Lorenz Curve and Gini Coefficient - Video Tutorials & Practice Problems

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Lorenz Curve and the Gini Coefficient

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So it wouldn't be economics if we didn't hit the graph. Right? So now let's talk about income inequality on the graph using the Lorenz curve and the Gini coefficient. So this Lorenz curve, it's gonna help us visualize the income inequality between the rich and the poor. Okay. So it's gonna show us how we see this, it's gonna show it to us on a graph. Okay So we talk about the Lorenz curve, it's gonna be a bit different than the graphs that we've talked about throughout this course. So on the horizontal axis we're gonna have the cumulative, so this is gonna be a cumulative percentage of households. Okay, cumulative means that we're gonna be adding more and more. It adds on to each other. Think about a cumulative exam, right? A cumulative exam. It takes the topics from chapter 12345. It adds everything that you've learned cumulatively. Alright. So cumulative amount of households and percentage on the horizontal axis, our X axis and the Y axis. The vertical axis is also gonna be cumulative percentage of income. Okay so the income is going to go on that Y axis. So when we when we create our Lorenz curve we're gonna break the population into what we call quintiles and each of these quintiles. Quint, meaning five each is gonna represent 20%. So we're gonna have 5, 20% sections of the of the economy of the households and we're gonna break it up in the income they're making. So we're gonna start with the poorest 20%. And we're gonna say how much income do the poorest 20% get what percentage of the total income are they getting? Then we'll look at the next 20%. How much income are they getting? Right? So we start with the poorest all the way to the riches. So let's go ahead and go to this graph and see how we calculate this um this Lorenz curve. So notice on the graph we've got our X and Y. Axis are labeled 20% 40% 60. These are the quintiles right? Going up and down there. So notice our X. Axis. That is our cumulative percentage of households, right? And our Y axis is the cumulative percentage of income. Cool. So let's go over here. We've got some information here about an economy where the lowest 20% they're earning 3.4% of the total income. 2nd 28.6, 3rd 4.7 and so on. Right? So we've got the percentage of income of that quintile And then we've got which quintile it is. Alright, but remember we need cumulative percentages of income. So even though we know the percentage of income of that quintile, we need to start adding them up. All right, so the lowest 20%, the cumulative amount is going to be 3.4. Okay, it's gonna be that percentage of their income. But now when we get to the 2nd, 20% we're gonna do cumulative, right? We're now talking about the bottom 40%. So this bottom 40%, they're earning 3.4% as well as the 8.6 right? We're cumulative. So we're gonna have to add the 3.4 plus the 8.6. Right? And that is going to give us see I can do this in my head, 12%, for our first cumulative there. So let me pull up my calculator as we continue here. Okay, so now we do the same thing with the 3rd% right? We have to keep adding cumulatively. So we could add the 3.4 plus the 8.6 plus the 14.7. Right? Well we already know that the 3.4 and the 8.6 they equal 12%. Right? So I'm just gonna go 12% plus 14.7%. Right? And that's going to equal 26.7%. I'm gonna put our final calculations over here, I'm gonna erase these and move them over just so that we can kind of see them all in place. So 3.4%. This was 12%. Alright, so now let's go on to the fourth. So now we're talking about the bottom 80% of the population. How much income are they getting. Cumulatively? Right, so we're gonna add up all these numbers 3.48.6 14.7 and 23.3. So we know that those first three numbers they equal 26.7. We just calculated that plus the next 1, 23.3. Right? So that's gonna equal 26.7 plus 23.3. That's 50%. So, notice what's happened here, we're talking about the bottom 80% of the nation. So we're talking about 80% of the nation is earning 50% of the income, right? So there you can see how we've got inequality, right? All of this is showing the inequality in this nation. And the last one, we've got the highest 20%. So now we're talking about everybody. So we should get to 100%. Right? We're talking about 100% of the population, we should have 100% of the income accounted for, right? And that's how it's always gonna happen, and that's what's happening here. So, if we add all those numbers together, well, that's just the 50% plus the 50% And that is 100%. Right? So 100% of the households are earning, earning 100% of the income. That should make sense. Alright, so let's go ahead and plot this Lorenz curve onto the graph. So what we have is we're gonna have as we add our cumulative percentages of households, we've got to put our cumulative percentages of total income that we've calculated over here. Alright, so let's start with the first one, the lowest 20% is earning 3.4%. So that's gonna be something way down here. Alright, we'll estimate it's right around there. Let's go on to the next one. So now we're gonna talk about the cumulative 40%, they're earning 12% of the income. That will be somewhere around there. Somewhere around there. All right, now let's go on to 60%. They're earning 26.7% of the income over there. So, they're gonna be somewhere around here, we can say. And