# Wealth Distribution Dynamics in rST¶

Note

In addition to what’s in Anaconda, this lecture will need the following libraries:

!pip install --upgrade quantecon


## Overview¶

This notebook gives an introduction to wealth distribution dynamics, with a focus on

• modeling and computing the wealth distribution via simulation,

• measures of inequality such as the Lorenz curve and Gini coefficient, and

• how inequality is affected by the properties of wage income and returns on assets.

The wealth distribution in many countries exhibits a Pareto tail

• See this lecture for a definition.

• For a review of the empirical evidence, see, for example, [BB18].

### A Note on Assumptions¶

The evolution of wealth for any given household depends on their savings behavior.

We will use the following imports.

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

import quantecon as qe
from numba import njit, jitclass, float64, prange


## Lorenz Curves and the Gini Coefficient¶

Before we investigate wealth dynamics, we briefly review some measures of inequality.

### Lorenz Curves¶

One popular graphical measure of inequality is the Lorenz curve.

The package QuantEcon.py, already imported above, contains a function to compute Lorenz curves.

To illustrate, suppose that

n = 10_000                      # size of sample
w = np.exp(np.random.randn(n))  # lognormal draws


is data representing the wealth of 10,000 households.

We can compute and plot the Lorenz curve as follows:

f_vals, l_vals = qe.lorenz_curve(w)

fig, ax = plt.subplots()
ax.plot(f_vals, l_vals, label='Lorenz curve, lognormal sample')
ax.plot(f_vals, f_vals, label='Lorenz curve, equality')
ax.legend()
plt.show()


This curve can be understood as follows: if point $$(x,y)$$ lies on the curve, it means that, collectively, the bottom $$(100x)\%$$ of the population holds $$(100y)\%$$ of the wealth.

a_vals = (1, 2, 5)              # Pareto tail index
n = 10_000                      # size of each sample
fig, ax = plt.subplots()
for a in a_vals:
u = np.random.uniform(size=n)
y = u**(-1/a)               # distributed as Pareto with tail index a
f_vals, l_vals = qe.lorenz_curve(y)
ax.plot(f_vals, l_vals, label=f'$a = {a}$')
ax.plot(f_vals, f_vals, label='equality')
ax.legend()
plt.show()


You can see that, as the tail parameter of the Pareto distribution increases, inequality decreases.

This is to be expected, because a higher tail index implies less weight in the tail of the Pareto distribution.

### The Gini Coefficient¶

The definition and interpretation of the Gini coefficient can be found on the corresponding Wikipedia page.

A value of 0 indicates perfect equality (corresponding the case where the Lorenz curve matches the 45 degree line) and a value of 1 indicates complete inequality (all wealth held by the richest household).

The QuantEcon.py library contains a function to calculate the Gini coefficient.

We can test it on the Weibull distribution with parameter $$a$$, where the Gini coefficient is known to be

$G = 1 - 2^{-1/a}$

Let’s see if the Gini coefficient computed from a simulated sample matches this at each fixed value of $$a$$.

a_vals = range(1, 20)
ginis = []
ginis_theoretical = []
n = 100

fig, ax = plt.subplots()
for a in a_vals:
y = np.random.weibull(a, size=n)
ginis.append(qe.gini_coefficient(y))
ginis_theoretical.append(1 - 2**(-1/a))
ax.plot(a_vals, ginis, label='estimated gini coefficient')
ax.plot(a_vals, ginis_theoretical, label='theoretical gini coefficient')
ax.legend()
ax.set_xlabel("Weibull parameter $a$")
ax.set_ylabel("Gini coefficient")
plt.show()


The simulation shows that the fit is good.

## A Model of Wealth Dynamics¶

Having discussed inequality measures, let us now turn to wealth dynamics.

