How would you change the U.S. tax system?

[jewish_scientist]

...the 'marginal tax rate' is the tax rate.

That is a really bad explanation of what I meant. Does someone want to keep score of how often I mess up? I think we are up to 5 by now. Anyway, this is what I want.

P = Money Paid as Taxes
x = Income [I was going to make this 'I', but all the 'I's started looking like '1's]
M = Marginal Tax Rate
P = x * f(x)

If f(x) = ∫g(x) / x, then P = ∫g(x), which implies M = g(x).

If f(x) =/= ∫g(x) / I, then P =/= ∫g(x), which implies that M has no acceptable value¹.

Asking the question 'What do you want to do with the M?' is limiting, because it assumes that there acceptable values for M. However, there are valid equations of f(x) that satisfy all the conditions you need to make sure weird stuff does not happen and cannot be rewritten such that the M has acceptable values. Although it is not what I would use, I believe that this is one such example.

1: I am not sure if this is the right way to express what I mean. Okay, I am very certain that this is not the right way to express what I mean. Maybe I can explain myself better if I use an example. Look at the equation x = 1/0. No matter what value x has, this equation cannot be true. Therefore, x has no acceptable value.

[Zohar]

You keep getting back to this idea that functions that are always differentiable are necessarily better and I think the issue everyone is having is you haven't explained why you think that. As far as pretty much everyone can tell they're not - it's harder to understand for a lot of people, they're more complicated to calculate... I don't see any benefit to doing this.

[ucim]

P = Money Paid as Taxes
x = Income [I was going to make this 'I', but all the 'I's started looking like '1's]
M = Marginal Tax Rate

M = dP/dx
It only requires that P be differentiable. In practical terms, this means that P should be continuous. M does not need to be continuous; where M is discontinuous, the graph of P=f(x) instantly changes slope. That slope is M.

The present system has M being discontinuous. That's ok. It does not have P being discontinuous (except to the resolution of the monetary unit itself).

M is what is published. It's easy: it consists of three (or several) constants, and a range to go with each. From this (and the complicated list of deductions based on source of income, what the money is spent on, and what your own x is), P is constructed. The complications do not arise from M, they arise from pretty much everything that isn't M.

Jose

[Soupspoon]

The 'ideal' line, anyway, is awkward.

For a given point of income representing the minimum wage at which you will demand (net¹) taxes, the line must rise only gradually, lest it discourage opportunities to progress into the taxable arena. Yet everyone seems to consider it fair to have the upper reaches of tax-take be hyperbolic to some final fraction-of-earnings asympotote (some would say that's 100%, many would not, but it offers rising personal rewards whilst to still being a rising contributor to the treasury in a fair-ish way, give or take the details).

Transitioning between the two, though, requires an S-bend inflection. An increasing rise of proportion-per-taxability from the softened lower end meets a declining proportion-per-taxibility as one moves down from the asymptote-kissing levels.

Variable earners in the crux of that curve get hit by more new-taxing-per-unit-earning-increase than anyone else. And they're (by all laws of statistics) likely among the most commonly represented public. Vote losing amongst the middle-classes.

Which is why I have maybe decided to preserve the 'soft lift off' out of the pre-taxable poverty zone, and not to inflect that line, but merely have its crossing of the "magic" top percentage wherever the toppest-of-all-tops tax-payer sits. It would seem to hurt them, individually, whilst more savvy near-top-earners deliberately hold their pay back (by hook or by crook) to enjoy the proportionally vastly lessened taxation level just shy from that top, but: a) The kudos of being the person paid enough to 'suffer' the full 50% (I'd recommend - no more!) is something to shout about; b) In order to avoid being top, or quite so near it, imagine a set of similarly rich multimillions-earners contriving to effectively donate half a million of their nominal take-homes (dressing it up as business deals/funds reallocations, all they want) to the nominated tax-scapegoat amongst them... Their loss of half a million in pre-tax is rewarded by the (buffed) scapegoater (still getting half of all the effective bonuses 'donated'!) pushing the tax curve further up and away from even their half-mill 'retreat point', giving them yet more relief.

