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In a 2-person economy, the owner of a firm pays 20% of the total output to the worker.

What is the Gini coefficient in this economy?

The answer is supposed to be 3/5 but I end up getting 3/10 instead.

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closed as off-topic by Bayesian, luchonacho, Adam Bailey, Herr K., Theoretical Economist Feb 27 '17 at 0:28

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Bayesian, luchonacho, Adam Bailey, Herr K., Theoretical Economist

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Gini Coefficient is the ratio of the area that lies between the 45 degree line (line of equality) and the Lorenz curve, and the area below the line of equality. Given the data in the problem Lorenz curve connects points $(0, 0)$, $(50, 20)$ and $(100, 100)$ in the graph. To find the Gini coefficient, we find the area between the line of equality and the Lorenz curve and divide it by the area of the triangular region that lies below the line of equality.

Gini coefficient $\displaystyle = 1 - \frac{\color{blue}{0.5\times 50 \times 20} + \color{red}{(100 - 50) \times 20} + \color{gray}{0.5\times (100-50) \times (100-20)}}{0.5\times 100 \times 100} = 0.3$

Here is the graph:

enter image description here

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The Gini coefficient is defined as the normalized Gini index. If $G$ is the Gini index, $0 \leq G \leq \frac{n-1}{n}$. The Gini coefficient $G^* = \frac{n}{n-1} G$ is a normalization such that $0\leq G^* \leq 1$. That way you can easier compare different populations with different $n$. In your example it should be that $G= 3/10$ and then $G^*=6/10=3/5$.

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  • $\begingroup$ How do you normalise it? I though the Gini was A/(A+B) $\endgroup$ – Dtomas Jan 8 '17 at 22:39
  • $\begingroup$ Just multiply with $n/(n-1)$ so that extreme inequality has a coefficient of 1. $\endgroup$ – Bayesian Jan 8 '17 at 22:56
  • $\begingroup$ See: en.wikipedia.org/wiki/Gini_coefficient You can define the Gini coefficient as the relative mean absolute difference divided by 2. Calculate your example with, say, $x_1=8, x_2=2$. You can see that extreme inequality $x_1=10,x_2=0$ would give you $G=0.5=(n-1)/n$ which can be normalized as above. $\endgroup$ – Bayesian Jan 9 '17 at 10:47

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