Income Inequality and the Gini Coefficient

Quote of the Day

The world is divided into people who do things and people who get the credit.

- Dwight Whitney Morrow


Introduction

Figure 1: A Common Example of Income Inequality.

Figure 1: A Common Example of Income Inequality.

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. The economy goes up and down as society changes, this can be positive or negative depending on the circumstances leading up to these outcomes. With so many using online currency now as a way to make money, this has shifted from the traditional 'wall street' that has been a staple in the economy. Websites such as https://kryptoszene.de/kryptowaehrungen-kaufen/ontology-kaufen/ are flooded with people trying to get into this new way of earning, but how will that affect people with limited money? Can it help them add to it? Support them? Only time will tell.

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

Figure M: Photograph of Corrado Gini.

Figure 2: Photograph of Corrado Gini.

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.

Figure 1: Illustration of Gini Coefficient and Its Calculation.

Figure 3: Illustration of Gini Coefficient and Its Calculation.

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 M: Interesting Cases of the Gini Coefficients.

Figure 4: Interesting Cases of the Gini Coefficients.

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.

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. This is most likely due to the introduction of technology and how easy it is for those already obtaining a portion of wealth to invest with forex apps uk among the likes of from any location with internet connection. Making money is no longer confined to office hours. Nowadays, money can be made from a number of different avenues, such as investments or even by playing some casino games. By using a live mobile casino, people can play different games in the hopes of winning some more money for themselves. This means that people can make more money from the comfort of their home.

Figure M: US Gini Ratio Over Time.

Figure 5: US Gini Coefficient Over Time.

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.

Figure M: Change in Gini Ratio for Various Countries Over Time.

Figure 6: Change in Gini Coefficient for Various Countries Over Time.

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 \displaystyle \text{Gini Coefficient}\triangleq \frac{{\int\limits_{{0\%}}^{{100\%}}{{x-f\left( x \right)\text{ }dx}}}}{{\int\limits_{{0\%}}^{{100\%}}{{x\text{ }dx}}}}

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.

Figure M: Lorenz Curves for the US in 1968 and 2010.

Figure 7: Lorenz Curves for the US in 1968 and 2010.

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.

Figure M: Gini Calculations for the US.

Figure 8: Gini Coefficient Calculations for the US.

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.

 
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4 Responses to Income Inequality and the Gini Coefficient

  1. Dr. Goulu says:

    Just to mention I made an spreadsheet to calculate Gini coefficients of any data you might have.
    I used it to calculate the Gini coefficient of tennis players, which ias bout 0.56, very unequal. My blog article about this (in french) is here : http://www.drgoulu.com/2013/10/26/112-le-prix-du-talent/

    The sheet is freely available for non commercial uses in Excel here http://www.box.net/shared/njhn3h3out and in Google Docs here : https://docs.google.com/spreadsheet/ccc?key=0Al_D4zS2T4QodGNwZmF0YlNhTFJGMmlVMFloM1ZzZ2c&usp=sharing

    I'm trying to promote the use of the Gini index in companies, where it is very easy calculate. How does your own company contribute to (in)equality ?

     
  2. Dr. Goulu says:

    Note that the Gini index of the whole world decreases steadily since the '70s ( http://www.voxeu.org/article/parametric-estimations-world-distribution-income ) : the globalized world becomes more equal as (hundreds of) millions of inhabitants from the BRICS reach higher levels of income. Paradoxically, this leads to more inequality in theses countries as other (hundreds of) millions remain poor.

     
  3. Dr. Goulu says:

    It is very interesting to compare the Gini index before and after faxes to see how money is redistributed . OECD has very nice graphs showing this for its member countries ( have them in french on http://www.drgoulu.com/2012/07/13/encore-plus-dinegalite/ ) . For example the Gini for the US shown in your post is the one before taxes. After taxes it is about 0.37.

    In France, they have about the same Gini before taxes (0.44), but after taxes they're much more equal with Gini =0.29

    In Switzerland (I'm swiss), we also reach a Gini after taxes of about 0.29, but it is "only" 0.34 before taxes : salary inequalities are low, so taxes do not need to redistribute much...

     
  4. Nikolai Zenevitsh says:

    Hi!

    I was wondering if you could help me out. How do you calculate the lorenz function for a give country? I canot find help from anywhere..

     

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