The actual intensity (hyperbolic eccentricity) of the curve used might need fine-tuning to avoid over-scapegoating, but it should be possible to gather everyone's 'bid' for personal incomes (or at least the expected top 25% of earners), create your curve and then demand your taxes (with or without menaces) whilst fining those that break from their 'bids' sufficient to be in the top 1% of specific deniers of expected monies. That's (specifically) going to be the near-top-enders who bid near-near-top but 'inadvertently' earned enough to be near-top (where the difference in pay most affects the treasury income) as well as the reference top-earner who earns less than predicted and thus forces the curve to adjust down.

Anyone (near-top or top in the original bidding) who breaks above the bid-top with their actual are now paying at >50% (could be >100%, even!), so are already punished by lower after-tax earnings, as are those who bid near-top but got near-near-top or less, and punitivism won't be useful.

And it doesn't matter how much income-obfuscation goes on, at the top levels where accountants can be hired to do such things, anything actually legal just tends to bring the curve down, to nobody's benefit, if everybody does it. Illegal evasion of declared income is just as punishable. At the low end of the tax curve (at the soft-start rate-increase of the curve) barely moves regardless of the top-flight's power games. No advantage to have accountants, there, where none could be afforded anyway. The taxes will be within the range of "same ol', same ol'" for a given income above the zero-crossing point, and the bulk of the middle-earners will lie within the elbow, not on as troublesomely steep increase-per-increase as in the S-bended version - aspiration will still be awarded, and a safety-net against down-sizing includes a relief.


Complicated (to get usdd to) , and would need to be set up from scratch, but if we are talking mad schemes based upon a single principle, then this is mine!


¹ Arguments about where the tax line wanders, prior to this value, aside. It could just be asking for zero tax, or it might be implementing a welfare credit/negative tax, and then you wander into "what is a leg-up, what is a reward for earning little?" territory with a vengeance.

[gmalivuk]

If f(x) =/= ∫g(x) / I, then P =/= ∫g(x), which implies that M has no acceptable value¹.

It's true that we probably don't want an M that isn't integrable, but fortunately integrability is a weaker condition than continuity. Our M (with tax brackets) is discontinuous but it is still integrable, so everything's fine.

[Zamfir]

It's true that we probably don't want an M that isn't integrable,

Axually, I see possibilities here.

Proposal for a change in the US tax code, as per the OP. In four easy steps:

1. The Vitali Tax Bracket Plan. This modifies the tax rate to some textbook non-Lebesque-integrable function. Obviously, this will initially cause difficulties. But as people say, the tax code is already mindbogglingly complicated. We can rely on the free market to solve the issues. Also, it relies on the axiom of choice, and would you oppose choice?

Once the tax software can deal with this new complication, we move to step 2

2. The Taxable Euclidean Dimension Xpansion law. This increase the taxable space form a single line to all three spatial dimensions, to deal with the full range of the modern economy. Again, it will take some effort before everyone is used to this.

Once the tax accountants are used to non-measurable sets and to 3 dimensions, we can move to step 3:

3. The Banach-Tarski Tax Schedule. This is a more complicated tax scheme than introduced in step 1, taking full advantage of the 3 dimensions. It has obvious advantages when it comes to deficit reduction.

4. Profit

[Soupspoon]

5. Tax that profit

[Thesh]

Well, if it's that or a flat tax, I'm going to have to go with a flat tax.

[ucim]

Well, if it's that or a flat tax, I'm going to have to go with a flat tax.

Why? Once we reach step three, we can generate as much income (and as much tax) as we like, just by duplicating spheres. Economics can be modelled as a sphere, and this way we can always have a growing economy. Bigly!

Jose

[Pfhorrest]

Assume a spherical economy of uniform density...

[jewish_scientist]

Zohar, it is not differentablity that a care about. I want no part of the tax system to be piece-wise.

Soupspoon, I made a mistake. That equation is not the ideal equation for effect tax, which I directly stated.

Although it is not what I would use, I believe that this is one such example.

I thought that that equation was an example of an equation for the effective tax that could not be rewritten in terms of a marginal tax. I recognize now that I made a mistake (thank you ucim), because any effective tax can be rewritten in terms of a marginal tax.

I have three questions.

