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https://en.wikipedia.org/wiki/Gini_coefficient

I says that if u people own f portion of the wealth, then the gini index is f-u.

They draw some triangle there

enter image description here

It seems that the area on top of the green line and to the left of the red line is simply .5(f-u)

However, I know no easy way to compute that. Pencil pushing confirm the calculation.

Are there simpler ways?

I know the formula hold if f is equal to 1. When f is less than 1 the area of the triangle shrink proportionally. Well, not really. So yea I am looking for better derivation here.

Note: f + u is NOT 1

It's u people have f incomes.

user4951
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3 Answers3

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The triangle of interest has coordinates $(0,0)$, $(1-u,1-f)$ and $(1,1)$ when visited in counterclockwise order. So compute its area as a determinant using the shoelace formula: $$ \frac12\left| \begin{matrix} 1&1&1\\ 0&1-u&1\\ 0&1-f&1\\ \end{matrix} \right|=\frac12\left((1-u) - (1-f)\right)=\frac12(f-u). $$ Using the shoelace formula you can prove that a triangle with one vertex at the origin and the others at $(a,b)$ and $(c,d)$ will have area $\frac12|ad-bc|$.

grand_chat
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enter image description here

First of all, note that $$ f+u = 1 $$ is assumed.

Now you will notice that the area $A$ can be calculated as $(A+B+C+D) - B - C - D$.

  1. $A+B+C+D$

It is nothing but a right triangle with width 1 and height 1.

Therefore, $$ A+B+C+D = \frac{1}{2}(1 \times 1) = 0.5 $$

  1. $B$ and $C$

It is a right triangle with width $u$ and height $f$.

$$ B = C = \frac{1}{2}(f \times u) $$

  1. $D$

It is square, therefore

$$ D = u^2 $$

Now we have

$$ A = \frac{1}{2} - fu - u^2 $$ $$ = \frac{1}{2} - u(f + u) $$ , and $$ = \frac{1}{2} - u $$ because $f+u =1$.

However, from $1=f+u$, $$ \frac{1}{2} = \frac{1}{2}(f+u) $$ , and we have However, from $1=f+u$, $$ \frac{1}{2} - u = \frac{1}{2}(f-u) $$

K. Ban
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Why didn't I notice this.

Take a line from top left of square D to the left hitting the hypotenuse. The length of that line is f-u. Let's called that segment e

enter image description here

e divides triangle A into 2 regions. e, is a base of those 2 triangles. The sum of their height is 1.

Hence, the area of A is simply .5(f-u)*1 which is what we want.

How do I modify the pictures and write math notation better? What software you use?

user4951
  • 1,714