Problem:
a) How large does a family's net worth have to be to be in the top $1$ percent in 2019?
b) a) How large does a family's net worth have to be to be in the top $2$ percent in 2019?
Answer:
I will work in units of thousands of dollars. My answer is based upon data from the following website.
Federal Reserve Survey
From this website we learn the following information:
For a family to be in the top 17.5% they need a net worth of $653.10$.
For a family to be in the top 5% they need a net worth of $2598.40$.
The mean net worth of a family in the top 10% is $5710.34$.
Let $n$ be the minimum net worth to be in the top $t$ percent. Let $c_1$ and $c_2$ be two constants. Now, I am looking for an equation of the following form:
$$ n t^{c_1} = c_2 $$
that satisfies the given two points. We have the following two equations:
\begin{align*}
(653.10) ( 17.5)^{c_1} &= c_2 \\
(2598.40) (5)^{c_1} &= c_2
\end{align*}
Now we solve the two equations:
\begin{align*}
(653.10) ( 17.5)^{c_1} &= (2598.40) (5)^{c_1} \\
( 17.5)^{c_1} &= (3.9785638) (5)^{c_1} \\
\ln{ ( 17.5)^{c_1} } &= \ln{ (3.9785638) (5)^{c_1} } = \ln{ 3.9785638 } + \ln{(5)^{c_1}} \\
\ln{ ( 17.5)^{c_1} } &= 1.3809209 + \ln{(5)^{c_1}} \\
c_1 \ln{ ( 17.5)} &= 1.3809209 + c_1 \ln{5} \\
c_1 \left( \ln{ ( 17.5) } - \ln 5 \right) &= 1.3809209 \\
c_1 &= 1.1023002 \\
c_2 &= (653.10) ( 17.5)^{1.1023002} \\
c_2 &= 15317.249 \\
\end{align*}
Now we have the following equation:
$$ n t^{1.1023002} = 15317.249 $$
Now we use this equation to solve for $n$ when $t = 1$.
\begin{align*}
n \left( 1^{1.1023002} \right) &= 15317.249 \\
n &= 15317.249 \\
\end{align*}
Now we use this equation to solve for $n$ when $2 = 1$.
\begin{align*}
n \left( 2^{1.1023002} \right) &= 15317.249 \\
n \left( 2.1469673 \right) &= 15317.249 \\
n &= 7134.3654
\end{align*}
Given this data, as a check, we find the mean of the top 10%. We know from the survey the mean
is $5710.34$. I am going to assume that there are $100e6$ families in the United States. Let $u_{10}$
be the mean net worth of families that are in the top 10%.
\begin{align*}
u_{10} &= \dfrac{ \left( \sum\limits_{t=1}^{1e7}
\left(
\dfrac{ 15317.249}{ { \left( \dfrac{t}{1e7} \right) }^{1.1023002} } \right) \right) }
{1e6} \\
%
u_{10} &= \sum\limits_{t=1}^{1e7}
\left(
\dfrac{ 0.00153172}{ { \left( \dfrac{t}{1e7} \right) }^{1.1023002} } \right)
\end{align*}
Using a Python script, I find that:
$$ u_{10} = 53381.96 $$
I will be happy to post the Python script upon request.
This means the mean of the top 10% is over 50 million dollars. This cannot be right. Where did I go wrong?
I am now thinking the model I am using is wrong. Please consider this:
Now let's find the net worth of the richest family in America using our model. We have: \begin{align*} t &= \dfrac{100}{100e6} = \dfrac{1}{e6} \\ n t^{1.1023002} &= 15317.249 \\ n &= \dfrac{ 15317.249 }{ t^{1.1023002} } = \dfrac{ 15317.249 }{ {\dfrac{1}{1e6}} ^{1.1023002} } \\ n &= \dfrac{ 15317.249 }{2.4333176148439223e-07 } \\ n &= 62948005252.41946 \end{align*} Since $n$ is in thousands, we are getting a value of about 63 trillion dollars which is not right. Therefore, I conclude that my model is wrong.
