Math help! x*2, y/3 in a formula

I'm writing about income distribution, and every time the wealth of a group doubles, the size of the group decreases with two-thirds.
So when there are 24,000 people that earn 400,000$....
24,000*1/3=8,000
400,000*2=800,000
There are 8,000 people that earn 800,000$.

I want to know the formula, so I can plot it in a graph and calculate stuff like "how many people earned more than 1,000,000' and 'how much does the richest guy earn'. Could you help me?

Thanks for the help! :D

I've posted this in the 'education & career talk' and asked the INTJs too. Sorry for the spam, but I really need an answer and I wonder where the mathematicians are.

Provide these answers:

Let wealth = x, size = f.

What are the interval sizes of your wealth data? I am assuming you are using continuous data, so it would be something like this:

0 <= x <= 10,000
10,000 <= x <= 20,000
20,000 <= x <= 40,000

etc...

Also give the size of the group (usually called frequency) for each of these. If you cannot, at least give the data that you have.

Originally Posted by Larxene

Provide these answers:

Let wealth = x, size = f.

What are the interval sizes of your wealth data? I am assuming you are using continuous data, so it would be something like this:

0 <= x <= 10,000
10,000 <= x <= 20,000
20,000 <= x <= 40,000

etc...

Also give the size of the group (usually called frequency) for each of these. If you cannot, at least give the data that you have.

Uhm... it has to do with power laws and Pareto's principle.

24,000 people earn more than 400,000$.
When we double the threshold (looking at people who earn more than 800,000$), only roughly one third of the old group is still 'there', so that's 8,000 people. So basically, if X doubles, F/3. I want to know the formula to extend this from 24,000 people with 400,000$ to 1 guy with x$. I think he earns roughly 250,000,000$.
And I want to know how many people earn more than 1,000,000$. 8000 people earn 800,000$, and 2667 people earn 1,600,000$, and want to have a quite exact answer for how many people earn more than 1,000,000$.

On your other thread it seemed like you already got your answer though. I'm not familiar with programming so I couldn't check the answerer's work. And notation on computers are atrocious T_T. I couldn't read the equation he gave.

Do you still need an answer?


Anyway, this is what I came up with:

Let X = income, S = size. Both variables are geometric progressions; so they have the form ar^n:

For X: 400,000 x 2^n
For S: 24,000 x (1/3)^n

They have the parameter n in common, so it's easier to create a relationship using n. So this is the equation:

S = f(n), where

f(n) = 24,000(1/3)^n and n = log2 (X/400,000). (NB: log2 refers to log base 2)

Check whether the equation is correct:

1. When X=400,000, n=0. So f(0) = 24,000(1/3)^0 = 24,000.
2. When X=1,600,000, n=2. So f(2) = 24,000(1/3)^2 = 2666.67.

It should be correct.

So when X=1,000,000, n=1.32... .

f(1.32...) = 24,000(1/3)^(1.32...) = 5616.8.

The size is roughly 5617 when X=1,000,000. Do you have the answer from a book or your teacher to verify this?


EDIT:

The answer for your 'highest possible income' question on the other thread:

1 = 24,000(1/3)^n
1/24,000 = (1/3)^n
log (1/24,000) = n log (1/3)
n = log(1/3) (1/24,000) (NB: log(1/3) refers to log with base 1/3)

Now, you know that n = log2 (X/400,000). Simply plug in the value of n and then move everything around until you get:

X = 2^n x 400,000

The answer I got was: $232,094,896.40. So, as an estimated answer, you say that the richest person will make 232.09 million dollars.

Originally Posted by Larxene

On your other thread it seemed like you already got your answer though. I'm not familiar with programming so I couldn't check the answerer's work. And notation on computers are atrocious T_T. I couldn't read the equation he gave.

Do you still need an answer?

Thanks for the answer! Yeah, I don't mind another answer/different formula. Thanks for all the help, I'm now going to try to understand this :P

Don't you have some grade 12 math knowledge if you are doing some economics question?