Using data from a nation's census, an economist produced the following Lorenz curves for the distribution of that nation's income in 1962 and 1972.
\[
\begin{array}{ll}
f(x)=\frac{3}{10} x+\frac{7}{10} x^{2} & \text { Lorenz curve for } 1962 \\
g(x)=\frac{3}{5} x+\frac{2}{5} x^{2} & \text { Lorenz curve for } 1972
\end{array}
\]
Find the Gini index of income concentration for each Lorenz curve and interpret the results.
Identify the integrand for the computation of the Gini index for 1962 and 1972.
The Gini index for 1962 is given by $2 \int_{0}^{1} \square d x$ and the Gini index for 1972 is given by $2 \int_{0}^{1}(2) d x$.
Answer
To calculate the Gini index from a Lorenz curve, we need to understand that the Gini index is a measure of income inequality within a population, ranging from 0 (perfect equality) to 1 (perfect inequality). It is calculated as the area between the line of perfect equality (which would be the line y=x in a Lorenz curve graph) and the Lorenz curve, divided by the total area under the line of perfect equality. For the Lorenz curve of 1962, represented by the function $f(x)=\frac{3}{10} x+\frac{7}{10} x^{2}$, the integrand for the Gini index calculation would be the difference between the line of perfect equality and the Lorenz curve, which is $x - f(x) = x - (\frac{3}{10} x+\frac{7}{10} x^{2})$. Therefore, the Gini index for 1962 is $2 \int_{0}^{1} (x - \frac{3}{10} x - \frac{7}{10} x^{2}) dx$. For the Lorenz curve of 1972, represented by the function $g(x)=\frac{3}{5} x+\frac{2}{5} x^{2}$, the integrand for the Gini index calculation would be $x - g(x) = x - (\frac{3}{5} x+\frac{2}{5} x^{2})$. Therefore, the Gini index for 1972 is $2 \int_{0}^{1} (x - \frac{3}{5} x - \frac{2}{5} x^{2}) dx$. To interpret the results, we would calculate the definite integrals for both equations. The resulting values would give us the Gini index for each year. A higher Gini index indicates greater income inequality. By comparing the Gini indices for 1962 and 1972, we can determine whether income inequality increased or decreased during that decade. It is important to note that while the Gini index is a widely used measure of inequality, it does not capture all aspects of income distribution, such as the depth or severity of poverty, and should be considered alongside other metrics for a comprehensive understanding of economic conditions.
Step 1 ：To calculate the Gini index from a Lorenz curve, we need to understand that the Gini index is a measure of income inequality within a population, ranging from 0 (perfect equality) to 1 (perfect inequality). It is calculated as the area between the line of perfect equality (which would be the line y=x in a Lorenz curve graph) and the Lorenz curve, divided by the total area under the line of perfect equality. For the Lorenz curve of 1962, represented by the function $f(x)=\frac{3}{10} x+\frac{7}{10} x^{2}$, the integrand for the Gini index calculation would be the difference between the line of perfect equality and the Lorenz curve, which is $x - f(x) = x - (\frac{3}{10} x+\frac{7}{10} x^{2})$. Therefore, the Gini index for 1962 is $2 \int_{0}^{1} (x - \frac{3}{10} x - \frac{7}{10} x^{2}) dx$. For the Lorenz curve of 1972, represented by the function $g(x)=\frac{3}{5} x+\frac{2}{5} x^{2}$, the integrand for the Gini index calculation would be $x - g(x) = x - (\frac{3}{5} x+\frac{2}{5} x^{2})$. Therefore, the Gini index for 1972 is $2 \int_{0}^{1} (x - \frac{3}{5} x - \frac{2}{5} x^{2}) dx$. To interpret the results, we would calculate the definite integrals for both equations. The resulting values would give us the Gini index for each year. A higher Gini index indicates greater income inequality. By comparing the Gini indices for 1962 and 1972, we can determine whether income inequality increased or decreased during that decade. It is important to note that while the Gini index is a widely used measure of inequality, it does not capture all aspects of income distribution, such as the depth or severity of poverty, and should be considered alongside other metrics for a comprehensive understanding of economic conditions.
