Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
use induction to prove the following for all natural numbers n:
![### Lorenz Curve and Gini Coefficient: Measurement of Income Inequality
#### Lorenz Curve
The Lorenz Curve is a graphical representation of income or wealth distribution in a society. It is used to depict the level of economic inequality within a population. The horizontal axis (X-axis) represents the cumulative percentage of total population (from poorest to richest), while the vertical axis (Y-axis) shows the cumulative percentage of total income or wealth.
In an ideal situation where everyone has equal income or wealth, the Lorenz Curve would be a 45-degree line known as the line of equality. However, in reality, the Lorenz Curve typically bows below this line, reflecting some level of inequality. The further the Lorenz Curve is from the line of equality, the greater the inequality in the distribution of income or wealth.
#### Gini Coefficient
The Gini coefficient is a numerical measure derived from the Lorenz Curve and is used to quantify income inequality. The coefficient ranges between 0 and 1:
- A Gini coefficient of 0 indicates perfect equality, where everyone has the same income.
- A Gini coefficient of 1 represents perfect inequality, where one person or household has all the income, and everyone else has none.
#### Calculating the Gini Coefficient
To calculate the Gini coefficient (G), the area between the Lorenz Curve and the line of equality (A) is divided by the total area under the line of equality (A + B):
\[ Gini Coefficient (G) = \frac{A}{A + B} \]
where:
- \( A \) is the area between the Lorenz Curve and the line of equality.
- \( B \) is the area below the Lorenz Curve.
The shaded area in the graph represents the area \( A \). The area below the Lorenz Curve up to the line of equality is the sum of \( A \) and \( B \).
The Lorenz Curve and Gini Coefficient are valuable tools for policymakers and economists to analyze and compare economic inequalities across different populations or over time. By understanding the extent of income disparity, strategies can be implemented to promote more equitable economic growth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14f49134-8e9b-4f9b-94e2-703a3b4a20e5%2F98f72f7d-f903-4681-bbd5-230a297f85cf%2F6daeuv_processed.png&w=3840&q=75)
Transcribed Image Text:### Lorenz Curve and Gini Coefficient: Measurement of Income Inequality
#### Lorenz Curve
The Lorenz Curve is a graphical representation of income or wealth distribution in a society. It is used to depict the level of economic inequality within a population. The horizontal axis (X-axis) represents the cumulative percentage of total population (from poorest to richest), while the vertical axis (Y-axis) shows the cumulative percentage of total income or wealth.
In an ideal situation where everyone has equal income or wealth, the Lorenz Curve would be a 45-degree line known as the line of equality. However, in reality, the Lorenz Curve typically bows below this line, reflecting some level of inequality. The further the Lorenz Curve is from the line of equality, the greater the inequality in the distribution of income or wealth.
#### Gini Coefficient
The Gini coefficient is a numerical measure derived from the Lorenz Curve and is used to quantify income inequality. The coefficient ranges between 0 and 1:
- A Gini coefficient of 0 indicates perfect equality, where everyone has the same income.
- A Gini coefficient of 1 represents perfect inequality, where one person or household has all the income, and everyone else has none.
#### Calculating the Gini Coefficient
To calculate the Gini coefficient (G), the area between the Lorenz Curve and the line of equality (A) is divided by the total area under the line of equality (A + B):
\[ Gini Coefficient (G) = \frac{A}{A + B} \]
where:
- \( A \) is the area between the Lorenz Curve and the line of equality.
- \( B \) is the area below the Lorenz Curve.
The shaded area in the graph represents the area \( A \). The area below the Lorenz Curve up to the line of equality is the sum of \( A \) and \( B \).
The Lorenz Curve and Gini Coefficient are valuable tools for policymakers and economists to analyze and compare economic inequalities across different populations or over time. By understanding the extent of income disparity, strategies can be implemented to promote more equitable economic growth.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning