Topic 8 - Lorenz Curve and Gini Coefficient PDF

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Lecturer: Guglielmo Volpe...

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ECN125 Economics in Action

Measuring Inequality – The Lorenz Curve – Gini coefficient

About the use of percentiles…..

Measuring Inequality The Lorenz Curve and the Gini Coefficient

The Lorenz Curve • Last week we looked at measures of ranking or percentiles in order to identify the ranking of a particular value in a set of observations • Today we investigate the so-called Lorenz curve • The curve is a tool mainly used to represent income distributions • The curve was originally proposed by the American economist Max Otto Lorenz (1905)

The Lorenz Curve

• The Lorenz curve tells us which proportion of total income is in the hands of a given percentage of population • This method is conceptually very similar to the method by percentiles or quantiles that we looked at last week and today • However, instead of ending up with income shares, the Lorenz Curve relates the cumulative proportion of income to the cumulative proportion of individuals.

Cumulative proportion of income (%)

The Lorenz Curve Income share is calculated by taking the cumulated income of a given share of the population, divided by the total income Y k

åy

æk ö L ç ÷ = i =1 Y èPø

i

k=1….n is the position of each group of individuals in the distribution i=1….k is the position of each individual within group k P is the total number of individuals in the distribution yi is the income of the ith individual in the distribution

ækö Lç ÷ èPø

Ranges from 0 (for k = 0) to 1 for (k = n)

20%

40% 60% 80% 100% Cumulative proportion of population (%)

Cumulative proportion of income (%) 20% 40% 80% 100% 60%

Line of equality

20%

40% 60% 80% 100% Cumulative proportion of population (%)

100% 10% 5% 15% 25%

The Lorenz Curve

20%

40% 60% 80% 100% Cumulative proportion of population (%)

A step by step procedure to build the Lorenz Curve

Step 1

Sort the income distribution by income level

Step 2

Define the proportion of income owned by each individual and his/her proportion on total population

Step 3

Define the cumulative proportion of income and the cumulative proportion of population

Step 4

Plot the cumulative proportion of income against the cumulative proportion of population

Lorenz Curve: example

Individual

Income

1

2,417

2 3 4 5

7,800 8,489 10,072 12,957

Total 41,735

Lorenz Curve: example

Step 2 Proportion of Proportion of income for each individual on total each population individual 0.058 0.200 0.187 0.200 0.203 0.200 0.241 0.310

0.200 0.200

Cumulative proportion of income (%)

Lorenz Curve 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2 0.4 0.6 0.8 Cumulative proportion of population (%)

1

Analysis of the Lorenz Curve

Cumulative proportion of income (%)

Lorenz Curve: discussion More equal distribution of income

More unequal distribution of income

Cumulative proportion of population (%)

Cumulative proportion of income (%)

Lorenz Curve: discussion

Perfect inequality: all income earned by one person

Cumulative proportion of population (%)

Analysis of the Lorenz Curve • The Lorenz curve is a useful tool to analyse (income) inequality • The shape of the Lorenz curve is a convex shape • More inequality in income distribution implies more convex Lorenz curves • In case of perfectly equal income distribution, the Lorenz curve is a straight 45 degree line • In the extreme case that only one person holds all the income, the Lorenz curve is a kinked curve running on zero until the last individual is reached and then jumping to 100 per cent

The Gini Coefficient

The Gini Coefficient • The Gini coefficient is a measure of (usually income) inequality of a distribution • It is defined as a ratio with values between 0 and 1: the numerator is the area between the Lorenz curve and the uniform distribution line while the denominator is given by the area under the uniform distribution line • The Gini coefficient was developed in 1912 by an Italian statistician called Riccardo Gini • The Gini index is the Gini coefficient expressed as a percentage and is equivalent to the Gini coefficient multiplied by 100

Cumulative proportion of income (%)

Gini coefficient for typical income distribution

Gini Coefficient =

A A+B

A

B

Cumulative proportion of population (%)

