## Overview

In this section we

In the lecture {doc}`long_run_growth` we studied how GDP per capita has changed

for certain countries and regions.

Per capital GDP is important because it gives us an idea of average income for

households in a given country.

However, when we study income and wealth, averages are only part of the story.

For example, imagine two societies, each with one million people, where

* in the first society, the yearly income of one man is $100,000,000 and the income of the

others is zero

* in the second society, the yearly income of everyone is $100

These countries have the same income per capita (average income is $100) but the lives of the people will be very different (e.g., almost everyone in the first society is

starving, even though one person is fabulously rich).

The example above suggests that we should go beyond simple averages when we study income and wealth.

This leads us to the topic of economic inequality, which examines how income and wealth (and other quantities) are distributed across a population.

In this lecture we study inequality, beginning with measures of inequality and

then applying them to wealth and income data from the US and other countries.

* provide motivation for the techniques deployed in the lecture and

* import code libraries needed for our work.

### Some history

Many historians argue that inequality played a key role in the fall of the

Roman Republic.

Many historians argue that inequality played a role in the fall of the Roman Republic (see, e.g., {cite}`levitt2019did`).

Following the defeat of Carthage and the invasion of Spain, money flowed into

Rome from across the empire, greatly enriched those in power.

Meanwhile, ordinary citizens were taken from their farms to fight for long

periods, diminishing their wealth.

The resulting growth in inequality caused political turmoil that shook the

foundations of the republic.

The resulting growth in inequality was a driving factor behind political turmoil that shook the foundations of the republic.

Eventually, the Roman Republic gave way to a series of dictatorships, starting

with Octavian (Augustus) in 27 BCE.

Eventually, the Roman Republic gave way to a series of dictatorships, starting with [Octavian](https://en.wikipedia.org/wiki/Augustus) (Augustus) in 27 BCE.

This history is fascinating in its own right, and we can see some

parallels with certain countries in the modern world.

This history tells us that inequality matters, in the sense that it can drive major world events.

Let's now look at inequality in some of these countries.

There are other reasons that inequality might matter, such as how it affects

human welfare.

### Measurement

With this motivation, let us start to think about what inequality is and how we

can quantify and analyze it.

Political debates often revolve around inequality.

One problem with these debates is that inequality is often poorly defined.

Moreover, debates on inequality are often tied to political beliefs.

### Measurement

This is dangerous for economists because allowing political beliefs to shape our findings reduces objectivity.

In politics and popular media, the word "inequality" is often used quite loosely, without any firm definition.

To bring a truly scientific perspective to the topic of inequality we must start with careful definitions.

To bring a scientific perspective to the topic of inequality we must start with careful definitions.

Hence we begin by discussing ways that inequality can be measured in economic research.

import plotly.express as px

```

## The Lorenz curve

One popular measure of inequality is the Lorenz curve.

---

mystnb:

figure:

caption: Lorenz curve of simulated data

caption: Lorenz curve of simulated wealth data

name: lorenz_simulated

---

n = 2000

ax.vlines([0.8], [0.0], [0.43], alpha=0.5, colors='k', ls='--')

ax.hlines([0.43], [0], [0.8], alpha=0.5, colors='k', ls='--')

ax.set_xlim((0, 1))

ax.set_xlabel("share of households (%)")

ax.set_xlabel("share of households")

ax.set_ylim((0, 1))

ax.set_ylabel("share of income (%)")

ax.set_ylabel("share of wealth")

ax.legend()

plt.show()

```

### Lorenz curves for US data

Next let's look at US data for both income and wealth.

