Modelling wealth with a Pareto distribution: how do I estimate the parameters? 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.
 A: 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.
