## Related questions with answers

Imagine an economy with only two people. Larry earns
$20,000 per year, while Roger earns$80,000 per year. As
shown in the following figure, the Lorenz curve for this twoperson economy consists of two line segments. The first runs
from the origin to point *a*, while the second runs from point *a*
to point *b*.

Solutions

Verified$\textbf{a.}$ We know that the Gini ratio is:

$\begin{align*}\text{Gini ratio} = \dfrac{\text{Area between Lorenz curve and diagonal}}{\text{Total area below the diagonal}}\end{align*}$

The total area below the diagonal is:

$\begin{align*}\dfrac{1}{2} \cdot (100) \cdot (100) = 5,000\end{align*}$

The area below the Lorenz curve is:

$\begin{align*}(\dfrac{1}{2} \cdot (50) \cdot (20)) + (\dfrac{1}{2} \cdot (50) \cdot (80)) + (50\cdot20) = 3,500\end{align*}$

Thus, the Gini ratio is:

$\begin{align*}\dfrac{5,000 - 3,500}{5,000} = \textbf{0.3}\end{align*}$

$\textbf{b.}$ The area below the Lorenz curve when $\$20,000$ taxed away from the Roger and given to the Larry:

$\begin{align*}(\dfrac{1}{2} \cdot (50) \cdot (40)) + (\dfrac{1}{2} \cdot (50) \cdot (60)) + (50\cdot40) = 4,500\end{align*}$

Thus, the Gini ratio is:

$\begin{align*}\dfrac{5,000 - 4,500}{5,000} = \textbf{0.1}\end{align*}$

$\textbf{c.}$ When the income doubled, the Larry income becomes $\$40,000$ and Rodger income becomes $\$160,000$:

$\begin{align*}(\dfrac{1}{2} \cdot (50) \cdot (20)) + (\dfrac{1}{2} \cdot (50) \cdot (80)) + (50\cdot20) = 3,500\end{align*}$

Thus, the Gini ratio is:

$\begin{align*}\dfrac{5,000 - 3,500}{5,000} = \textbf{0.3}\end{align*}$

The Gini ratio stays the same even though the income is doubling.

The Gini ratio is the area between the diagonal and the Lorenz curve divided by the whole area under the diagonal.

Let's see the Gini ratio in an economy with only two people.

In this task, we need to analyze the Lorenz curve and the economy for Larry and Roger.

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