Spoiler:

B_n = upper limit of the n^th tax bracket, inclusive
M_n = tax rate of the n^th tax bracket
E = Effective tax

First, what is the equation for P in terms of x, M_n and B_n that does not call another function?

Second, if I change the current system such that the M₁ = 0.15 and M₂ = 0.20, then what is f(x) when /\E = f(x)?

The third question is just me venting, so it is probably best if you ignore it:
Why is Zamfir's plan SO great?! At almost every step he is like 'this is hard to work into the current system' or 'this is complicated'. He is literally adding another dimension to taxes! But no, that's okay. People will understand that, perfectly. But when I say I want to write the marginal tax as an effective tax, the freaking world is about to fall apart! Literally every result on the first page of Google when you google Banach-Tarski is about the paradox. HOW ON EARTH IS ADDING A PARADOX INTO THE TAX SYSTEM A GOOD IDEA! And why does simplicity even matter! Think about a librarian types a book's title into a search bar and the computer gives her the location of it. Even if she does not know anything about arrays and/or never heard the term 'bubble sort' before. She does not need to know that information as long as the programmer who made the search engine does. It just accord to me that urim and Pfhorrest were being sarcastic. *shakes fist in the air* Damn you Poe! Damn you and your law to HELL!

[Zohar]

Zohar, it is not differentablity that a care about. I want no part of the tax system to be piece-wise.

Why? Why do you hate piece-wise functions so much? What's wrong about their aesthetic that deters you so much?

[Zamfir]

Why is Zamfir's plan SO great?!

Because the plan can generate as much tax income as you want, from a finite tax base.

Edit: I'll add a serious part of this empty post:

First, what is the equation for P in terms of x, Mn and Bn that does not call another function?

k is the largest value such that B[k]<x
B[0] =0

P = (x-B[k])*M[k+1] + sum( Mi*(Bi -B[i-1]) )for i= 1 to k

[CorruptUser]

There's also the consideration that poor, desperate people may work more, not less, as a result of an increased tax burden, assuming no benefits. The tax results in relative poverty, and in order to maximize utility between money and leisure the person has to work more.
You know the old expression, give a man a fish and feed him for a day, take away that fish and tell him he's lucky that's all you took and he'll figure out how to get you another fish next week.

[sardia]

There's also the consideration that poor, desperate people may work more, not less, as a result of an increased tax burden, assuming no benefits. The tax results in relative poverty, and in order to maximize utility between money and leisure the person has to work more.
You know the old expression, give a man a fish and feed him for a day, take away that fish and tell him he's lucky that's all you took and he'll figure out how to get you another fish next week.

What the fuck? Do you even have a citation for that?

[Soupspoon]

Zohar, it is not differentablity that a care about. I want no part of the tax system to be piece-wise.

Why? Why do you hate piece-wise functions so much? What's wrong about their aesthetic that deters you so much?

I think, though largely not voiced, one of the problems considered with it is the "less than X; between X and Y; between Y and Z; above Z" thing seems to use values of X, Y and Z (and S, T, U, V, W below, probably) that are neither linearly nor geometrically nor exponentially spaced, being seemingly arbitrarilyspaced, 'chosen' maybe by either random drawing out of a hat or by petitioning by edge-case persons who wanted not to be across the edge.

I mean, what's the progression behind $0<(10%)<$9,325<(15%)<$37,950<(25%)<$91,900<(28%)<$191,650<(33%)<$416,700<(35%)<$418,400<(39.6%)<$∞?

In particular, that $416,700 to $418,400 gap (never mind the non-round values, they're likely created from prior values adjusted by official inflation) for 4.6% 'relief' on the top bracket and/or 2% extra penalty on the bracket below. And 5, 10, 3, 5, 2, 4.6% is odd and not obviously mappable to the dollar-gaps.

If it were "every doubling over $10k asks for 2% more", or similar, then that's two values (or three, if you count the 'doubling' and anticipate changing that to "every 1.9x" or 2.1x if that's what you want to do). Change the $10k base and the percentage over time ($11k and 2.5%, for example?) and it's at least unarbitrary. The figures can still be chosen to benefit the donor pestering your Administration for a useful old-taxes-vs-new-taxes reciprocal benefit to the beneficiary(/ies), with sufficient selective restructuring. But it wouldn't be as simple as adding in an extra thin bracket with no discernable reason, or making a fractional rate-change likely as a weird gift

(Does that sort of thing sound like it would preferentially float your boat, as describing your instincts, JS?)

