# Wealth Distribution Dynamics in MyST
```{note}
You can {download}`Download the source file for this page <./wealth_dynamics_md.md>`
```
```{contents}
:depth: 2
```
In addition to what's in Anaconda, this lecture will need the following
libraries:
```{code-block} ipython
---
class: hide-output
---
!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 {doc}`this lecture ` for a
definition.
- For a review of the empirical evidence, see, for example,
{cite}`md-benhabib2018skewed`.
### A Note on Assumptions
The evolution of wealth for any given household depends on their savings
behavior.
We will use the following imports.
```{code-block} python
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](https://en.wikipedia.org/wiki/Lorenz_curve).
The package [QuantEcon.py](https://github.com/QuantEcon/QuantEcon.py),
already imported above, contains a function to compute Lorenz curves.
To illustrate, suppose that
```{code-block} python
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:
```{code-block} python
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.
```{code-block} python
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](https://en.wikipedia.org/wiki/Gini\_coefficient).
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](https://github.com/QuantEcon/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 $a$.
```{code-block} python
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
```{math}
---
label: md:wealth_dynam_ah
---
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
$\mathbb R^3$.
(md:sav_ah)=
```{math}
---
label: md:sav_ah
---
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.
```{code-block} python
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.
```{code-block} python
@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.
```{code-block} python
@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[0] = 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.
```{code-block} python
@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.
```{code-block} python
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
{doc}`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.
```{code-block} python
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.
```{code-block} python
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.
(htop_again)=
```{image} htop_again.png
---
scale: 80
---
```
Now let's check the Gini coefficient.
```{code-block} python
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.
```{code-block} python
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.
```{bibliography} references.bib
:labelprefix: md
:keyprefix: md-
```