Chapter 23, Problem 2P

Sub part (a):

To determine

The Gini ratio.

Explanation of Solution

Figure 1 illustrates the Lorenz curve.

In Figure 1, the horizontal axis measures the percentage of population and the vertical axis measures the percentage of income. In this, we find the area under the Lorenz curve. Thus, the area of triangle can be calculated as follows:

$\begin{array}{l}\text{Area of the triangle }=\left(\frac{\text{1}}{\text{2}}\right)\times \text{ Base }\times \text{ Height}\\ =\left(\frac{\text{1}}{\text{2}}\right)\times \text{ 50 }\times \text{ 20}\\ =\text{ 500}\end{array}$

Thus, the area of triangle is 500.

The base of the triangle is 50 (at 50%) because there are only two people in the economy. The height of the triangle is 20 (at 20%) because the lowest income is $20,000, which is 20% of the total income earned in society (= ($20,000 / $100,000) x 100).

The area of the rectangle of B can be calculated as follows:

$\begin{array}{l}\text{The area of rectangle B }=\text{ Base }\times \text{ Height}\\ =\text{ 50 }\times \text{ 20}\\ =\text{ 1,000}\end{array}$

Thus, the area of the rectangle B value is 1,000.

The area of the rectangle of C can be calculated as follows:

$\begin{array}{l}\text{The area of triangle C = }\left(\frac{\text{1}}{\text{2}}\right)\times \text{ Base }\times \text{ Height}\\ =\left(\frac{\text{1}}{\text{2}}\right)\times \text{ 50 }\times \text{ 80}\\ =\text{ 2,000}\end{array}$

The area of triangle C is 2,000.

The total area under the Lorenz curve can be calculated as follows:

$\begin{array}{c}\text{Total area }=\text{ Area of triangle A }+\text{ Area of triangle B }+\text{ Area of triangle C}\\ =\text{500 }+\text{ 1,000 }+\text{ 2,000}\\ =\text{3,500}\end{array}$

The total area value is 3,500.

The Gini ratio can be calculated as follows:

$\begin{array}{c}\text{Gini ratio}=\frac{\left(\begin{array}{l}\text{The area under the equality curve}\\ -\text{The area under lorenz curve}\end{array}\right)}{\text{The area under the equality curve}}\\ =\frac{\text{5,000}-3,\text{500}}{\text{5,000}}\\ =\frac{1,\text{500}}{\text{5,000}}\\ =\text{0}\text{.3}\end{array}$

Thus, the Gini ratio is 0.300.

Economics Concept Introduction

Concept introduction:

Gini ratio: It is a measure of statistical scattering that is intended to represent the income distribution of a nation's residents and is most commonly used to measure inequality.

Lorenz curve: It is the graphical illustration of the income distribution or of the wealth.

Sub Part (b):

To determine

The Gini ratio in the scenario 2.

Explanation of Solution

After the tax and redistribution program Larry's new income is $40,000 and Roger's new income is $60,000. This implies that Larry (50% of the population) now controls 40% of the society's income rather than the 20% before the program.

The area of the triangle A can be calculated as follows:

$\begin{array}{c}\text{Triangle A}=\left(\frac{\text{1}}{\text{2}}\right)\times \text{Base}\times \text{Height}\\ =\left(\frac{\text{1}}{\text{2}}\right)\times \text{50}\times \text{40}\\ =\text{1,000}\end{array}$

The value of the triangle area is 1,000.

The value of rectangle B can be calculated as follows:

$\begin{array}{c}\text{Area of rectangle B }=\text{ Base}\times \text{Height}\\ =\text{50}\times \text{40}\\ =\text{2,000}\end{array}$

The area of rectangle B is 2,000.

The area of rectangle C can be calculated as follows:

$\begin{array}{c}\text{Aea of triangle C}=\left(\frac{\text{1}}{\text{2}}\right)\times \text{Base}\times \text{Height}\\ =\left(\frac{\text{1}}{\text{2}}\right)\times \text{50}\times \text{60}\\ =\text{1,500}\end{array}$

The area of triangle C can be calculated as follows:

The total area under the Lorenz curve can be calculated as follows:

$\begin{array}{c}\text{Total area }=\text{ Area of triangle A }+\text{ Area of triangle B }+\text{ Area of triangle C}\\ =\text{1,000 }+\text{ 2,000 }+\text{ 1,500}\\ =\text{4,500}\end{array}$

The total area value is 4,500.

The Gini ratio can be calculated as follows:

$\begin{array}{c}\text{Gini ratio}=\frac{\left(\begin{array}{l}\text{The area under the equality curve}\\ -\text{The area under lorenz curve}\end{array}\right)}{\text{The area under the equality curve}}\\ =\frac{\text{5,000}-4\text{,500}}{\text{5,000}}\\ =\frac{500}{\text{5,000}}\\ =\text{0}\text{.1}\end{array}$

$\begin{array}{l}\text{Then the area between the two curves, which is calculated by}\\ =\left(\begin{array}{l}\text{The area under the equality curve }-\\ \text{The area under the Lorenz curve}\end{array}\right)\\ =\text{ 5,000 - 4,500}\\ =\text{ 500}\end{array}$

Thus the Gini ratio here is 0.100.

Economics Concept Introduction

Concept introduction:

Gini ratio: It is a measure of statistical scattering that is intended to represent the income distribution of a nation's residents and is most commonly used to measure inequality.

Lorenz curve: It is the graphical illustration of the income distribution or of the wealth.

Sub part (c):

To determine

Change in the Gini ratio with change in income.

Explanation of Solution

If the income is double, Larry will earn $40,000 and Roger will earn $160,000. The total income in society is now $200,000. Larry still only controls 20% of the society's income ((=$40,000/$200,000) x 100).

Since the doubling of income does not change the percentage of income that is controlled by Larry (or Roger) , the Gini ratio will not change. Thus, the answer will be the same as in part 'a' at 0.300. Thus, the Gini ratio remains the same as it was before and after the income doubles.