A Question about Pareto Distribution

Question

Below is a problem that I made up and my attempt to solve it. I am thinking I have the wrong answer and somebody here can tell me where I went wrong.
Thanks,
Bob

Problem:
The median net worth of a certain population is $100K$. To be in the top $10\%$ you need to have a net worth of $1000K$. Assuming the population follows the Pareto distribution, what is the minimum net worth you need to be in top $2\%$?
Answer:
Answer: \newline For the Pareto distribution we have: \begin{eqnarray*} P(X > x) &=& \Big( \frac{x_m}{x} \Big) ^ \alpha \\ \end{eqnarray*} Now we can setup two equations with two unknowns. \begin{eqnarray*} \Big( \frac{x_m}{100} \Big) ^ \alpha &=& 0.5 \\ \Big( \frac{x_m}{1000} \Big) ^ \alpha &=& 0.1 \\ \end{eqnarray*} Now we solve the two equations. \begin{eqnarray*} \Big( \frac{x_m}{100} \Big) ^ \alpha \Big( \frac{1}{10} \Big) ^ \alpha &=& 0.9 \\ 0.5 \Big( \frac{1}{10} \Big) ^ \alpha &=& 0.1 \\ \Big( \frac{1}{10} \Big) ^ \alpha &=& 0.2 \\ \alpha \ln{ 0.1 } &=& \ln{0.2} \\ \alpha &=& 0.69897\\ \Big( \frac{x_m}{100} \Big) ^ { 0.69897} &=& 0.5 \\ \Big( \frac{x_m}{100} \Big) &=& 0.5 ^ { \frac{1}{ 0.69897 } } = 0.3709569 \\ x_m &=& 37.09569 \\ \end{eqnarray*} Now we need to find the minimum net worth of the top $2\%$. \begin{eqnarray*} \Big( \frac{37.09569}{x} \Big) ^ { 0.69897 } &=& 0.98 \\ \Big( \frac{37.09569}{x} \Big) &=& 0.98 ^ { \frac{1}{0.69897} } = 0.9715102 \\ 0.9715102 x &=& 37.09569 \\ x &=& 38.183531 \\ \end{eqnarray*}

Answer 1

Note that you are using the tail (complement CDF) $P(X > x)$ already.

The second of your starting equations

Now we can setup two equations with two unknowns. \begin{eqnarray*} \Big( \frac{x_m}{100} \Big) ^ \alpha &=& 0.5 \\ \Big( \frac{x_m}{1000} \Big) ^ \alpha &=& \color{red}{0.9} \\ \end{eqnarray*}

should be just directly the "top $10\%$"

$$ \Big( \frac{x_m}{1000} \Big) ^ \alpha = 0.1$$

The solution is $x_m \approx 37.095689$ and $\alpha \approx 0.69897$. The left end cutoff $x_m$ is now a reasonably low value (hmm? totally a different universe from ours), and $\alpha > 0 $ gives the typical $1/x$ like shape.

The minimum of the top $2\%$ is exactly $10^4$K, and this is not a coincidence. By having the top $10\%$ to be ten times (in wealth) the top $50\%$, you are setting a power law that goes 10 times in wealth when you go one-fifth in top percentage. Thus $2\%$ is another 10 times of the $10\%$.