 — Network Models for Financial Contagion and Systemic Risk