Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

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 \( 100,000\).)

Step-by-Step Solution

Answer

(a) M = 13,333.33, C = 7,926,535,867.\n(b) Variance = 88,888,889.\n(c) 0.237% earn more than 100,000.

Step 1 - Understanding Pareto Parameters

For a Pareto distribution, the probability density function is defined as: \[ f(x; k, M) = \frac{kM^k}{x^{k+1}} \quad \text{for } x \ge M, \ k, M > 0 \]The mean \( \mu \) is \( \mu = \frac{kM}{k-1} \) for \( k > 1 \). Given that \( k = 3 \) and \( \mu = 20,000 \), we need to find \( M \).

Step 2 - Solving for M

Using the equation \( \mu = \frac{kM}{k-1} \), substitute the given values:\[ 20,000 = \frac{3M}{3-1} = \frac{3M}{2} \]Solving for \( M \):\[ M = \frac{20,000 \times 2}{3} = 13,333.33 \]

Step 3 - Parametrizing constant C

The constant \( C \) in the density function is \( kM^k \). With \( k = 3 \) and \( M = 13,333.33 \), calculate \( C \):\[ C = 3 \times (13,333.33)^3 = 7,926,535,867 \]

Step 4 - Calculating Variance

The variance \( \sigma^2 \) of a Pareto distribution is given by:\[ \sigma^2 = \frac{kM^2}{(k-1)^2(k-2)} \quad \text{for } k > 2 \]Substitute the known values:\[ \sigma^2 = \frac{3 \times (13,333.33)^2}{2^2 \times 1} = 88,888,889 \]

Step 5 - Calculating Probability Income Greater Than 100,000

To find the fraction of earners making more than 100,000, we calculate \( P(X > 100,000) \):\[ P(X > x) = \left(\frac{M}{x}\right)^k \]Substituting the values:\[ P(X > 100,000) = \left(\frac{13,333.33}{100,000}\right)^3 = 0.00237 \]

Key Concepts

Income Distribution and Pareto Distribution

The Pareto distribution is commonly used in economics to model income distribution. It helps explain how wealth is distributed among individuals within an economy.
The Pareto distribution is characterized by the "80-20 rule," which implies that, typically, 20% of the population owns 80% of the wealth. In this context, the distribution is defined by parameters such as shape parameter \(k\) and scale parameter \(M\).

• The scale parameter \(M\) is the minimum possible value or threshold, which in income distribution can be viewed as the lowest income level of interest.
• The shape parameter \(k\) affects the curvature of the distribution. Higher values of \(k\) indicate a more tilted distribution, where fewer people hold about the same amount of wealth.

Understanding these parameters helps us model real-world scenarios using the Pareto distribution, and knowing the mean income allows us to determine these parameters effectively.

Variance Calculation in Pareto Distribution

Variance is a crucial measure used to understand the spread of a distribution. For a Pareto distribution, calculating variance can be more complex than simply squaring the standard deviation because it considers how likely different outcomes are based on parameter values.
The variance \( \sigma^{2} \) for a Pareto distribution is given by the formula: \[ \sigma^{2} = \frac{kM^2}{(k-1)^2(k-2)} \] where \( k > 2 \). This formula requires understanding the balance between the shape and scale parameters to calculate how widespread incomes are from the average.
In the exercise, using the Pareto parameters \(k=3\) and \(M=13,333.33\), we find that the variance is large. This indicates high diversity within income distribution, which is typical for real-world economic systems.

Understanding the Probability Density Function

A probability density function (PDF) describes the likelihood of a random variable taking on a specific value. For continuous distributions like Pareto, the PDF gives us insights into how incomes are spread across the economy.
In the Pareto model, the PDF is given by:\[ f(x; k, M) = \frac{kM^k}{x^{k+1}} \] for \( x \ge M\), where \( k \) and \( M \) are the parameters influencing the curve. This function tells us the relative likelihood of various incomes within the economy.
The constant \( C \), found in the PDF as \( kM^k \), acts as a normalization factor ensuring the total probability over all possible income values sums to one. In practical terms, this means understanding the PDF helps model how different income levels contribute to total economic output.

Mathematical Modeling of Income Distribution

Mathematical modeling is the process of using mathematical expressions to represent real-world situations. In income distribution, models like the Pareto distribution enable us to predict and analyze economic patterns.
By setting parameters \(k\) and \(M\), we simulate the distribution of incomes and can calculate critical measures like mean, variance, and probabilities for earning above a certain income.

• The Pareto distribution provides a framework that aligns with observed economic phenomena, such as wealth concentration and inequality.
• Mathematical models also allow economists to conduct simulations and policy analysis to understand potential impacts of changes within the economy.

These models are indispensable for economists as they provide insights that are crucial for designing informed economic policies and strategies.