The model we will study is

(1)$w_{t+1} = (1 + r_{t+1}) s(w_t) + y_{t+1}$

where

• $$w_t$$ is wealth at time $$t$$ for a given household,

• $$r_t$$ is the rate of return of financial assets,

• $$y_t$$ is current non-financial (e.g., labor) income and

• $$s(w_t)$$ is current wealth net of consumption

Letting $$\{z_t\}$$ be a correlated state process of the form

$z_{t+1} = a z_t + b + \sigma_z \epsilon_{t+1}$

we’ll assume that

$R_t := 1 + r_t = c_r \exp(z_t) + \exp(\mu_r + \sigma_r \xi_t)$

and

$y_t = c_y \exp(z_t) + \exp(\mu_y + \sigma_y \zeta_t)$

Here $$\{ (\epsilon_t, \xi_t, \zeta_t) \}$$ is IID and standard normal in $$\mathbb R^3$$.

(2)$s(w) = s_0 w \cdot \mathbb 1\{w \geq \hat w\}$

where $$s_0$$ is a positive constant.

## Implementation¶

Here’s some type information to help Numba.

wealth_dynamics_data = [
('w_hat',  float64),    # savings parameter
('s_0',    float64),    # savings parameter
('c_y',    float64),    # labor income parameter
('μ_y',    float64),    # labor income paraemter
('σ_y',    float64),    # labor income parameter
('c_r',    float64),    # rate of return parameter
('μ_r',    float64),    # rate of return parameter
('σ_r',    float64),    # rate of return parameter
('a',      float64),    # aggregate shock parameter
('b',      float64),    # aggregate shock parameter
('σ_z',    float64),    # aggregate shock parameter
('z_mean', float64),    # mean of z process
('z_var', float64),     # variance of z process
('y_mean', float64),    # mean of y process
('R_mean', float64)     # mean of R process
]


Here’s a class that stores instance data and implements methods that update the aggregate state and household wealth.

@jitclass(wealth_dynamics_data)
class WealthDynamics:

def __init__(self,
w_hat=1.0,
s_0=0.75,
c_y=1.0,
μ_y=1.0,
σ_y=0.2,
c_r=0.05,
μ_r=0.1,
σ_r=0.5,
a=0.5,
b=0.0,
σ_z=0.1):

self.w_hat, self.s_0 = w_hat, s_0
self.c_y, self.μ_y, self.σ_y = c_y, μ_y, σ_y
self.c_r, self.μ_r, self.σ_r = c_r, μ_r, σ_r
self.a, self.b, self.σ_z = a, b, σ_z

# Record stationary moments
self.z_mean = b / (1 - a)
self.z_var = σ_z**2 / (1 - a**2)
exp_z_mean = np.exp(self.z_mean + self.z_var / 2)
self.R_mean = c_r * exp_z_mean + np.exp(μ_r + σ_r**2 / 2)
self.y_mean = c_y * exp_z_mean + np.exp(μ_y + σ_y**2 / 2)

# Test a stability condition that ensures wealth does not diverge
# to infinity.
α = self.R_mean * self.s_0
if α >= 1:
raise ValueError("Stability condition failed.")

def parameters(self):
"""
Collect and return parameters.
"""
parameters = (self.w_hat, self.s_0,
self.c_y, self.μ_y, self.σ_y,
self.c_r, self.μ_r, self.σ_r,
self.a, self.b, self.σ_z)
return parameters

def update_states(self, w, z):
"""
Update one period, given current wealth w and persistent
state z.
"""

# Simplify names
params = self.parameters()
w_hat, s_0, c_y, μ_y, σ_y, c_r, μ_r, σ_r, a, b, σ_z = params
zp = a * z + b + σ_z * np.random.randn()

# Update wealth
y = c_y * np.exp(zp) + np.exp(μ_y + σ_y * np.random.randn())
wp = y
if w >= w_hat:
R = c_r * np.exp(zp) + np.exp(μ_r + σ_r * np.random.randn())
wp += R * s_0 * w
return wp, zp


Here’s function to simulate the time series of wealth for in individual households.

@njit
def wealth_time_series(wdy, w_0, n):
"""
Generate a single time series of length n for wealth given
initial value w_0.

The initial persistent state z_0 for each household is drawn from
the stationary distribution of the AR(1) process.

* wdy: an instance of WealthDynamics
* w_0: scalar
* n: int

"""
z = wdy.z_mean + np.sqrt(wdy.z_var) * np.random.randn()
w = np.empty(n)
w = w_0
for t in range(n-1):
w[t+1], z = wdy.update_states(w[t], z)
return w


Now here’s function to simulate a cross section of households forward in time.

