Question

The Pareto distribution is often used to model income distribution. Suppose that in some economy the income distribution does follow a Pareto distribution with \(k=3 .\) Suppose that the mean income is $$\$ 20,000$$. (a) Find \(M\) and \(C\). (b) Find the variance \(\sigma^{2}\). (c) Find the fraction of income earners who earn more than $$\$ 100,000$$. (Note: This is the same as asking what is the probability that a randomly chosen person has an income of more than $$\$ 100,000 .$$ )

Short answer

(a) \( M \approx \$13,333.33 \), \( C = 3 \times (13,333.33)^3 \). (b) \( \sigma^2 \approx 133,333,333.5 \). (c) About 3.125% earn more than \$100,000.

Step-by-step solution

Understanding the Pareto Distribution

In a Pareto distribution with parameter \( k \), the probability density function (PDF) is given by \( f(x; k, x_m) = \frac{kx_m^k}{x^{k+1}} \) for \( x \geq x_m \) and 0 otherwise. Here, \( M \) is the minimum income threshold \( x_m \). We have \( k = 3 \) and the mean income \( \mu = \frac{k M}{k - 1} \).

Solve for M using the Mean Income

Given the mean income is \$20,000, we use the formula for the mean: \( 20,000 = \frac{3M}{2} \). Solving for \( M \), we get \( M = \frac{20,000 \times 2}{3} = 13,333.33 \). Thus, \( M \approx 13,333.33 \).

Determine Constant C

Since \( C \) in the Pareto distribution can be represented as \( C = kM^k \), substituting \( k = 3 \) and \( M = 13,333.33 \), we get \( C = 3 \times (13,333.33)^3 \).

Find the Variance

The variance for the Pareto distribution is given by \( \sigma^2 = \frac{k M^2}{(k - 1)^2 (k - 2)} \) for \( k > 2 \). Substituting \( k = 3 \) and \( M = 13,333.33 \), \( \sigma^2 = \frac{3 \times (13,333.33)^2}{(2)^2 \times 1} = \frac{3 \times 177777778}{4} \approx 133333333.5 \).

Calculate the Probability of Earning More than $100,000

For a Pareto distribution, \( P(X > x) = \left(\frac{M}{x}\right)^k \). Substituting \( M = 13,333.33 \), \( x = 100,000 \), and \( k = 3 \), we find: \( P(X > 100,000) = \left(\frac{13,333.33}{100,000}\right)^3 \approx 0.03125 \). Thus, approximately 3.125% of income earners earn more than \$100,000.

Key concepts

Income Distribution

The Pareto distribution is a popular model for representing income distribution in an economy. It is particularly useful because it captures the reality that a small percentage of people often earn significantly more than the rest. This mathematical concept is derived from the work of economist Vilfredo Pareto, who observed that wealth in society followed a predictable distribution pattern.
A key feature of the Pareto distribution is the parameter \(k\), which indicates how incomes are spread. In this exercise, \(k = 3\) suggests a moderately skewed income distribution. Another important element is \(M\), the minimum income threshold, estimated here as approximately \$13,333.33.
Understanding the distribution of income is crucial for analyzing economic inequality and making policy decisions. By knowing how income is distributed, economists and policymakers can design better welfare programs and tax systems.

Probability Density Function

In the context of the Pareto distribution, the probability density function (PDF) is expressed as \( f(x; k, x_m) = \frac{kx_m^k}{x^{k+1}} \) for \(x \geq x_m\). This formula defines the likelihood of different income levels within the distribution. The PDF is essential for understanding how probabilities are assigned to various income intervals.
Here, we observe that \(k = 3\) and \(x_m = M = 13,333.33\). These values are plugged into the PDF equation to describe the specific income distribution for this economy.
PDFs are a core concept in probability and statistics as they provide complete information about the distribution characteristics, allowing one to determine probabilities related to specific income brackets.

Variance

Variance is a statistical measurement that quantifies the spread or dispersion of data points in a dataset. In the Pareto distribution, the variance is given by \( \sigma^2 = \frac{k M^2}{(k - 1)^2 (k - 2)} \). For \(k = 3\), the variance can be computed using this specific formula, considering \(M = 13,333.33\).
Through calculation, we find the variance to be approximately 133,333,333.5. This high number indicates a broad spread in income, signifying significant income inequality within this particular distribution. A higher variance in income data usually signifies a larger gap between the poorest and richest income earners.
Variance is crucial for understanding the level of inequality within an economic distribution and is an important parameter when modeling income data with a Pareto distribution.

Probability Calculation

Probability calculations in the Pareto distribution involve finding the likelihood of certain income thresholds. We use the formula \( P(X > x) = \left(\frac{M}{x}\right)^k \) to determine the probability of earning above a specific amount.
In this exercise, we are interested in finding the fraction of income earners making more than \\(100,000. Substituting \(M = 13,333.33\), \(k = 3\), and \(x = 100,000\), we calculate \( P(X > 100,000) \). The result is approximately 3.125%, indicating that only a small portion of the population earns more than \\)100,000.
This kind of probability analysis is helpful in recognizing economic disparities and planning targeted interventions to address gaps in income distribution.