The Gini Index

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# The Gini Index - PowerPoint PPT Presentation

The Gini Index. Paige Stillwell and Tanya Picinich. Overview . Lorenz Curve Perfect Income Equality / Complete Income Inequality Gini Index Calculation Examples United States Gini Index Riemann Sum and Trapezoidal Rule Potential Issues with the Gini Index

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## The Gini Index

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### The Gini Index

Paige Stillwell

and

Tanya Picinich

Overview
• Lorenz Curve
• Perfect Income Equality / Complete Income Inequality
• Gini Index
• Calculation Examples
• United States Gini Index
• Riemann Sum and Trapezoidal Rule
• Potential Issues with the Gini Index
• Why is the Gini Index Important?
• Gini Index Comparison Across Countries
• United States Gini Index Over Time
The Lorenz Curve
• Shows the share of total income of the population from 0 to t where t is the rank of a household’s income as a percentage of the total population
Complete Inequality
• One person makes all the money. Everyone else makes nothing
Gini Index
• Gives information about the income inequality of a country in one number
• Ranges from 0 to 1
• Calculated as the area between perfect equality (y=x) and the Lorenz curve
The Gini Index for perfect equality is 0
• The Gini Index for total inequality is 1
Example: L(t) = t2
• What’s the Gini Index for a country that has a Lorenz curve of L(t) = t2 ?
Example: L(t) = t3
• Will the Gini Index increase or decrease from the previous example if the equation changes to L(t) = t3?
Reality: How the Gini Index is Calculated
• In real life we are not given functions
• We must use data points to find the Gini Index
• We use income quintiles which are made available by the U.S. Census Bureau
The United States Gini Index 2007
• The United States Gini Index for 2007 is quoted by the U.S. Census Bureau as G(t)=.463
• This value may vary depending on what is considered as income and whether individuals or households are examined
Riemann Sum and Trapezoidal Rule
• Approximates an integral when the equation of the function is unknown
• Using the 2007 data we can approximate the Gini Index
• Riemann sums use either right or left endpoints to form rectangles
• The trapezoidal rule is the average of the right and left endpoint approximations
Riemann Sum
• The definition of an integral of f from a to b is:
• Using right end points:
• Using left end points:
Riemann Sum
• Using right end points:
• Using left end points:
Trapezoidal Rule
• Basically an average of the right and left endpoints from the Riemann Sum
• Using the 2007 Income Quintiles as data points we get:
Trapezoidal Rule

Actual Gini Index = .463

Potential Issues
• The Gini Index glosses over many details
• Gives a more accurate picture of the relationship between the upper class and middle class than the relationship between the upper class and lower class
• Does not reflect unreported income and money received through welfare systems
Jackknife

Calculate the Gini Index many times, but remove one data point each time

Produces a mean distribution and a standard deviation for the Gini Index

Bootstrap

Calculate the Gini Index from a random sample of the income data many times

Produces a mean distribution and a standard deviation for the Gini Index

Ways to calculate a more accurate Gini Index:
Why is the Gini Index so important?
• Compiles information about income inequality into 1 number
• Allows for comparisons with other countries
• Shows how income inequality changes over time
• This information has great social, political, and economic implications
Comparisons Across Countries
• The Gini index for the entire world = 56 - 66
• Tolerance for inequality of income varies between countries
• US = higher tolerance for income inequality
• European countries = lower tolerance for income inequality
• Underdeveloped countries have a higher income inequality