The following income distribution data define points on Lorenz curves, where x represents the fraction of a certain country's population and L(x) the cumulative percent of income held by fraction x. Use this data and a numerical method to evaluate and hence to find the Gini coefficient of income for this country in 1995 and 2003. Round your answers to three decimal places.
A)
B)
C)
D)
E)

Question

The following income distribution data define points on Lorenz curves, where x represents the fraction of a certain country's population and L(x) the cumulative percent of income held by fraction x. Use this data and a numerical method to evaluate  and hence to find the Gini coefficient of income for this country in 1995 and 2003. Round your answers to three decimal places. 
A)​ ​ 
B)​ ​ 
C)​ ​ 
D)​ ​ 
E)​ ​​ ​

Quizplus · Accepted Answer

Using the trapezoidal rule to estimate the definite integral of L(x) from 0 to 1, we get:

- For 1995: 0.5 * [(0+0.1) * 0.04 + (0.1+0.2) * 0.09 + ... + (0.9+1) * 0.36] = 0.605
- For 2003: 0.5 * [(0+0.1) * 0.034 + (0.1+0.2) * 0.086 + ... + (0.9+1) * 0.366] = 0.581

Then, we use the formula:

G = 2 * (0.5 - AUC)

where AUC is the area under the Lorenz curve, to calculate the Gini coefficient:

- For 1995: G = 2 * (0.5 - 0.605) = 0.190
- For 2003: G = 2 * (0.5 - 0.581) = 0.166

Therefore, the best choice is C, with the correct values for the Gini coefficient in 1995 and 2003 rounded to three decimal places.

The Following Income Distribution Data Define Points on Lorenz Curves