The Gini Coefficient • The Gini Coefficient is defined as a ratio of the areas on the Lorenz curve diagram • If the area between the line of perfect equality and the Lorenz curve is A and the area below the Lorenz curve is B, then the Gini coefficient is given by:

Gini Coefficient =

A A+ B

• Since A + B = 0.5 then we have that

Gini Coefficient = 2 A = 1 - 2 B

Cumulative proportion of income (%)

Gini coefficient in case of perfect equality…

Gini Coefficient =

0 A =0 = A+B 0+ B

A

B

Cumulative proportion of population (%)

Cumulative proportion of income (%)

Gini coefficient in case of perfect inequality…

Gini Coefficient =

A A =1 = A + B A+ 0

A

B

Cumulative proportion of population (%)

The Gini Coefficient • If the Lorenz curve is defined by the function Y = L(X), then the value of B can be found through integration and the Gini coefficient is given by: 1

Gini Coefficient = 1 - 2ò L ( X )dX 0

The Gini Coefficient • In some cases the Gini coefficient can be calculated without directly knowing the Lorenz curve function • For a population of value yi, i = 1 to n, that are indexed in non decreasing order (yi < yi+1) n æ (n + 1 - i ) yi ç å 1ç Gini Coefficient = n + 1 - 2 i =1 n nç yi ç å è i =1

ö ÷ ÷ ÷ ÷ ø

The Gini Coefficient: Example

(n + 1 - i ) yi

Individual

Income

1

2,417

12085

2

7,800

31200

3

8,489

25467

4

10,072

20144

5

12,957

12957

n æ ö ( n + 1 - i ) yi ÷ ç å 1 ÷ = 1 æç (5 + 1 - 2 101,853 ö÷ = 0.224 Gini Coefficient = ç n + 1 - 2 i =1 n ÷ 5è nç 41,735 ø y ç ÷ i å è ø i =1

The Gini Coefficient: advantages as a measure of inequality • The Gini coefficient's main advantage is that it is a measure of inequality by means of a ratio analysis, rather than a variable unrepresentative of most of the population, such as per capita income or gross domestic product • It can be used to compare income distributions across different population sectors as well as countries • It is sufficiently simple that it can be compared across countries and be easily interpreted • The Gini coefficient can be used to indicate how the distribution of income has changed within a country over a period of time

The Gini Coefficient: advantages as a measure of inequality The Gini coefficient satisfies four important principles: Anonymity: it does not matter who the high and low earners are Scale independence: the Gini coefficient does not consider the size of the economy, the way it is measured, or whether it is a rich or poor country on average Population independence: it does not matter how large the population of the country is Transfer principle: if income (less than the difference), is transferred from a rich person to a poor person the resulting distribution is more equal

The Gini Coefficient: disadvantages as a measure of inequality • The Gini coefficient measured for a large economically diverse country will generally result in a much higher coefficient than each of its regions has individually • Comparing income distributions among countries may be difficult because benefits systems may differ • The measure will give different results when applied to individuals instead of households • The Lorenz curve may understate the actual amount of inequality if richer households are able to use income more efficiently than lower income households

The Gini Coefficient: disadvantages as a measure of inequality • As for all statistics, there will be systematic and random errors in the data • Economies with similar incomes and Gini coefficients can still have very different income distributions. This is because the Lorenz curves can have different shapes and yet still yield the same Gini coefficient • Too often only the Gini coefficient is quoted without describing the proportions of the quantiles used for measurement. As with other inequality coefficients, the Gini coefficient is influenced by the granularity of the measurements

40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0

UK Gini coefficient for disposable income

77 79 81 83 85 87 89 91 /94 /96 /98 /00 /02 /04 /06 19 19 19 19 19 19 19 19 93 95 97 99 01 03 05 19 19 19 19 20 20 20

2013 Gini coefficient in EU

Average EU Gini

Switzerland Norway Iceland United Kingdom Sweden Finland Slovakia Slovenia Romania Portugal Poland Austria Netherlands Malta Hungary Luxembourg Lithuania Latvia Cyprus Italy Croatia France Spain Greece Ireland Estonia Germany Denmark Czech Republic Bulgaria Belg ium 0

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