<p>With my very limited knowledge of development economics:</p>

<p><span class="math-container">$\left(\frac{x_m}{x}\right)^\alpha$</span> represents the proportion of the population that has an income larger or equal to <span class="math-container">$x$</span> where <span class="math-container">$x\geq x_m&gt;0$</span> and <span class="math-container">$x_m$</span> is the minimum income amount. </p>

<p><strong>Example 1:</strong> </p>

<p>Suppose <span class="math-container">$\alpha\rightarrow 1$</span> and the minimum income in the economy is <span class="math-container">$50,000$</span>. We may ask the question, what proportion of the population has an income larger or equal to <span class="math-container">$50,000$</span>? </p>

<p><span class="math-container">$$\lim_{\alpha \rightarrow 1} \left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{50,000}\right)^1=1$$</span> </p>

<p>This suggests that <span class="math-container">$100\%$</span> of the population has an income larger than or equal to <span class="math-container">$50,000$</span>.</p>

<p><strong>Example 2:</strong></p>

<p>Suppose <span class="math-container">$\alpha \rightarrow 1$</span> and the minimum income in the economy is <span class="math-container">$50,000$</span>. We may ask the question, what proportion of the population has an income larger or equal to <span class="math-container">$500,000$</span>? </p>

<p><span class="math-container">$$\lim_{\alpha \rightarrow 1}\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^1=0.1$$</span> </p>

<p>This suggest that <span class="math-container">$10\%$</span> of the population has an income larger than or equal to <span class="math-container">$500,000$</span>.</p>

<p>The story does not end there. What about <span class="math-container">$\alpha$</span>? It must be the case that <span class="math-container">$\left(\frac{x_m}{x}\right)^\alpha$</span> is bounded between 0 and 1 (it is a proportion). As such, it must be the case that <span class="math-container">$\alpha&gt;1$</span>. In the literature, <span class="math-container">$\alpha$</span> is refered to as the <strong>Pareto Index</strong>. Let us consider another example.</p>

<p><strong>Example 3:</strong></p>

<p>Suppose <span class="math-container">$\alpha=2$</span> and the minimum income in the economy is <span class="math-container">$50,000$</span>. We may ask the question, what proportion of the population has an income larger or equal to <span class="math-container">$500,000$</span>? </p>

<p><span class="math-container">$$\left(\frac{x_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^2=0.01$$</span> </p>

<p>This suggest that <span class="math-container">$1\%$</span> of the population has an income larger than or equal to <span class="math-container">$500,000$</span>. From this, a high <span class="math-container">$\alpha$</span>, or a high <strong>Pareto Index</strong>, suggests a small proportion of high-income individuals.</p>

<p>What is a reasonable value of <span class="math-container">$\alpha$</span>? We can normalize <span class="math-container">$x_m = 1$</span>. Suppose we wanted to know the share of total income that is received by those individuals with an income above, say <span class="math-container">$\tilde{x}$</span>. We take the following integrals:</p>

<p><span class="math-container">$$\int_{x_m}^{\infty} x^{-\alpha} dx = \int_{1}^{\infty} x^{-\alpha} dx = \frac{1}{\alpha-1}$$</span></p>

<p><span class="math-container">$$\int_{\tilde{x}}^{\infty} x^{-\alpha} dx = \frac{\tilde{x}^{1-\alpha}}{\alpha-1}$$</span></p>

<p>Therefore the share of total income that is received by those individuals with an income above <span class="math-container">$\tilde{x}$</span>, call it <span class="math-container">$q$</span>, is the ratio of the two integrals:</p>

<p><span class="math-container">$$q = \tilde{x}^{1-\alpha}$$</span></p>

<p>We also want to define the proportion of the high-income population with an income larger than <span class="math-container">$\tilde{x}$</span> as <span class="math-container">$p$</span>: </p>

<p><span class="math-container">$$p = \tilde{x}^{-\alpha}$$</span></p>

<p>The next logical question would be what value of <span class="math-container">$\alpha$</span> allows us to reconcile the empirical observation that <span class="math-container">$20\%$</span> of the population has <span class="math-container">$80\%$</span> of the total income? This implies that <span class="math-container">$p=.2$</span> and <span class="math-container">$q=.8$</span>. </p>

