I wish to create a function that will estimate the wealth of a person in the United States. It would be used to make a table with each decile and their estimated wealth.

This estimate will be based on very rudimentary data, and is only for personal interest. The data is:

  • The total wealth of the bottom 90% is equal to the total wealth of the top 0.1%.
  • Both proportions have 22% of the total wealth.
  • The total wealth under the distribution is $80 trillion.
  • The total population is 160 million households.

Given this data, how would I create parameter estimates for the exponent and scale of a pareto distribution? What would be $f(x)$ where $x$ is from $(0, 1)$, and the solution is the wealth of someone richer than that proportion of people? For example $f(0.1)$ is someone richer than or equal to exactly 10% of the least wealthy, and could equal 1,000 dollars. $F(0.5)$ is the median wealth, and could be 200,000 dollars. $F(0.9999)$ is richer than 99.99% and would be somewhere in the tens or hundreds of millions of dollars.

  • 2
    $\begingroup$ Some practical questions: 1) Why do you want to use a Pareto distribution and how do you know it's appropriate? 2) What is the end goal of what you're trying to achieve with fitting your data to a Pareto distribution? Of course, we could ignore these questions entirely, but these are practical considerations to consider when you're in this situation. $\endgroup$ – Clarinetist Jul 26 '18 at 12:07
  • $\begingroup$ I'm using the Pareto distribution as it often comes up in discussions about wealth inequality and the 80/20 rule. I suspect it will not be a very good estimator based on such limited data, but I'd like to see if there is any correlation with further data. For instance, if F(0.5) is anywhere in the ballpark of the median net worth, that would be interesting. There is no goal other than curiosity as to the exact method of finding such a function based on such information. $\endgroup$ – Paul Kusmanoff Jul 26 '18 at 12:25
  • $\begingroup$ I'm not going to have time to write a detailed answer, but you should look into maximum likelihood estimation for the Pareto distribution. $\endgroup$ – Clarinetist Jul 26 '18 at 12:28

First of all, the Pareto distribution is defined on non-negative real numbers. Thus it doesn't really make sense to force it to the interval $ (0, 1) $.

Suppose, however, you knew the wealth of two people: someone at the 90th percentile and the 99.90th percentile. Then, if you assume that wealth is truly Pareto-distributed, you could take a method-of-moments approach.

Letting $ A $ be the wealth of the 90th percentile person, and $ B $ the wealth of the 99.90th percentile person, and $ F(x) $ be the cumulative distribution function of a Pareto($x_m, \alpha$) random variable, we have that

$$ F(x) = 1 - (\frac{x_m}{x})^\alpha \iff F^{-1}(y) = \frac{x_m}{(1 - y)^{1/\alpha}} \iff x_m = x(1-y)^{1/\alpha} $$

From two statements that you provided, we have that the person with wealth $ B $ has greater wealth than 78% of the population, and that the person with wealth $ A $ has greater wealth than 22% of the population. You could then estimate the Pareto distribution parameters by solving the equations

$$ x_m = A \cdot (1-0.22)^{1/\alpha} $$ $$ x_m = B \cdot (1-0.78)^{1/\alpha} $$

If you had the full dataset, however, you could certainly perform MLE (@Clarinetist's suggestion). You might get better theoretical guarantees that way.

  • $\begingroup$ Thank you very much Kevin, this is exactly what I was looking for! $\endgroup$ – Paul Kusmanoff Jul 27 '18 at 7:11

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