Question

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 projected income distribution after enactment of the tax changes are as follows.
$f(x)=x^{2.2}$ current lorenz curve
$g(x)=0.7x + 0.3x^{2}$ 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 $2\int_{0}^{1}(\square)dx$ and the projected gini index is given by $2\int_{0}^{1}(\square)dx$.

Explanation:

Step1: Recall the formula for Gini index

The formula for the Gini index is \( G = 2\int_{0}^{1}(x - L(x))dx\), where \(L(x)\) is the Lorenz curve.

Step2: Find the integrand for the current Gini index

For the current Lorenz curve \(f(x)=x^{2.2}\), the integrand is \(x - f(x)=x - x^{2.2}\).

Step3: Find the integrand for the projected Gini index

For the projected Lorenz curve \(g(x)=0.7x + 0.3x^{2}\), the integrand is \(x - g(x)=x-(0.7x + 0.3x^{2})=x - 0.7x-0.3x^{2}=0.3x - 0.3x^{2}\)

Answer:

The integrand for the current Gini index is \(x - x^{2.2}\) and for the projected Gini index is \(0.3x - 0.3x^{2}\)