Note the use of parallelization to speed up computation.

@njit(parallel=True)
def update_cross_section(wdy, w_distribution, shift_length=500):
"""
Shifts a cross-section of household forward in time

* wdy: an instance of WealthDynamics
* w_distribution: array_like, represents current cross-section

Takes a current distribution of wealth values as w_distribution
and updates each w_t in w_distribution to w_{t+j}, where
j = shift_length.

Returns the new distribution.

"""
new_distribution = np.empty_like(w_distribution)

# Update each household
for i in prange(len(new_distribution)):
z = wdy.z_mean + np.sqrt(wdy.z_var) * np.random.randn()
w = w_distribution[i]
for t in range(shift_length-1):
w, z = wdy.update_states(w, z)
new_distribution[i] = w
return new_distribution


Parallelization is very effective in the function above because the time path of each household can be calculated independently once the path for the aggregate state is known.

## Applications¶

Let’s try simulating the model at different parameter values and investigate the implications for the wealth distribution.

### Time Series¶

Let’s look at the wealth dynamics of an individual household.

wdy = WealthDynamics()

ts_length = 200
w = wealth_time_series(wdy, wdy.y_mean, ts_length)

fig, ax = plt.subplots()
ax.plot(w)
plt.show()


Notice the large spikes in wealth over time.

Such spikes are similar to what we observed in time series when we studied Kesten processes.

### Inequality Measures¶

Let’s look at how inequality varies with returns on financial assets.

The next function generates a cross section and then computes the Lorenz curve and Gini coefficient.

def generate_lorenz_and_gini(wdy, num_households=100_000, T=500):
"""
Generate the Lorenz curve data and gini coefficient corresponding to a
WealthDynamics mode by simulating num_households forward to time T.
"""
ψ_0 = np.ones(num_households) * wdy.y_mean
z_0 = wdy.z_mean

ψ_star = update_cross_section(wdy, ψ_0, shift_length=T)
return qe.gini_coefficient(ψ_star), qe.lorenz_curve(ψ_star)


Now we investigate how the Lorenz curves associated with the wealth distribution change as return to savings varies.

The code below plots Lorenz curves for three different values of $$\mu_r$$.

If you are running this yourself, note that it will take one or two minutes to execute.

This is unavoidable because we are executing a CPU intensive task.

In fact the code, which is JIT compiled and parallelized, runs extremely fast relative to the number of computations.

fig, ax = plt.subplots()
μ_r_vals = (0.0, 0.025, 0.05)
gini_vals = []

for μ_r in μ_r_vals:
wdy = WealthDynamics(μ_r=μ_r)
gv, (f_vals, l_vals) = generate_lorenz_and_gini(wdy)
ax.plot(f_vals, l_vals, label=f'$\psi^*$ at $\mu_r = {μ_r:0.2}$')
gini_vals.append(gv)

ax.plot(f_vals, f_vals, label='equality')
ax.legend(loc="upper left")
plt.show()


The Lorenz curve shifts downwards as returns on financial income rise, indicating a rise in inequality.

Now let’s check the Gini coefficient.

fig, ax = plt.subplots()
ax.plot(μ_r_vals, gini_vals, label='gini coefficient')
ax.set_xlabel("$\mu_r$")
ax.legend()
plt.show()


Once again, we see that inequality increases as returns on financial income rise.

Let’s finish this section by investigating what happens when we change the volatility term $$\sigma_r$$ in financial returns.

fig, ax = plt.subplots()
σ_r_vals = (0.35, 0.45, 0.52)
gini_vals = []

for σ_r in σ_r_vals:
wdy = WealthDynamics(σ_r=σ_r)
gv, (f_vals, l_vals) = generate_lorenz_and_gini(wdy)
ax.plot(f_vals, l_vals, label=f'$\psi^*$ at $\sigma_r = {σ_r:0.2}$')
gini_vals.append(gv)

ax.plot(f_vals, f_vals, label='equality')
ax.legend(loc="upper left")
plt.show()


We see that greater volatility has the effect of increasing inequality in this model.

BB18

Jess Benhabib and Alberto Bisin. Skewed wealth distributions: theory and empirics. Journal of Economic Literature, 56(4):1261–91, 2018.