Find the Gini index of income concentration for the Lorenz curve with equation y=x^2 e^x-1

Name: Problem 70, Chapter 6: Calculus for Business, Economics, Life Sciences, and Social Sciences
Uploaded: 2021-04-06T16:56:57-08:00
Duration: 4 min 28 s
Channel: Angela Guo
Description: Find the Gini index of income concentration for the Lorenz curve with equation y=x^2 e^x-1

Find the Gini index of income concentration for the Lorenz...

Key Concepts

Lorenz Curve

A Lorenz curve is a graphical representation of the cumulative distribution of income or wealth, showing the proportion of total income earned by the bottom x% of the population. It is a crucial tool in economics for visualizing inequality: a perfectly equal distribution is represented by the 45-degree line, while deviations from this line indicate inequality.

Gini Index

The Gini index (or Gini coefficient) is a numerical measure of income inequality within a population. It is calculated as the ratio of the area between the line of perfect equality and the Lorenz curve over the total area under the line of perfect equality, with values ranging from 0 (perfect equality) to 1 (maximum inequality).

Integral Calculus in Inequality Measurement

Integral calculus is used to compute the areas required to calculate the Gini index. By integrating the Lorenz curve function over the interval from 0 to 1, one can determine the area under the curve, which is then compared with the area under the line of equality to quantify income inequality.

Transcript

00:01 We're given the lorenz curve with the equation y is equal to x squared times e to the x minus 1.

00:11 Now we're asked to find the genie index of income concentration.

00:15 So let's denote this equation as a function f of x.

00:21 Then the jinny index is going to be so g is equal to 0 .5 minus the integral from the gney index is going to be, so g is equal to 0 .5 minus the integral from 0 to 1 of our function, so which is our lorentz curve, and divided by 0 .5.

00:40 So this tells us we need to find this integral first.

00:43 So let's go ahead and do that on the side.

00:47 So the integral from 0 to 1 of f of x, which is x squared e to the x minus 1, is equal to, and to do that we're going to need to do integration by parts twice.

01:00 So the first time we're going to let u be equal to x squared, which means dv is going to be e to the x minus 1 dx.

01:09 So now du is 2x dx and v is e to the x minus 1.

01:17 So this gives us using our integration by parts formula, this will be x squared, e to the x minus 1 evaluated from 0 to 1, minus the integral from 0 to 1, minus the integral from 0 to 1, of v times d u so that would be you can pull the two to the front and that would be x e to the x minus 1 d x okay so now we can see that this part requires us to do integration by parts again so let's do this part so we're going to let you be equal to x here which means dv is still e to the x minus 1 d x and now du is equal to just the x, and v is still e to the x minus 1 dx.

02:06 So now putting that in, we're going to get, so we can actually evaluate from 0 to 1 here.

02:13 So when we put in 1, this is 1 times e to the 0, which is just 1.

02:18 And then if you put in 0, that's just 0.

02:20 So this, this part here, is equal to 1.

02:23 And then this is minus two times, and then we'll put this in bracket.

02:29 So now integration by parts tells us we need u times v.

02:33 So that's xb to the x minus 1 from 0 to 1 minus the integral of vdu.

02:42 So that's e to the x minus 1 d x...