ax.plot(f_vals_ti[-1], l_vals_ti[-1], label=f'total income')

ax.plot(f_vals_li[-1], l_vals_li[-1], label=f'labor income')

ax.plot(f_vals_nw[-1], f_vals_nw[-1], label=f'equality')

ax.set_xlabel("share of households (%)")

ax.set_ylabel("share of income/wealth (%)")

ax.set_xlabel("share of households")

ax.set_ylabel("share of income/wealth")

ax.legend()

plt.show()

```

## The Gini coefficient

The Lorenz curve is a useful visual representation of inequality in a distribution.

The Lorenz curve provides a visual representation of inequality in a distribution.

Another way to study income and wealth inequality is via the Gini coefficient.

In this section we discuss the Gini coefficient and its relationship to the

Lorenz curve.

In this section we discuss the Gini coefficient and its relationship to the Lorenz curve.

---

mystnb:

figure:

caption: Shaded Lorenz curve of simulated data

caption: Gini coefficient (simulated wealth data)

name: lorenz_gini

---

fig, ax = plt.subplots()

f_vals, l_vals = lorenz_curve(sample)

ax.plot(f_vals, l_vals, label=f'lognormal sample', lw=2)

ax.plot(f_vals, f_vals, label='equality', lw=2)

ax.vlines([0.8], [0.0], [0.43], alpha=0.5, colors='k', ls='--')

ax.hlines([0.43], [0], [0.8], alpha=0.5, colors='k', ls='--')

ax.fill_between(f_vals, l_vals, f_vals, alpha=0.06)

ax.set_ylim((0, 1))

ax.set_xlim((0, 1))

ax.text(0.04, 0.5, r'$G = 2 \times$ shaded area')

ax.set_xlabel("share of households (%)")

ax.set_ylabel("share of income/wealth (%)")

ax.set_ylabel("share of wealth (%)")

ax.legend()

plt.show()

```

ax.set_xlim((0, 1))

ax.text(0.55, 0.4, 'A')

ax.text(0.75, 0.15, 'B')

ax.set_xlabel("share of households (%)")

ax.set_ylabel("share of income/wealth (%)")

ax.set_xlabel("share of households")

ax.set_ylabel("share of wealth")

ax.legend()

plt.show()

```

```{seealso}

The World in Data project has a [nice graphical exploration of the Lorenz curve and the Gini coefficient](https://ourworldindata.org/what-is-the-gini-coefficient)

The World in Data project has a [graphical exploration of the Lorenz curve and the Gini coefficient](https://ourworldindata.org/what-is-the-gini-coefficient)

```

### Gini coefficient of simulated data

---

mystnb:

figure:

caption: Histogram of Gini coefficients

caption: Histogram of Gini coefficients across countries

name: gini_histogram

---

# Fetch gini data for all countries

trended upward from 1980 to 2020 and then dropped following at the start of the COVID pandemic.

(compare-income-wealth-usa-over-time)=

### Gini coefficient for wealth (US data)

### Gini coefficient for wealth

In the previous section we looked at the Gini coefficient for income using US data.

In the previous section we looked at the Gini coefficient for income, focusing on using US data.

Now let's look at the Gini coefficient for the distribution of wealth.

We can use the {ref}`Survey of Consumer Finances data <data:survey-consumer-finance>` to look at the Gini coefficient

computed over the wealth distribution.

We will use US data from the {ref}`Survey of Consumer Finances<data:survey-consumer-finance>`

```{code-cell} ipython3

df_income_wealth.year.describe()

```

**Note:** This code can be used to compute this information over the full dataset.

This code can be used to compute this information over the full dataset.

```{code-cell} ipython3

:tags: [skip-execution, hide-input, hide-output]

The time series for the wealth Gini exhibits a U-shape, falling until the early

1980s and then increasing rapidly.

One possibility is that this change is mainly driven by technology.

However, we will see below that not all advanced economies experienced similar growth of inequality.

### Cross-country comparisons of income inequality

Earlier in this lecture we used `wbgapi` to get Gini data across many countries and saved it in a variable called `gini_all`

Earlier in this lecture we used `wbgapi` to get Gini data across many countries

and saved it in a variable called `gini_all`

In this section we will use this data to compare several advanced economies, and

to look at the evolution in their respective income Ginis.

Another popular measure of inequality is the top shares.

In this section we show how to compute top shares.