<p><span class="math-container">$$q = p^{\frac{\alpha-1}{\alpha}}$$</span></p>

<p>Taking the log of both sides:</p>

<p><span class="math-container">$$\ln(q) = \frac{\alpha-1}{\alpha}\ln(p)$$</span></p>

<p>Solving for <span class="math-container">$\alpha$</span>:</p>

<p><span class="math-container">$$\alpha = \frac{\ln(p)}{\ln(p)-\ln(q)} = \frac{\ln(.2)}{\ln(.2)-\ln(.8)} = 1.16$$</span></p>

<p>Therefore, an <span class="math-container">$\alpha=1.16$</span> is consistent with what is observed empirically. To answer the question, <span class="math-container">$x_m$</span> is the minimum income amount and is usually normalized to <span class="math-container">$1$</span>. <span class="math-container">$\alpha$</span> is the Pareto Index which is a measure of the proportion of high-income individuals. A high <span class="math-container">$\alpha$</span> suggests a low number of high-income individuals. </p>

<p>The transformation <span class="math-container">$L(F)$</span> simply allows you to compute what % of the total wealth is controlled by those with an income below <span class="math-container">$\tilde{x}$</span>.</p>

CC BY-SA 4.0
2019-11-27T11:00:08.207Z
Hi! Firstly, thank you for the excellent answer. It clarified many of my questions. However, assuming $X$ is Pareto distributed, in order for the $EX$ to take a finite value it has to be that $\alpha>1$. Based on this, why is $\alpha =1$ allowed?
Joe
CC BY-SA 4.0
2019-11-27T13:57:06.887Z
Thank you for pointing out this issue. I have updated the answer to demonstrate that I am talking about alpha approaching 1 in the limit.
lunar_props
CC BY-SA 4.0
2019-11-26T19:48:10.803Z
2019-11-27T13:46:54.873Z
2019-11-27T13:46:54.873Z
lunar_props
<p>I am currently studying Pareto distributions and their relation to Lorenz curves but I am having a hard time understanding the concept. If I am understanding correctly, Lorenz curve describes the wealth distribution and Lorenz curve of a Pareto distribution means that we are assuming that the income is Pareto distributed. Is this correct? But if Pareto distribution itself describes wealth distribution, why do we need to take a Lorenz curve of it? </p>

<p>Also, assume the cumulative distribution function for a Pareto distributed random variable <span class="math-container">$X$</span> with parameters <span class="math-container">$\alpha$</span> and <span class="math-container">$x_m$</span> is</p>

<p><span class="math-container">$F_X(X)=\begin{cases} 1-(\frac{x_m}{x})^{\alpha}, \quad x\ge x_m \\ 0, \quad x&lt;x_m \end{cases}$</span></p>

<p>and the Lorenz curve of it to be </p>

<p><span class="math-container">$L(F)=1-(1-F)^{1-\frac{1}{\alpha}}$</span>.</p>

<p>What do the parameters <span class="math-container">$\alpha$</span> and <span class="math-container">$x_m$</span> mean in the Lorenz curve? Is <span class="math-container">$\alpha$</span> the income level? What does the variable <span class="math-container">$x$</span> mean?</p>

<p>Apologies if this is a sort of stupid question but I cannot seem to understand it. </p>

<microeconomics><wealth>
Pareto distribution and Lorenz curve
Joe
# Pareto distribution and Lorenz curve

(Asked by: username_0 on 2019-11-25)

I am currently studying Pareto distributions and their relation to Lorenz curves but I am having a hard time understanding the concept. If I am understanding correctly, Lorenz curve describes the wealth distribution and Lorenz curve of a Pareto distribution means that we are assuming that the income is Pareto distributed. Is this correct? But if Pareto distribution itself describes wealth distribution, why do we need to take a Lorenz curve of it?

Also, assume the cumulative distribution function for a Pareto distributed random variable $X$ with parameters $\alpha$ and $x\_m$ is

$F\_X(X)=\begin{cases} 1-(\frac{x\_m}{x})^{\alpha}, \quad x\ge x\_m \\ 0, \quad x<x\_m \end{cases}$

and the Lorenz curve of it to be

$L(F)=1-(1-F)^{1-\frac{1}{\alpha}}$.

