Gini Coefficient and Lorenz Curve

In this video, we're going to discuss income inequality, which is something that is often debated. Thinking about comparing countries, thinking about whether it's an issue or not, and how to address it. To appreciate what income inequality is, let's imagine two different countries.

Let's imagine first Country A, and there are two people in Country A. So you have Person One here, who makes $1,000 a year—that's their income. And then there's Person Two in Country A that makes $99,000 a year. So what is going to be the average income in Country A? If these are the only two people, you could think of it as the per capita national income.

Well, the average income here—average income—to figure that out, you would just have to average the $1,000 and the $99,000. So you have a total income of $1,000 plus $99,000, which equals $100,000, divided by two. Folks, we're going to have an average income of $50,000 per year.

Now, let me construct another country that has the same average income, but the distribution is very different. So, in Country B, let's say the first citizen of Country B makes $50,000 a year, and let's say there's a second person in Country B, and they also make $50,000 per year. Well, what's the average income now? Well, this is even easier to compute: $50,000 plus $50,000 divided by 2. Your average income is $50,000 per year.

So what you see here are two countries that, if you just looked at the average income, they seem similarly wealthy. But that doesn't give you the full picture—they seem to have similar average income, so you say, "Oh, maybe they're similarly prosperous." But when you go a step deeper, you see that they are very different. Country A is a lot more unequal than Country B when it comes to income.

So the question is, above and beyond looking at things like average income, or average GDP, or per capita GDP, how do you measure inequality? This is something that this Italian statistician Corrado Gini tried to address, and he came up with something called a Gini coefficient to measure income inequality for a nation.

The way he approached it is actually pretty intuitive. What he did is he set up two axes. So this axis right over here is going to be the cumulative percentage of the population. You start at 0% and then you get all the way to 100% of the population. So this is the cumulative percent of the population in a country. And then on this axis, you have the cumulative percentage of the income in a country.

So this would be zero down here, and then this would be 100 up here. And so this is cumulative percent of income in a country. Then he said, "Well, what would a perfectly equal society look like?" In a perfectly equal society, as you add a percentage on your cumulative percentage of population, you should add that exact same percentage to your cumulative percentage of income.

So as you go up, you really should just have a slope of 1 going up like this. One way to think about it is when you're at 0% of the population, you should have 0% of the income. If you have a total of 10% of the population, they should have 10% of the national income. If you were to go to 50% of the population—which looks like it's around there—if it was perfectly distributed, the income should be 50% of the national income.

But no nation is actually there, and so then we have to compare that to the reality. So let's say you look at a country, and what you do is when you're looking at the cumulative percentage of the population, you start at the left with the lowest income. As you add percentages to the cumulative population, you're not adding the same percent to the cumulative income.

So it might—you might have a curve that looks like this. And then, as you add percentages in the wealthier population, for every 1% you add, you're adding more than 1% of national income. So this curve variety over here, which you could view as describing the reality for a certain nation, this is known as a Lorenz curve.

What Gini said is, well, the difference between the Lorenz curve and this line right over here that would be a measure of income inequality. He would look at this area right over here and say what percentage is this area between this line and the Lorenz curve? What percentage is this of this total area under the line?

And this percentage is called the Gini coefficient, and it's typically quoted as being a value from 0 to 1, or sometimes you might see the scale as being from 0 to 100. So what would a Gini coefficient of 0 represent? Well, if you have a Gini coefficient of 0, that means that this area right over here between the Lorenz curve and this line is 0. So that means we are dealing with a perfectly equal income distribution.

At the 0 end, this is perfect equality, perfect income equality. And then what does 1 or 100 mean? That means that the area between the line and the Lorenz curve is 100% of the area under this line. So that would look like something like this—a country whose Lorenz curve looks like something like this where all of these people keep adding more and more and more population, but I'm not adding more and more income.

Then all of a sudden, you get to the very last person, and that person has all of the income. In that case, the Gini coefficient would be the percentage of this area, which would be 100%. We could view that as a 1 or a 100.

So an interesting thing to do is look at Gini coefficients for various countries and compare them, and that's exactly what we have here on this map. You can see that the countries that are shaded red—these are countries that have high Gini coefficients. This is where you have more income inequality.

The ones that are shaded green are the ones where you have relatively low Gini indices or Gini coefficients, and so that would be indicative of reasonably low income inequality. Now, it's important to point out, you might think that red is always bad and green is always good, but this just tells you inequality.

It does not tell you on average how prosperous folks are or what average income is in that country. This is an indication that in places like Latin America and Sub-Saharan Africa, you definitely have very high inequality. In places like Canada and Europe, you seem to have very low inequality.

But it doesn't tell you that people are for sure better off in Canada than, say, the United States, for example. You could have a higher average income in the United States than you have in Canada. One can have a very spirited debate about which one you would rather be in: would you rather be in a country that has higher inequality and higher average income, or one that has lower income and lower inequality?