Problem 69 - Precalculus Mathematics for Calculus

Question

Vilfredo Pareto \((1848-1923)\) observed that most of the wealth of a country is owned by a few members of the population. Pareto's Principle is $$\log P=\log c-k \log W$$ where \(W\) is the wealth level (how much money a person has) and \(P\) is the number of people in the population having that much money. (a) Solve the equation for \(P\). (b) Assume that \(k=2.1, c=8000,\) and \(W\) is measured in millions of dollars. Use part (a) to find the number of people who have \(\$ 2\) million or more. How many people have \(\$ 10\) million or more?

Step-by-Step Solution

Answer

For \$2 million: \(P \approx 1834\). For \$10 million: \(P \approx 63\).

Step 1: Rearrange the Equation

Start with the given equation \(\log P = \log c - k \log W\). We need to solve this equation for \(P\). To do this, isolate \(\log P\) on one side.

Step 2: Simplify Using Logarithm Properties

Apply the logarithm properties to combine the terms: \(\log P = \log \left(\frac{c}{W^k}\right)\). This utilizes the property \(\log a - \log b = \log \left(\frac{a}{b}\right)\).

Step 3: Remove Logarithm with Exponentiation

Convert the logarithmic form to exponential form: \(P = \frac{c}{W^k}\). This is done by raising 10 to the power of both sides because \(\log_{10}\) was assumed.

Step 4: Calculate Number of People for \(\$2\) Million

Substitute \(k = 2.1, c = 8000, W = 2\) into \(P = \frac{8000}{W^{2.1}}\). Calculate \(P = \frac{8000}{2^{2.1}}\).

Step 5: Calculate Number of People for \(\$10\) Million

Similarly, substitute \(k = 2.1, c = 8000, W = 10\) into \(P = \frac{8000}{W^{2.1}}\). Calculate \(P = \frac{8000}{10^{2.1}}\).

Key Concepts

Logarithmic Equations

Logarithmic equations involve expressions where the unknown variable is under a logarithm. In solving these equations, one often uses key properties of logarithms to simplify them. The basic property employed is that a logarithm of a quotient can be written as the difference of two logarithms: \( \log a - \log b = \log \left(\frac{a}{b}\right) \).
This was used in the original problem where

• \( \log P = \log c - k \log W \)

was simplified to \( \log P = \log \left(\frac{c}{W^k}\right) \).
The next step is to solve for \(P\), and we do this by converting the logarithmic form to exponential form. If \( \log_{10} x = y \), then \( x = 10^y \). Applying this to isolate \(P\), we get:

• \( P = \frac{c}{W^k} \)

Understanding logarithmic equations in this way allows us to rewrite them in a form that is easier to compute.

Wealth Distribution

Wealth distribution is a concept similar to what Pareto observed, where wealth among a population is not evenly spread. According to Pareto's principle, a small percentage of people control most of the wealth in a society. This is represented mathematically in the exercise by the equation:

• \( \log P = \log c - k \log W \)

Here,

• \(P\) represents the number of people with a specific level of wealth \(W\).
• \(c\) and \(k\) are constants that adjust based on the society being observed.

The calculation process will show that as \(W\) increases, \(P\) decreases, indicating fewer people have more wealth.
In our practical example, substituting values allows us to find out precisely how many individuals hold wealth of $2 million or more, \( W = 2 \), for instance. This principle highlights that wealth concentration increases at higher wealth levels.

Exponential Functions

When working with logarithms, one frequently encounters exponential functions since they are essentially inverse operations. After simplifying the logarithm equation in the exercise, we ended up with an exponential function:

• \( P = \frac{c}{W^k} \)

This function tells us that the number of people \( P \) is inversely related to \( W^k \), meaning as wealth \( W \) increases, \( P \) decreases significantly. The variable \( k \) plays a role in determining how steep this relationship is, with higher values indicating a sharper decline.

Using Exponential Functions in Calculations

In the calculations for \( \\(2 \) million and \( \\)10 \) million, substituting the respective values for \( W \) into \( P = \frac{8000}{W^{2.1}} \) helps us determine how many people are at each wealth level. Exponential decay, which is what this function represents, is useful for modeling real-world situations where quantities diminish sharply over time or due to certain factors.