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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.

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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 $\tilde{x}$.

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  • $\begingroup$ 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? $\endgroup$ – Joe Nov 27 '19 at 11:00
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    $\begingroup$ 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. $\endgroup$ – lunar_props Nov 27 '19 at 13:57

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