Chapter 4.4, Problem 69E

Wealth Distribution 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.

1. (a) Solve the equation for P.
2. (b) Assume that k = 2.1 and c = 8000, and that 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?

To determine

To solve: The equation for P in the Pareto’s Principle $\log P=\log c-k\log W$.

Answer to Problem 69E

The equation of P is $P=\frac{c}{W^k}$.

Explanation of Solution

Law used:

Let $a$ be positive no, with $a\ne 1$. Let $A,B$ and C are any real no with $A>0$ and $B>0$.

${\log }_a\left(\frac{A}{B}\right)={\log }_{a}A-\log B$

$\log \left(A^C\right)=C\log A$.

Calculation:

Consider the Pareto’s Principle, $\log P=\log c-k\log W$.

Use the law of logarithm, ${\log }_a\left(\frac{A}{B}\right)={\log }_{a}A-\log B$ and $\log \left(A^C\right)=C\log A$ then express $\log P=\log c-k\log W$ in the following way.

$\begin{array}{c}\log P=\log c-k\log W\\ =\log c-\log W^k\\ =\log \left(\frac{c}{W^k}\right)\end{array}$

That is, $\log P=\log \left(\frac{c}{W^k}\right)$.                                                                                         ...... (1)

Take exponential function on both side of the equation (1).

Then, equation (1) becomes,

$\begin{array}{l}{e}^{\log P}=e^{\log \left(\frac{c}{W^k}\right)}\\ P=\frac{c}{W^k}\end{array}$

Therefore, the equation for P in the Pareto’s Principle $\log P=\log c-k\log W$ is $P=\frac{c}{W^k}$.

To determine

To compute: The number of peoples that have money $\$2$ million or more and the number of peoples that have money $\$10$ million or more.

Answer to Problem 69E

The number of peoples that have money $\$2$ million or more is approximately $1866$.

The no of peoples that have money $\$10$ million or more is approximately $64$.

Explanation of Solution

Given:

Consider the value of $k=2.1$ and $c=8000$.

Use the equation of part (a), $P=\frac{c}{W^k}$, which gives the number of persons that have more money.

That is, $P=\frac{c}{W^k}$, where W is a wealth level.

Then, number of peoples that have money $\$2$ million or more is $P=\frac{c}{2^k}$.

$\begin{array}{c}P=\frac{c}{2^k}\\ =\frac{8000}{2^{2.1}}\\ =1866.0659\end{array}$

That is, the number of peoples that have money $\$2$ million or more is approximately $1866$.

Similarly, number of peoples that have money $\$10$ million or more is $P=\frac{c}{{10}^k}$.

$\begin{array}{c}P=\frac{c}{{10}^k}\\ =\frac{8000}{{10}^{2.1}}\\ =63.546\end{array}$

That is, the number of peoples that have money $\$10$ million or more is approximately $64$.