What do the parameters $\alpha$ and $x\_m$ mean in the Lorenz curve? Is $\alpha$ the income level? What does the variable $x$ mean?

Apologies if this is a sort of stupid question but I cannot seem to understand it.

**Tags:** microeconomics, wealth


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## 1 answers

### Answer by username_1 on 2019-11-26. Score: 4 | Accepted answer

With my very limited knowledge of development economics:

$\left(\frac{x\_m}{x}\right)^\alpha$ represents the proportion of the population that has an income larger or equal to $x$ where $x\geq x\_m>0$ and $x\_m$ is the minimum income amount.

**Example 1:**

Suppose $\alpha\rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $50,000$?

$$\lim\_{\alpha \rightarrow 1} \left(\frac{x\_m}{x}\right)^\alpha = \left(\frac{50,000}{50,000}\right)^1=1$$

This suggests that $100\%$ of the population has an income larger than or equal to $50,000$.

**Example 2:**

Suppose $\alpha \rightarrow 1$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?

$$\lim\_{\alpha \rightarrow 1}\left(\frac{x\_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^1=0.1$$

This suggest that $10\%$ of the population has an income larger than or equal to $500,000$.

The story does not end there. What about $\alpha$? It must be the case that $\left(\frac{x\_m}{x}\right)^\alpha$ is bounded between 0 and 1 (it is a proportion). As such, it must be the case that $\alpha>1$. In the literature, $\alpha$ is refered to as the **Pareto Index**. Let us consider another example.

**Example 3:**

Suppose $\alpha=2$ and the minimum income in the economy is $50,000$. We may ask the question, what proportion of the population has an income larger or equal to $500,000$?

$$\left(\frac{x\_m}{x}\right)^\alpha = \left(\frac{50,000}{500,000}\right)^2=0.01$$

This suggest that $1\%$ of the population has an income larger than or equal to $500,000$. From this, a high $\alpha$, or a high **Pareto Index**, suggests a small proportion of high-income individuals.

What is a reasonable value of $\alpha$? We can normalize $x\_m = 1$. Suppose we wanted to know the share of total income that is received by those individuals with an income above, say $\tilde{x}$. We take the following integrals:

$$\int\_{x\_m}^{\infty} x^{-\alpha} dx = \int\_{1}^{\infty} x^{-\alpha} dx = \frac{1}{\alpha-1}$$

$$\int\_{\tilde{x}}^{\infty} x^{-\alpha} dx = \frac{\tilde{x}^{1-\alpha}}{\alpha-1}$$

Therefore the share of total income that is received by those individuals with an income above $\tilde{x}$, call it $q$, is the ratio of the two integrals:

$$q = \tilde{x}^{1-\alpha}$$

We also want to define the proportion of the high-income population with an income larger than $\tilde{x}$ as $p$:

$$p = \tilde{x}^{-\alpha}$$

The next logical question would be what value of $\alpha$ allows us to reconcile the empirical observation that $20\%$ of the population has $80\%$ of the total income? This implies that $p=.2$ and $q=.8$.

$$q = p^{\frac{\alpha-1}{\alpha}}$$

Taking the log of both sides:

$$\ln(q) = \frac{\alpha-1}{\alpha}\ln(p)$$

Solving for $\alpha$:

$$\alpha = \frac{\ln(p)}{\ln(p)-\ln(q)} = \frac{\ln(.2)}{\ln(.2)-\ln(.8)} = 1.16$$

Therefore, an $\alpha=1.16$ is consistent with what is observed empirically. To answer the question, $x\_m$ is the minimum income amount and is usually normalized to $1$. $\alpha$ is the Pareto Index which is a measure of the proportion of high-income individuals. A high $\alpha$ suggests a low number of high-income individuals.

The transformation $L(F)$ simply allows you to compute what % of the total wealth is controlled by those with an income below $\tilde{x}$.

> **Comments:**
> **username_0**: Hi! Firstly, thank you for the excellent answer. It clarified many of my questions. However, assuming $X$ is Pareto distributed, in order for the $EX$ to take a finite value it has to be that $\alpha>1$. Based on this, why is $\alpha =1$ allowed?  
> **username_1**: Thank you for pointing out this issue. I have updated the answer to demonstrate that I am talking about alpha approaching 1 in the limit.  

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