Math Help - For the life of me I can't resolve the correct inversion of this simple thing...

I have a simple formula, but I also need the inverse of it. Basically, it is:
y / x * (y / a * b)
An example of numbers would be
25000 / 52000 * (25000 / 260000 * 65000).
*(this is exactly what I need for y <= x)

When I try to invert it, I get something like this, which is incorrect..:
(1-x / y+1) * (y / a * b)
So an example would be (1-52000/75000+1)*(75000 / 260000 * 65000)
(and this is too high for y > x)

Thanks, and sorry to trouble you. I just spent two days trying to resolve this and was (some freaking how) unable to.

I must have forgotten a lot of math.....

Originally Posted by jeremy000000

I have a simple formula, but I also need the inverse of it. Basically, it is:
y / x * (y / a * b)
An example of numbers would be
25000 / 52000 * (25000 / 260000 * 65000).
*(this is exactly what I need for y <= x)

When I try to invert it, I get something like this, which is incorrect..:
(1-x / y+1) * (y / a * b)
So an example would be (1-52000/75000+1)*(75000 / 260000 * 65000)
(and this is too high for y > x)

Thanks, and sorry to trouble you. I just spent two days trying to resolve this and was (some freaking how) unable to.

I must have forgotten a lot of math.....

We need more information. What is this a formula for? And how did you "invert" that? None of this makes any sense.

-Dan

Originally Posted by jeremy000000

I have a simple formula, but I also need the inverse of it. Basically, it is:
y / x * (y / a * b)

It is not clear what you mean by the inverse of this formula. Also note that
\[
y/a*b=\frac{y}{a}\cdot b
\]
and
\[
y/(a*b)=\frac{y}{ab}.
\]

The correct form would be y/a * b, instead of y/(a*b).....

What y/a*b represents is just another way to do an absolute flat tax (like what the US wants to move to), only now we are taking a snap-shot of national income, personal income, and the amount of tax we need, isntead of just going 'yeah - 25%'...

This is like trying to figure out a progressive tax code without resorting to 'tax brackets'.

The variables are as follows: y = income (10,000-100,000 ex). a = national income. (260000 ex) b = necessary national tax (65,000 ex), and x = flat average (52,000 ex - 260000 / 5 people)
So what I want to do is give a 0-100% of all tax for incomes 0-52,000, and 100%-200% for incomes 52,000+.

Comparing to actual numbers, people can do the 0-100% part, and the equasion holds:

income / 260,000 * 65,000 = .25 (avg tax rate if everyone were making 52,000). y / x then becomes the 'adjustment' to that 'avg "tax"'... I need to be able to invert it so instead of 0-1, it is 1-2...
*note* for any income, this does not come out to .25 if you do y/(a*b).... the actual one we need is y/a*b

Then I want to apply the 0-200% rule. For y <= x, this is what I need. But I just can't get y > x right - I always end up with too much. It's just an adjustment to the .25 flat-rate...

So for the lower incomes, it's 0-100% Of .25%..
For higher incomes, it should be 100-200% of .25%.

The problem is, the way I had it the higher incomes, overall they pay too much "tax" and the national tax then doesn't equal 65,000.

As a simple example, assume we have 5 'groups' that we use to represent "5 people", and their incomes are 10,000, 25,000, 50,000, 75,000 and 100,000. The total is 260,000, and we need 65,000 in funding. We have to arrive at 65,000 regardless, but instead of a flat .25, we want to make it so everyone pulls their fair 'load' based on deviation in incomes from the flat average of 52,000.

I guess for a little background on why this poc is necessary is this:

Right now, congress wants a flat 25% tax rate. What that would do, if you did actual numbers with 50% of the population making under $30,000/yr, the median middle class making $41,000 and the median lower class making $21,000, is that the median lower class would immediately fall to extreme poverty by today's measure and the median middle class falls to the poverty line by today's measure - all because of a flat tax rate.

If GDP is stalling now, it's sure to vomit at this 'tax code'.

So then the answer becomes how do we fix that once it goes wrong? And here is the reason why I am trying to work out this problem. It should be mathematically feasible for any income earner in relation to the flat-average income. Once we know how to do that, we can make further adjustments on other measurements such as 'median income'.

In affect, a proper continuous tax code would reverse the problems in a flat tax code, without re-introducing a progressive tax or tax brackets......

Thanks, (sorry this algebraic problem was moved to business. It's more math than business, but whatever...)

I do not know of any politically serious proposal for a "flat tax rate" that implies what you seem to think is implied. Virtually all such proposals continue to be progressive in this sense: tax is applied at a flat percentage above some threshold. It reduces the number of brackets to two, one being a zero tax bracket. Furthermore, most of these proposals involve the elimination of deductions that benefit primarily higher income families (such as the special rate on income from capital gains and the absence of tax on interest from obligations of states and municipalities.)

