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# Gini Coefficient and Lorenz Curve

## Video transcript

- [Instructor] 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. And to appreciate what income inequality is, let's imagine two different countries. Let's imagine first, country A, and there's two people in country A. So, you have person one here, who makes $1000 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? Or 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 1000 and the 99,000. So you have a total income of one plus 99, 100,000 divided by two folks, well you're gonna 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, they make$50,000 a year, and let's say there's a second person in country B, and they also make $50,000 per year, per year. Well, what's the average income now? Well, this is even easier to compute, 50 plus 50 divided by two, your average income is$50,000 per year. So, what you see here is two countries that if you just looked at the average income, they seem similarly wealthy, but that doesn't give you, they seem to have similar average incomes. So you would 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, question is, is above and beyond looking at things like average income or average GDP or per capita GDP, how do you measure inequality? And this is something that this Italian statistician, Corrado Gini, tried to address, and he comes up with something called a Gini coefficient to measure income inequality for a nation. And the way he approached it is actually pretty intuitive. What he did is he sets up two axes. So, this axis right over here is going to be the cumulative percentage of the population. So, 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, 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 there. And so, this is cumulative percent of income in a country, and then he said, well what would a perfectly equal society look like? Well 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 one going up like this. So, 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, well then that should be 50% of the national income. But no nation is actually there, and so then we have to then 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, and as you add percentages of the population, you get to higher and higher income folks. So, let's say we're looking at a country that for the poorest folks, as you add percentages to the cumulative population, you're not adding the same percent to the cumulative income. And so, 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. And so, this curve right over here, which you could view as describing the reality for a certain nation, this is known as a Lorenz curve, Lorenz curve, and what Gini said is, well the difference between the Lorenz curve and this line right over here, that that would be a measure of income inequality. And so, 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 zero to one, or sometimes you might see the scale as being from zero to 100. So, what would a Gini coefficient of zero represent? Well, if you have a Gini coefficient of zero, that means that this area right over here between the Lorenz curve and this line is zero. So that means that we are dealing with a perfectly equal income distribution. So at the zero end, this is perfect equality, perfect income equality, and then what does one 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, I keep adding more and more and more population, but I'm not adding more and more income, and then all of a sudden, you get to the very last person, and then that person has all of the income. So, that person has all of the income. Well in that case, the Gini coefficient would be the percentage of this area, which would be 100%, which we could view as a one or 100. And so, an interesting thing to do is, is look at Gini coefficients for various countries and compare them. And that's exactly what we have here on this map, and you can see that the countries that are shaded red, these are countries that have high Gini coefficients. So, this is where you have more income inequality, and 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, what average income is in that country, and so, this is an indication that in places like Latin America and Sub-Saharan Africa, you definitely have very high inequality, and places like Canada and Europe, you seem to have very low inequality, but it doesn't tell you that people for sure are 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, and one can have a very spirited debate, which one you would rather be. Would you rather be in a country that has higher inequality and higher average income or one that has lower income and lower inequality?