Question

Consider a country with a population of 10: 5 blue people and 5 green people. Each green
person has an income of $1 per year. Each blue person has an income of $3 per year.
(a) Draw the Lorenz curve for this country.
(b) On your diagram for part (a), indicate clearly what area you would divide by what other
area to calculate the Gini coefficient.
(c) Calculate the Gini coefficient. [Hint: the easiest way to do this is to draw some triangles
in your graph of the Lorenz curve and compute the parts of the Gini coefficient based
on the areas of the triangles.]

Answer 1

Total Population = 10

5 blue people = 50% of the people

5 green people = 50% of the people

1 Green person income = $ 1 per year

5 Green Persons total income = $1 x 5 = $5 per year

1 Blue person income = $ 3 per year

5 Blue Persons total income = $3 x 5 = $15 per year

Total income in the country = $5 + $15 = $20 per year

50% of the population has 25% of the income (represented by Green people) which are the poorest

Remaining 50% of the population has 75% of the income (represented by Blue people)

Lorenz curve is a graphical representation of the distribution of income, consumption in the country whereas Gini coefficient is a mathematical indicator of inequality, which can be easily related to the Lorenz curve.

(a) Let's plot this on a graph as in the attached image. We always plot the poorest first and then the ones better off so in the graph we have plotted Green people first (marked ad G in the image) and then the blue people (marked as R in the image) G has (25,50) as cordinates and R has (100,100) The reason R has 100,100 as coordinates is because it is a cumulative figure that is taken as we move from poor to rich in a country so R = (50+50) as polulation and (25+75) as income.

Just for your knowledge, the stratight line (45degree) represents perfect equality or the line of absolute equality and the farther the lorenz curve is from the straight line the more the inequality in the country.

(b) It is clearly indicated in the graph as area A and area B

Section A highlighted as Grey and Section B is the area below that.

Gini Coefficient = Section A/ Section (A+B)

we can say that Gini coefficient is the area that lies between the line of equality and the Lorenz curve (marked A in the diagram) over the total area under the line of equality (marked A and B in the diagram)

(c) See image 2 for clarity parallel to the below details:

Based on the abover formula, let's calculate each of the areas:

Section (A+B) is a right angle triangle and area of an right angle = (hxb)/2 (as reflected by blue lines in the image2)

From the diagram above it is clear that h = 100 , b = 100

Area of Section (A+B) = (100 x 100)/ 2 = 5000

Now let's calculate area of Section A

We will draw a triangle to ease of the calculation as reflected by pink lines.

It is an acute triangle and area of an actute triangle = (hxb)/2

Where height is 25 (see red dotted lines) and base is 100

Area of Section A = (25x100)/2 = 1250

Gini Coefficient = Section A/ Section (A+B)

= 1250/5000

Gini's coeffient will be 0.250

Images attached, Image-1, answers part a and b and Image 2 answers part c