To find the Gini index for the given Lorenz curve L(x) = x^3.4, we first need to calculate the area between the line of perfect equality (y = x) and the given Lorenz curve defined by L(x) = x^3.4.
Start by understanding that the line of equality essentially denotes a completely egalitarian society, where each percentage of the population controls the same percentage of total income, forming a straight line from the origin (0,0) to the point (1,1) on a plot.
The Lorenz curve, on the other hand, illustrates the distribution of income. The more bowed the curve, the higher the level of income inequality.
So let's name the area between these two curves as A.
The formula for calculating this area is the integral from 0 to 1 of (x - x^3.4) dx. Use this formula to calculate the value of area A.
Next, use the Gini index formula which is G = 1 - 2 * A to calculate the Gini index.
In this case, A is the area calculated before.
After inserting A into the Gini formula, round your answer to three decimal places.
The final result our calculations give us is approximately 0.455.
That is, the Gini index for the given Lorenz curve L(x) = x^3.4 is approximately 0.455. The Gini index ranges from 0 to 1, where 0 represents perfect equality and 1 represents perfect inequality. So a Gini index of 0.455 shows a moderate level of income inequality.