# Pareto distribution and Lorenz curve

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

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.

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 $$\tilde{x}$$.

• 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 Nov 27 '19 at 11:00
• 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 Nov 27 '19 at 13:57