[Thesh]

The brackets should be based on economic analysis about what the most effective policies will be. This will usually result in people being grouped into buckets based on income so we can see how policies will affect different groups. The cut-offs will then be determined by how you choose those groups.

[CorruptUser]

There's also the consideration that poor, desperate people may work more, not less, as a result of an increased tax burden, assuming no benefits. The tax results in relative poverty, and in order to maximize utility between money and leisure the person has to work more.
You know the old expression, give a man a fish and feed him for a day, take away that fish and tell him he's lucky that's all you took and he'll figure out how to get you another fish next week.

What the fuck? Do you even have a citation for that?

Yes, this one shows a fairly obvious example in moving from upper to middle class.

Wikipedia on this, check out the part about "Inverted S shaped supply curve" for the case of poverty. There's a reason people worked 14 hour days when the pay was shit.

Economics is weird.

[Pfhorrest]

I wonder if there might be a bit of miscommunication here. To my reading, CorruptUser is saying that imposing a tax on the poor (without giving them any benefit in return) could induce them to work more, which makes straightforward sense to me at least, for the reasons he outlines now. But I wonder if sardia perhaps thinks CorruptUser was saying something about raising taxes on the rich (to fund things that benefit the poor) would induce the poor to work harder, which is, yeah, wtf. I'm not sure why sardia might think that, but I'm also not sure why CorruptUser brings up the point to begin with, so...

[CorruptUser]

I wonder if there might be a bit of miscommunication here. To my reading, CorruptUser is saying that imposing a tax on the poor (without giving them any benefit in return) could induce them to work more, which makes straightforward sense to me at least, for the reasons he outlines now. But I wonder if sardia perhaps thinks CorruptUser was saying something about raising taxes on the rich (to fund things that benefit the poor) would induce the poor to work harder, which is, yeah, wtf. I'm not sure why sardia might think that, but I'm also not sure why CorruptUser brings up the point to begin with, so...

I don't think I was ambiguous enough for Sardia to think that I was stating that raising taxes on the rich to increase benefits for the poor would cause the poor to work harder. So no.

The reason I brought it up was that taxes are weird in how they impact the economy.

[jewish_scientist]

I'll get to what everyone else said in a while. This is just something that I happen to know off the top of my head that is semi-relevant.

The "Inverted S Shaped Supply Curve" fails to take into account employee productivity. I feel safe in assuming that people who have to work 14 hours/ day just to get food on the table are unhappy. Unhappiness causes a remarkably sharp decline in productivity. Everything taken into account, making employees happy increases work/ day despite the decrease in hours/ day.

Sources:
http://fortune.com/2015/10/29/happy-productivity-work/
https://www.theguardian.com/science/201 ... productive
https://www.forbes.com/sites/martinzwil ... 13888c1efa

P.S. This actually creates a big problem in Marxism, but that is neither here nor there.

[CorruptUser]

Happiness matters more for complex mental tasks than simple but physically demanding tasks. Dishwashers don't tend to be paid well, but they wouldnt double prodictivity with double the wage. Where happiness becomes critical for unskilled labor is when it comes to retention and loyalty.

There's other reasons to pay loving wages, mostly involving ethics and societal benefits, but from a sociopathic employer point of view...

[jewish_scientist]

Do you have a citation?

[jewish_scientist]

Zohar, it is not differentablity that a care about. I want no part of the tax system to be piece-wise.

Why? Why do you hate piece-wise functions so much? What's wrong about their aesthetic that deters you so much?

I need the answer to my second question to answer this. But, I am not smart enough to figure it out.

[Zamfir]

@ jewish scientist, let's see if I understand your question:

Second, if I change the current system such that the M1 = 0.15 and M2 = 0.20, then what is f(x) when /\E = f(x)?

