The government of a small country is planning sweeping changes in the tax structure in order to provide a more equitable distribution of income. The Lorenz curves for the current income distribution and for the projocted income distribution after enactment of the tax changes are as follows.
\[
\begin{array}{l}
f(x)=x^{2.3} \\
g(x)=0.7 x+0.3 x^{2}
\end{array}
\]
Current Lorenz curve
Projected Lorenz curve after changes in tax laws
Find the Gini index of income concentration for each Lorenz curve. Will the proposed changes provide a more equitable income distribution? Explain.
Identify the integrand for the computation of the current and projected Gini indices.
The current Gini index is given by $2 \int_{0}^{1}(\square) d x$ and the projected Gini index is given by $2 \int_{0}^{1}(\square) d x$
Answer
Since the Gini index for the projected income distribution is lower than the current one, the proposed changes will provide a more equitable income distribution.
Step 1 ：The Lorenz curves for the current income distribution and for the projected income distribution after enactment of the tax changes are given by the functions \(f(x) = x^{2.3}\) and \(g(x) = 0.7x + 0.3x^{2}\) respectively.
Step 2 ：The Gini index is a measure of inequality of a distribution. It is defined as a ratio with values between 0 and 1: a low Gini coefficient indicates more equal income or distribution, while a high Gini coefficient indicates more unequal distribution.
Step 3 ：The Gini index is calculated as twice the area between the Lorenz curve and the line of perfect equality. This can be calculated as 1 minus twice the integral of the Lorenz curve from 0 to 1.
Step 4 ：Let's calculate the Gini indices for the current and projected income distributions.
Step 5 ：The Gini index for the current income distribution is calculated as \(1 - 2 \int_{0}^{1} f(x) dx\), where \(f(x) = x^{2.3}\).
Step 6 ：The Gini index for the projected income distribution is calculated as \(1 - 2 \int_{0}^{1} g(x) dx\), where \(g(x) = 0.7x + 0.3x^{2}\).
Step 7 ：The Gini index for the current income distribution is approximately \(\boxed{0.394}\).
Step 8 ：The Gini index for the projected income distribution is approximately \(\boxed{0.1}\).
Step 9 ：Since the Gini index for the projected income distribution is lower than the current one, the proposed changes will provide a more equitable income distribution.
