Income Inequality and the Gini Coefficient (2024)

Introduction

I recently have been hearing politicians lamenting the growth of income inequality in the United States. These politicians frequently talk about the number of homeless people (Figure 1), but they never talk about how income inequality is defined, measured, or has been changing with time.

I was raised in a small, working-class, agricultural community and I never saw any homeless people when I was growing up. I now live in an affluent community and I see homeless people on some street corners. The “Great Recession” had a devastating effect on many parts of our economy and it is not hard to imagine that it expanded the income gap between rich and poor.

Since I do not believe anything our politicians say without independent confirmation, I have to ask the following questions:

• What is income inequality and how is it measured?
• How has income inequality been changing over time in the US?
• How does the US income inequality compare with other countries?

These are all questions a little math can help with. Let’s dig in …

Background

Definitions

Income Inequality

Economic inequality is the state of affairs in which assets, wealth, or income are distributed unequally among individuals in a group, among groups in a population, or among countries (source).

Lorentz curve

In economics, the Lorenz curve is a graphical representation of the cumulative distribution function of the empirical probability distribution of wealth or income. The curve is a graph showing the proportion of overall income or wealth assumed by the bottom x% of the people (source).

Overview

As I began researching income inequality, I quickly discovered that the most commonly used metric for income inequality is called the Gini coefficient, which was developed in 1912 by the Italian economist Corrado Gini (Figure 2). You can find tables and graphs of the Gini coefficient for various countries in a number of places (e.g. OECD).

I subscribe to an excellent economics blog by Jodi Beggs that gives an excellent description of the Gini coefficient. Since she gives an excellent detailed description, I will just give a quick description below and I will refer you to her blog if you want more details. She also has a good Youtube video on the subject, which I embed here.

My Quick Gini Coefficient Briefing

Suppose we examine the income of every individual of a country and arrange the individuals in order of increasing income. If we plot the percentage of national income versus the population percentage (Lorenz curve), we will get a curve like that shown in Figure 3.

The area of part B represents the total actual national income.Part A by itself represents the loss of national income because everyone does not earn income as the same as the wealthiest people. The areas of parts A and B together represents the national income that would be realized if everyone earned the same amount as the wealthiest person. The closer the country is to an equal income distribution, the smaller the Gini coefficient.

While the Gini coefficient is well-defined and its calculation is straight-forward, the definition of national income in not. Agencies use different measures of national income − here are a few examples:

• pre-tax income
• post-tax income
• income without government transfer payments
• income with government transfer payments

The difference between pre- and post-tax results is particularly dramatic for high-tax countries. The Wikipedia actually lists both post- and pre-tax Gini coefficients from a number of agencies: World Bank, UN, and CIA.

Figure 4 presents a number of interesting special cases of the Gini coefficient. No country is perfectly egalitarian nor perfectly totalitarian. However, it is interesting to look at the Gini coefficients for places like Sweden (23%, close to egalitarian) and South Africa (63.1%, close to totalitarian). It does not surprise me that the Scandinavian countries all seem to be close to egalitarian. I live in Minnesota, which has strong connection with Scandinavian culture, and our politics has a strong egalitarian feel.

The Gini coefficient is not the only metric for income inequality. Here are a few other metrics that economists and politicians may use to educate or confuse us.

• Hoover Index (aka Robin Hood Index)
• Theil Index
• Atkinson Index

There are many other income inequality indices.

US Gini Coefficient over Time

Figure 5 shows how the Gini coefficient of the US has varied over time. We have definitely seen a significant rise over the last 40 years.

Figure 6 shows how the US Gini coefficient compares with other countries and how it has varied over time. Notice China’s rise in income inequality. When I travel in China, I have heard concerns expressed by the people there about the rising income inequality.

Mathematical Definition of Gini Coefficient

The Wikipedia has an excellent discussion of how you can compute the Gini coefficient, but I am going to use a very simple approach with no simplifications.

Eq. 1

where

• f(x) is the Lorentz curve for country of interest.
• x is the population percentage

Analysis

Economic Data

Figure 7 shows the Lorenz curves for the US in 1968 and 2010. I will digitize this graph using Dagra and evaluate the Gini coefficient using Mathcad.

Calculations

Figure 8 show how I computed the Gini coefficient. There are numerous simplifications that can be made when computing the Gini coefficient − I used none of them.

My computed values have an error of about 1% from the reported Gini coefficients for 1968 and 2010 – not bad agreement considering the errors in digitizing off of an image.

Conclusions

I was able to understand how income inequality is defined, computed, and how it has changed for various countries over time. The US has clearly seen a rise in income inequality.