Chapter 6.2, Problem 75E

a.

To determine

To explain: The reason why the line $y=x$ is called the line of equality.

b.

To determine

To explain: The reason why the Lorentz curve satisfies the conditions $L\left(0\right)=0$, $L\left(1\right)=1$, $L\left(x\right)\le x$ and $L^{\prime }\left(x\right)\ge 0$ on $\left[0,1\right]$.

c.

To determine

To sketch: The graph of the Lorentz curves $L\left(x\right)=x^p$ corresponding to $p=1.1,1.5,2,3,4$ and find the value of p which corresponds to the most equitable and to the least equitable distribution of wealth.

d.

To determine

To show: The Gini Index is $G=2A=1-2{\displaystyle \underset{0}{\overset{1}{\int }}L\left(x\right)dx}$.

e.

To determine

To compute: The Gini Index for the cases $L\left(x\right)=x^p$ corresponding to $p=1.1,1.5,2,3,4$

f.

To determine

To find: The smallest interval $\left[a,b\right]$ on which values of Gini Index lie for $L\left(x\right)=x^p$ with $p\ge 1$ and the endpoints of $\left[a,b\right]$ correspond to the least and most equitable distribution of wealth.

g.

To determine

To find: The Gini Index lie for $L\left(x\right)=\frac{5x^2}{6}+\frac{x}{6}$ and show that it satisfies the conditions $L\left(0\right)=0$, $L\left(1\right)=1$, $L\left(x\right)\le x$ and $L^{\prime }\left(x\right)\ge 0$ on $\left[0,1\right]$.