Note that the information in the Lorenz curve is often summarized in a single measure called Gini index. Consider the diagram of Lorenz curve. Suppose the area between the curve and is A and the area of the region between and is Therefore, the Gini index is given by

According to the diagram, consider the area This is the area under the line and within the interval So, the area is

Sol8:

To calculate the Gini index using the diagram of the Lorenz curve, you first need to determine the areas mentioned.

Let\'s denote the x-axis as the cumulative percentage of the population and the y-axis as the cumulative percentage of income or wealth.

The area between the curve and the line of perfect equality (the diagonal line from the origin to the point [1, 1]) represents the extent of income or wealth inequality in the population. This area is denoted as A.

The area between the line of perfect equality and the curve represents the remaining portion of the population, indicating the level of income or wealth distribution that would need to be redistributed to achieve perfect equality. This area is denoted as B.

To calculate the Gini index, you divide the area B by the sum of areas A and B:

Gini Index = B / (A + B)

Therefore, the area B is given by:

B = 0.5 - A

Since the area mentioned in your question refers to the area under the line between two specific points within the interval, you can calculate it by finding the area of the trapezoid formed by those points.

Remember that the Lorenz curve is a plot of cumulative proportions. Thus, if the interval you mentioned is between x1 and x2, you need to find the corresponding y-values on the Lorenz curve at those points. Let\'s denote them as y1 and y2.

Then, the area you are referring to, denoted as C, is given by:

C = (x2 - x1) * (y1 + y2) / 2

Now that you have the value of C, you can substitute it into the equation for B:

B = 0.5 - C

Finally, you can calculate the Gini index using the formula mentioned earlier:

Gini Index = B / (A + B)