I doubt anyone except a specialist can figure out the likely initial effects of the elimination of many deductions. But if I understand what you are trying to do on a simple basis, you need to be looking at both the level of income on which no tax is levied and the flat rate applied above that level. You could analyze as an example the actual proposal made by Steve Forbes, who called for a 17% flat rate on income, with no deductions except a general deduction of 46,000 per household. So households with income of 46,000 or less would pay no federal income tax at all. A household with an income of 100,000 would pay about 9% of their income in federal income tax, and a household with an income of 200,000 would pay about 13%. It is in fact a progressive tax.

The effective tax rate on the Forbes proposal would be $\dfrac{(I - 46000) * 0.17}{I}, where\ I\ is\ annual\ income.$

A general formula would be

$I \le D \implies e = 0;$

$I > D \implies e = \dfrac{(I - D) * f}{I};$

where e = effective tax rate,

I = annual household income,

D = income not subject to tax, and

f = flat tax rate.

I = annual household

Thanks Jeff! I can figure the rest of what I wanted out from here, with just the information you gave. I will apply these to a realistic sample distribution on the web to see how it does.

On a further note, if you want to know where I obtained the 25% flat tax rate, which was on the GOP agenda after the 2012 elections, started by Rick Perry as a 20% flat tax idea:

Rick Perry?s Flat Tax: A Bold Challenge to 9-9-9 | The Fiscal Times

That idea has been floating around the GOP ever since.

I can also see what you are saying that I missed, which is a newer version of GOP tax overhaul:

House Republicans unveil tax reform plan -- does it stand a chance? | Fox News

However, I have to wonder under this new tax code on the second link, if the government will still get the same level of funding when it has dropped almost 3 trillion in taxes by 10%, giving us probably an additional 700 billion dollar further deficit than today. However, it does not limit taxes for poverty programs, which is essential in maintaining the viability of those programs.

Here's what I mean: 1. If I can solve this further with real numbers on realistic data, I can solve both healthcare (medicare, medicaid, private insurance, public insurance) with a national one, and I can give the government each year exactly the funding it budgeted for that year. This at least solves two of our biggest fiscal problems. Then we need to replace the unsustainable social security as a retirement option, which if we can just give ourselves 65 years before it is put into action, we can do without any interruption. Solving the first two problems gives us the fiscal footing we need to be able to solve that last one (buys us time). Then we wait for the education bubble to pop and only then step in. It may fix itself by doing nothing about it at all, but I would hate to end up with only trade schools. At that time, extra money could be used to give a budget to state colleges just to keep them afloat during collapse.

All the economic bubbles burst within 15 years, and by my measure the two biggest are all of healthcare (including medicare and medicaid, hospitals, doctors, and pharma - leaving us with clinics and generic medication), and then higher education. They are bubbles not because of 'perceived value', but rather because we are pumping money that isn't realistic into them, allowing them to think we still have the ability to pay for it, when in fact we already cannot. In 15 years, a person newly attending any 4 year degree will not be able to mathematically pay it back (going by inflation, wage stagnation, and average student loan amount) at all. They will default almost no matter what in only a year. Also in 15 years, that is when medicare and medicaid will no longer be acceptable, and at this time public and private insurance cannot be paid for, again going by income vs. necessity costs. Other sectors are similar.

Want to know why? Simply unsustainable growth patterns. We have a lot of them, but these are the big ones. If we solve these, the damage is dramatically reduced.

And it starts with properly and realistically addressing the national budget and budgets of these areas.

I can already see a flaw in Forbe's tax code.

Assume the nation's poverty level is 10,000. Assume 5 people in our little 'economy' make incomes. The distribution is the same as I stated above, 10,000, 25,000, 50,000, 75,000, and 100,000.

the 'national income' is 260,000. The nation has a budget of 25% of national income, or (3.5 trillion in 2013 dollars of 14 trillion in income), and it needs 80% of that from income tax. In affect, it needs 20% of national income as an income tax.

Assuming Steve Forbes' assumption, we would need a flat tax of 20% (overdoing it just to show you), to give us the needed income tax we budgeted.

So here's what this comes out to with the Forbes' tax:

10,000 = e = 0. 25,000 = e .12 = 3,000. 50,000 = e = .16 = 8,000. 75,000 = e = .17 = 13,000. 100,000 = e = .18 = 18,000.

Total income tax for year = 42,000. Expected and Needed (target): 52,000.

If you continue with trying to come up with the exact target flat-tax rate according to Forbes', you'll not only see that according to total national income it needs to change every year according to how much is budgeted, but you can also notice that it is only progressive up to 20%. At least it gives a tax break to the poor, I guess, but if you worked out the realistic numbers, you would see that this creates a similar tax scenario that is not exact (how do you know who pays what?), has to change every year (the effective tax rate needs to be variable every year in this scenario, which becomes a guessing game - 20%? 25%? 24%? Who knows?), and is relatively unpredictable (how can both the government and the individual at tax season have any reasonable expectation?).

However, that doesn't mean it isn't at least a start. I'll show you what Forbes' failed to accomplish....