Chapter 5.3, Problem 69E

(a)

To determine

To show: The coefficient of inequality is equal to $2{\displaystyle \underset{0}{\overset{1}{\int }}\left[x-L\left(x\right)\right]dx}$ .

Explanation of Solution

Given information: Economists use a cumulative distribution called Lorenz curve to describe the distribution of income between households in a given country. Typically, a Lorenz curve is defined on $\left[0,1\right]$ with end points $\left(0,0\right)$ and $\left(1,1\right)$ , and is continuous, increasing and concave upward. The points on this curve are determined by ranking all households by income and then computing the percentage of households whose income is less than or equal to a given percentage of the total income of the country. Absolute equality of income distribution would occur if the bottom $a\%$ of the households receive $a\%$ of income, in which case the Lorenz curve would be the line $y=x$ . The area between the Lorenz curve and the line $y=x$ measures how much the income distribution differs from absolute equality. The coefficient of inequality is the ratio of the area between the Lorenz curve and the line $y=x$ to the area under $y=x$ .

  

Proof:

The coefficient of inequality is the ratio of the area between the Lorenz curve and the line $y=x$ to the area under $y=x$ that is $\frac{{\displaystyle \underset{0}{\overset{1}{\int }}\left(x-L\left(x\right)\right)dx}}{{\displaystyle \underset{0}{\overset{1}{\int }}xdx}}$ .

Simplify the integral.

  $\begin{array}{c}\frac{{\displaystyle \underset{0}{\overset{1}{\int }}\left(x-L\left(x\right)\right)dx}}{{\displaystyle \underset{0}{\overset{1}{\int }}xdx}}=\frac{{\displaystyle \underset{0}{\overset{1}{\int }}\left(x-L\left(x\right)\right)dx}}{{\frac{x^2}{2}|}_0^1}\\ =\frac{{\displaystyle \underset{0}{\overset{1}{\int }}\left(x-L\left(x\right)\right)dx}}{\frac{1}{2}-\frac{0}{2}}\\ =\frac{{\displaystyle \underset{0}{\overset{1}{\int }}\left(x-L\left(x\right)\right)dx}}{\frac{1}{2}}\\ =2{\displaystyle \underset{0}{\overset{1}{\int }}\left(x-L\left(x\right)\right)dx}\end{array}$

Hence, the coefficient of inequality is equal to $2{\displaystyle \underset{0}{\overset{1}{\int }}\left[x-L\left(x\right)\right]dx}$ .

(b)

To determine

To find: The percentage of the total income received by the bottom $50\%$ of the households and also the coefficient of inequality.

Answer to Problem 69E

The percentage of the total income received by the bottom $50\%$ of the households is $39.5\%$ and the coefficient of inequality is equal to $0.138$ .

Explanation of Solution

Given information: The income distribution for a particular country is represented by the Lorenz curve defined by the equation $L\left(x\right)=\frac{5}{12}{x}^2+\frac{7}{12}x$ .

Calculation:

The percentage of total income received by the bottom $50\%$ of the households is $b\%$ if the point $\left(\frac{50}{100},\frac{b}{100}\right)=\left(0.5,0.01b\right)$ lies on the Lorenz curve.

Substitute the point in the Lorenz curve.

  $\begin{array}{c}L\left(x\right)=\frac{5}{12}{x}^2+\frac{7}{12}x\\ 0.01b=\frac{5}{12}{\left(0.5\right)}^2+\frac{7}{12}\left(0.5\right)\\ 0.01b=\frac{5}{12}\left(0.25\right)+\frac{3.5}{12}\\ 0.01b=\frac{19}{48}\end{array}$

Further simplify.

  $\begin{array}{c}0.01b=\frac{19}{48}\\ b=\frac{19}{48\times 0.01}\\ b=39.5\end{array}$

So, the percentage of the total income received by the bottom $50\%$ of the households is $39.5\%$ .

As the coefficient of inequality is given by integral $2{\displaystyle \underset{0}{\overset{1}{\int }}\left[x-L\left(x\right)\right]dx}$ .

Substitute $\frac{5}{12}{x}^2+\frac{7}{12}x$ for $L\left(x\right)$ in the integral.

  $\begin{array}{c}2{\displaystyle \underset{0}{\overset{1}{\int }}\left[x-L\left(x\right)\right]dx}=2{\displaystyle \underset{0}{\overset{1}{\int }}\left[x-\left(\frac{5}{12}{x}^2+\frac{7}{12}x\right)\right]dx}\\ =2{\displaystyle \underset{0}{\overset{1}{\int }}\left[\frac{5}{12}x-\frac{5}{12}{x}^2\right]dx}\\ =2\left({\frac{5}{12}\times \frac{x^2}{2}-\frac{5}{12}\times \frac{x^3}{3}|}_0^1\right)\\ =2\left(\frac{5}{12}\left(\frac{1}{2}\right)-\frac{5}{12}\left(\frac{1}{3}\right)-\frac{5}{12}\left(0\right)+\frac{5}{12}\left(0\right)\right)\end{array}$

Further simplify.

  $\begin{array}{c}2{\displaystyle \underset{0}{\overset{1}{\int }}\left[x-L\left(x\right)\right]dx}=2\left(\frac{5}{12}\left(\frac{1}{2}\right)-\frac{5}{12}\left(\frac{1}{3}\right)-\frac{5}{12}\left(0\right)+\frac{5}{12}\left(0\right)\right)\\ =2\left(\frac{5}{24}-\frac{5}{36}\right)\\ =2\left(\frac{15-10}{72}\right)\\ =0.138\end{array}$

So, the coefficient of inequality is equal to $0.138$ .

Therefore, the percentage of the total income received by the bottom $50\%$ of the households is $39.5\%$ and the coefficient of inequality is equal to $0.138$ .