The Gini Coefficient of wealth distribution

In a "perfect" society, everyone would share wealth (or income) evenly. (Don't worry - this is an article about math, not communism!)

The Gini Coefficient is one way to measure how evenly the income (or wealth) is distributed throughout a country.

The Gini Coefficient is calculated as follows. We find out the income of all the people in a country and then express this information as a cumulative percentage of people against the cumulative share of income earned. This gives us a Lorenz Curve which typically looks something like the following.

In plain English, the graph above indicates the proportion of the income going to the poorest people, middle-income people and richest people.

There will always be rich and poor, but we are interested in how evenly wealth is distributed and most governments put effort into keeping this coefficient as low as possible.

The Gini Coefficient ranges between 0 and 1 (or it can also be expressed as a number from 0 to 100) and is given by the ratio of the areas:

If A = 0, it means the Lorenz Curve is actually the Line of Equality. In this case, the Gini Coefficient is 0 and it means there is "perfect" distribution of income (everyone earns the same amount).

If A is a very large area (making B very small), then the Gini Coefficient is large (almost 1) and it means there is very uneven distribution of income. Countries with a high Gini Coefficient are more likely to become unstable, since there is a large mass of poor people who are jealous of the small number of rich people.

What Does it Mean?

Let's try to understand the above graph.

For example, say we have 10 people in a village and the income for the village is $100 per day. If every person shares this income evenly, they get $10 each per day.

So the income distribution would be as follows. ("Cumulative" just means add up the number you have so far for each step.)

Person	Proportion of population (%)	Cumulative proportion of population (%)	Income (%)	Cumulative income (%)
A	10%	10%	10%	10%
B	10%	20%	10%	20%
C	10%	30%	10%	30%
D	10%	40%	10%	40%
E	10%	50%	10%	50%
F	10%	60%	10%	60%
G	10%	70%	10%	70%
H	10%	80%	10%	80%
I	10%	90%	10%	90%
J	10%	100%	10%	100%

So for this society with perfectly-distributed income, we could draw a graph of the cumulative proportuion of population (on the horizontal axis) against the cumulative percentage of income (on the vertical axis) as follows.

In the above case, A = 0 so the Gini Coefficient is 0.

Now, people being people, some of the villagers decide they should be paid more because they work harder, or because they are older, or because they have more children, or whatever. So three of them (persons H, I and J) decide to keep 15% of the income each, and distribute the rest evenly among the others. However, that doesn't work out evenly, so they decide the laziest 3 people in the village (persons A, B and C) should only get 5% of the income. Our table now looks like this:

Person	Proportion of population (%)	Cumulative proportion of population (%)	Income (%)	Cumulative income (%)
A	10%	10%	5%	5%
B	10%	20%	5%	10%
C	10%	30%	5%	15%
D	10%	40%	10%	25%
E	10%	50%	10%	35%
F	10%	60%	10%	45%
G	10%	70%	10%	55%
H	10%	80%	15%	70%
I	10%	90%	15%	85%
J	10%	100%	15%	100%

Let's graph it and see what it looks like.

In summary, the bottom 30% of the population earns 15% of the income, while the top 30% earns 45% of the income.

I've shaded 2 regions in the above graph, region A (with light magenta shading) and region B (with light green shading).

Recall the Gini Coefficient is the ratio of the areas:

Area A = 0.095 (from calculating area B - one triangle and 2 trapezoids - and subtracting it from 0.5)

Area (A + B) = 0.5 (this is half of the rectangle)

So the Gini Coefficient in this case is:

Let's take it another step. The three richer guys (H, I and J) have a fight and J wins. He demands 50% of the income and leaves it to H and I to distribute the rest.

Then H and I have a fight and I wins. He wants 33% and gives 10% to H and they decide to give what's left (1% or $1 a day) to each of the rest of the village.

(Millions of people live on less than $1 per day.)

Person	Proportion of population (%)	Cumulative proportion of population (%)	Income (%)	Cumulative income (%)
A	10%	10%	1%	1%
B	10%	20%	1%	2%
C	10%	30%	1%	3%
D	10%	40%	1%	4%
E	10%	50%	1%	5%
F	10%	60%	1%	6%
G	10%	70%	1%	7%
H	10%	80%	10%	17%
I	10%	90%	33%	50%
J	10%	100%	50%	100%

Now we have a very uneven income distribution. The bottom 70% of the population earn only 7% of the income, while the top 30% earn 93% of the income.

Here's the graph.

The Gini Coefficient for this situation is very high:

Finally, let's take the extreme case, where "Person J" becomes a dictator and decides all the income should go to him and everyone else gets nothing.

The cumulative income is 0% for Persons A to I, then it jumps up to 100% for Person J. Here's the graph.

This time area A is very large and the Gini Coefficient is:

Why isn't it equal to 1?

The highest possible Gini Coefficient is 1 and this implies 1 person gets all the income.

In our story, we only have 10 people in our example population. If there were, say, 100 million people in the country, and one person had all the income, then the Gini Coefficient would be 0.999999, or very close to 1.

Using Calculus to find the Gini Coefficient

The above story is simplified and with a large data set, the Lorenz Curve will appear to be a curve, not a series of straight lines.

This time I have modeled the Lorenz curve using:

Cumulative share of income = (cumulative share of people)⁵

(I've used this since the curve y = x⁵ is close to the shape of the curve we have in the above graph.)

If we use I (for income) and P (for people), this would be written

I = P⁵

We find the area A using the following:

This gives:

So the Gini Coefficient in this case is very high, at:

Gini Coefficients in Various Countries

These are sorted highest (worst equality) to lowest (best equality).

Country	UN Gini Coefficient	Rank
Namibia	0.743	1
Sierra Leone	0.629	3
Haiti	0.592	7
South Africa	0.578	10
China	0.469	34
Singapore	0.425	51
United States	0.408	58
India	0.368	76
United Kingdom	0.360	83
Australia	0.352	86
Japan	0.249	110
Denmark	0.247	111

China's coefficient is quite high and this is causing a lot of concern. The Eastern provinces are now well-developed and responsible for most of the income growth, whereas the rural west is still quite poor.

You can see the full list here: Gini Coefficient by Country.

Singapore's Case

Here's the coefficient for Singapore over the last decade. The rapid rise from 2002 and spike in 2007 were due to several factors, including rapid population increases (through immigration) of higher-income people, and a subsequent boost in the overall economy.

The drop in 2008 and 2009 is due to the Global Financial Crisis, where many high-paying jobs either disappeared, or bonuses were slashed.

Information source for graph: Straits Times, Singapore

Further Reading

You may also be interested in: Database of Happiness.

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