now let's go on to 80% of the population is earning 50% of the income. It's gonna be right here and last, but not least, we've got 100% earning 100%. Right? So that's gonna go always up in that corner. So, we're ready to plot our Lorenz curve and let's see how well I can do this on my first try. Let's go for it. So, it's gonna look something like that, right? It's gonna look something like that. Or Lorenz curve right there. And when we plot this, we're always gonna want to plot one other line here. Okay. And it's called as you see underneath the line of perfectly equal distribution. Okay, in this line of perfectly equal distribution, it's gonna depict the situation with no income inequality. Okay. A situation with no income inequality. So, how would we consider that to be, how would we consider that to be on the graph? Well, that would be a situation if there's no income inequality, 20% of the lowest 20% would be earning 20% of the income, right? The lowest 40% would be earning 40% of the income, right? So it would look something like this. So 20% of the population would earn 20% of the income right here, 40% would earn 40% 60 would earn 60, right? So this is totally equal. Everyone is earning an equal amount here. And we end up with a line that looks straight like this, right? It's just got a slope of one. It's just going straight diagonal, just like that. Okay, So now we've basically plotted everything here. This blue line being our Lorenz curve and this green line being the line of perfect, I'm gonna say of equal distribution line of equal distribution. Okay, so that's how we plot this on the graph. Let's talk about one other extreme situation. So we've got the line of equal distribution there. I just want to draw this in real quick. We don't usually talk about this too much, but the complete inequality. If we were in the other end of the spectrum, complete inequality, that would mean a situation where there's one household earning all the income, right? That's as that's as unequal as it could get, everyone gets zero and one person gets everything. Well, I'm gonna draw it real quick and this is what it would look like. It would be the 1st 20% get zero next 20. Get zero, next 20. Get zero next get zero then finally that last house would get us up to there. So it would be this shape where we're just going up the graph just like that. That would be our line of unequal distribution. Right? Completely complete inequality There. One last topic when we talk about the Lorenz curve is the gini coefficient. Okay, so the gini coefficient, it's a ratio. And this ratio it shows the level of income inequality in the economy. Alright. So it's gonna help us further uh find what the inequality is, right? It's gonna give us a number that we can compare across different nations. All right, So let's think about this geni coefficient. When we have a Gini coefficient of zero, That means that we have completely equal distribution of income. Okay, When the gini coefficient equal zero and when it equals one, we have completely unequal distribution. Okay, zero is equal, one is completely unequal. And you can imagine any number in the middle is going to be that range of inequality. Right? As it gets closer to one, we're getting more unequal and as we get closer to zero we get more equal. Cool. And this is how we're going to calculate the Gini coefficient. It's very simple calculation. It's just A over A plus B. Okay. And let me show you where A. And B are on this graph. So A A. Is gonna be this area right here, in between the Lorenz curve and the line of equal distribution. Okay, so that area is our A and R. B. Is gonna be this area down here, below the Lorenz curve, but under the axis there. Right? So that's gonna be area A and area B. Now for this class, you're not gonna have to calculate these areas at all. Right. This is gonna take some calculus actually to get this these areas um correct. So you're generally just gonna be given a number for a A number for B. And you go ahead and use our formula down here, we're giving this formula A over A plus B is our jenny Kofi. All right. So if you have a number for a if you have a number for B, this is gonna be really, really simple to calculate, right? Just A over A plus B. Cool. Alright, so that's quite a lot of information. We took all of our income inequality discussion and put it on the graph there. Alright, so let's do a little bit of practice and then we'll move on to the next topic

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Problem

Problem

Which of the following would represent the Lorenz curve of an nation where everyone earned equal income?

A

The Lorenz curve would have a negative slope

B

The Lorenz curve would be U-shaped.

C

The Lorenz curve would be a straight line with a slope of 1.

D

The Lorenz curve would not exist.

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Problem

Problem

Which of the following is true regarding the Gini coefficient?

A

The Gini coefficient will only fall when population rises quickly

B

As income inequality rises, the Gini coefficient will fall

C

As income inequality falls, the Gini coefficient will fall

D

The Gini coefficient breaks the population into quintiles based on income

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Problem

Problem

The country of Newland has a Lorenz curve where the area between the line of equal distribution and the Lorenz curve is 0.22 and the area below the Lorenz curve is .46. What is the Gini coefficient for Newland?