<p>Let our variable of interest be <span class="math-container">$x$</span> that takes values <span class="math-container">$x_i, ...., x_n$</span> for a sample <span class="math-container">$1, ...., n$</span>. Let <span class="math-container">$\bar x$</span> be the sample mean.</p>
<p>The Gini coefficient can be defined as:</p>
<p><span class="math-container">$$\frac{\sum_{i=1}^n \sum_{j=1}^n |x_i-x_j|}{2 n^2 \bar x}$$</span></p>
<p>This is simply the mean of all the pair-wise absolute differences between individual observations, divided by the sample mean, normalised to have a maximum value of 1 byt dviding further by 2.</p>
<p>As you can note, the units in the numerator and denominator are the same, so the ratio as a whole is <em>unitless</em>. This is a strong indication that scaling the variable <span class="math-container">$x$</span> wont affect it.</p>
<p>To be more precise, if the data set is <span class="math-container">$kx_1, ...., kx_n$</span>, where <span class="math-container">$k&gt;0$</span> instead, then the coefficient is</p>
<p><span class="math-container">$$\frac{\sum_{i=1}^n \sum_{j=1}^n |kx_i-kx_j|}{2 n^2 k \bar x}$$</span></p>
<p><span class="math-container">$$= \frac{\sum_{i=1}^n \sum_{j=1}^n k|x_i-x_j|}{2 n^2 k \bar x}$$</span></p>
<p><span class="math-container">$$= \frac{\sum_{i=1}^n \sum_{j=1}^n |x_i-x_j|}{2 n^2  \bar x}$$</span></p>
<p>Which is the same as the original, the <span class="math-container">$k$</span>s just cncel out as the numerator and the denominator are scaled by the same factor. This property is called <strong>scale invariance</strong>.</p>
<p>For the second question, let the new data be <span class="math-container">$x_i + k, ...., x_n + k$</span>. This time, the Gini coefficient is</p>
<p><span class="math-container">$$\frac{\sum_{i=1}^n \sum_{j=1}^n |x_i + k -x_j - k|}{2 n^2 (\bar x + k)}$$</span></p>
<p><span class="math-container">$$= \frac{\sum_{i=1}^n \sum_{j=1}^n |x_i -x_j|}{2 n^2 (\bar x + k)}$$</span></p>
<p><span class="math-container">$$= \frac{\sum_{i=1}^n \sum_{j=1}^n |x_i -x_j|}{2 n^2 \bar x} \frac{\bar x}{\bar x + k}$$</span></p>
<p><span class="math-container">$$=OriginalGini * \frac{\bar x}{\bar x + k}$$</span></p>
<p>Thus, when everybody's income rises by a constant, the Gini coefficient falls. This effect is more pronounced when <span class="math-container">$\bar x$</span> is smll to begin with. And when everybody's income falls by a constant, the Gini coefficient rises. This effect is more pronounced for distributions where we had a large <span class="math-container">$\bar x$</span> initially.</p>

CC BY-SA 4.0
2022-12-18T07:48:26.390Z
2022-12-18T07:48:26.390Z
Ishan Kashyap Hazarika
<p>If income of all the individuals in a country doubles, what will happen to the Gini coefficient?</p>
<p>What happens to Gini coefficient when income of all the individuals in a country increases by 1000?</p>

<public-economics><gini>
Gini Coefficient
user43114
# Gini Coefficient

(Asked by: username_0 on 2022-12-18)

If income of all the individuals in a country doubles, what will happen to the Gini coefficient?

What happens to Gini coefficient when income of all the individuals in a country increases by 1000?

**Tags:** public-economics, gini

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## 1 answers

### Answer by username_1 on 2022-12-18. Score: 2 | Accepted answer

Let our variable of interest be $x$ that takes values $x\_i, ...., x\_n$ for a sample $1, ...., n$. Let $\bar x$ be the sample mean.

The Gini coefficient can be defined as:

$$\frac{\sum\_{i=1}^n \sum\_{j=1}^n |x\_i-x\_j|}{2 n^2 \bar x}$$

This is simply the mean of all the pair-wise absolute differences between individual observations, divided by the sample mean, normalised to have a maximum value of 1 byt dviding further by 2.

As you can note, the units in the numerator and denominator are the same, so the ratio as a whole is *unitless*. This is a strong indication that scaling the variable $x$ wont affect it.

To be more precise, if the data set is $kx\_1, ...., kx\_n$, where $k>0$ instead, then the coefficient is

$$\frac{\sum\_{i=1}^n \sum\_{j=1}^n |kx\_i-kx\_j|}{2 n^2 k \bar x}$$

$$= \frac{\sum\_{i=1}^n \sum\_{j=1}^n k|x\_i-x\_j|}{2 n^2 k \bar x}$$

$$= \frac{\sum\_{i=1}^n \sum\_{j=1}^n |x\_i-x\_j|}{2 n^2 \bar x}$$

Which is the same as the original, the $k$s just cncel out as the numerator and the denominator are scaled by the same factor. This property is called **scale invariance**.

For the second question, let the new data be $x\_i + k, ...., x\_n + k$. This time, the Gini coefficient is

$$\frac{\sum\_{i=1}^n \sum\_{j=1}^n |x\_i + k -x\_j - k|}{2 n^2 (\bar x + k)}$$

$$= \frac{\sum\_{i=1}^n \sum\_{j=1}^n |x\_i -x\_j|}{2 n^2 (\bar x + k)}$$

$$= \frac{\sum\_{i=1}^n \sum\_{j=1}^n |x\_i -x\_j|}{2 n^2 \bar x} \frac{\bar x}{\bar x + k}$$

$$=OriginalGini \* \frac{\bar x}{\bar x + k}$$

Thus, when everybody's income rises by a constant, the Gini coefficient falls. This effect is more pronounced when $\bar x$ is smll to begin with. And when everybody's income falls by a constant, the Gini coefficient rises. This effect is more pronounced for distributions where we had a large $\bar x$ initially.


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