Looking at a post from you above, then you define f(x) like this:
P= x*f(x)
with
P = income taxes owed
x = pre-tax income

Now you are considering a system with two (or more) income tax brackets, like this:

P(0) = 0
dP/dx =
{M1 = 0.15 for 0<x<=B1
{M2= 0.20 for B1<x<=B2
{M3 = ...


And you are looking for a formula for f(x), correct? If so:
------------------------------------------------
f(x) = M1 + (M2-M1)* max(0, 1 - [B1/x] ) + (M3-M2)*max(0, 1- [B2/x]) + ...

------------------------------------------------
It might help if you don't look at it as brackets with different rates, but as "tax layers" stacked on top of each other, with constant rate in every layer.

There is a first layer with rate M1, which you pay over all income from 0 to x.
So Pa = M1*x

Then a second layer on top, with rate (M2-M1), over all income from B1 to x.
Pb = (M2-M1)*max(0,x-B1).

The max function means that you don't get money back from this layer if x<B1

Then a third layer, Pc = (M3-M2)*max(0, x-B2), etc

Therefore, P = Pa + Pb+Pc +...
and f(x) = P/x = Pa/x + Pb/x +...

Like this picture, it's supposed to be two times the same function

Spoiler:

[jewish_scientist]

@ jewish scientist, let's see if I understand your question:

Second, if I change the current system such that the M1 = 0.15 and M2 = 0.20, then what is f(x) when /\E = f(x)?

Looking at a post from you above, then you define f(x) like this:

That is not what I meant. I should have made myself clearer. I meant f(x) as in 'a function in terms of x'. Seeing how I was still referencing variables I defined in that post, this miscommunication is definitely my fault. Here is the least ambiguous way to ask the question that I can think of:

If we change the current U.S. marginal tax system such that the first bracket is taxed at 15% and the second is taxed at 20%, then what is a function that gives the change in effective tax for a given income?

[ucim]

If we change the current U.S. marginal tax system such that the first bracket is taxed at 15% and the second is taxed at 20%, then what is a function that gives the change in effective tax for a given income?

Uh... the tax bracket itself? I'm not sure you're asking what you think you're asking.

"the first bracket is taxed at 15% and the second is taxed at 20%" means that, if your income is in the first bracket, then you are taxed at 15%. This is a constant. Within this bracket, the rate of change of tax vs income is 15%. The rate of change of tax rate vs income is zero.
...and...
if your income is in the second bracket, then you are taxed at 20% on the "excess" income, plus being taxed at 15% for the previous income. This is also a constant. Within this bracket, the rate of change of tax vs income is 20%. The rate of change of tax rate vs income is zero.

At the boundary between the brackets, there is a discontinuity in tax rate, but not in tax paid.

Now, the effective tax rate is a blend of all applicable brackets, but the rate of change of tax doesn't care about that. It only cares about the tax bracket. That's what the "marginal tax rate" means... the tax on the last dollar. The rate of change of total tax at that point is the marginal tax rate.

So, it's still not clear (to me) what you are asking.

If you are asking for a function for the change in effective tax rate vs income, consider that the total tax paid is the sum of the taxes on each of the brackets. "Effective tax rate" is just the total tax paid divided by income. Take the derivative and you have the change in effective tax rate vs income.

Jose

[Zamfir]

If we change the current U.S. marginal tax system such that the first bracket is taxed at 15% and the second is taxed at 20%, then what is a function that gives the change in effective tax for a given income?

Ok, you want a function that describes the difference( in effective tax rate )between the current system, and a hypothetical system that is identical except for a 5% higher rate on the first two brackets?

That's
f(X) =
{ 0.05 for X=< B2
{ 0.05*B2/X for X> B2

[measure]

If we change the current U.S. marginal tax system such that the first bracket is taxed at 15% and the second is taxed at 20%, then what is a function that gives the change in effective tax for a given income?

Uh... the tax bracket itself? I'm not sure you're asking what you think you're asking.

It sounds like jewish scientist is asking for the function describing the rate of change of the effective tax rate as a function of gross income, which, for the bracketed system, is nontrivial. It decreases locally with increasing income, but jumps up a discontinuity at each bracket change.

[jewish_scientist]

Zamfir, I meant as a percentage, not percentage points.

[Zamfir]

Now you've lost me, sorry.

[jewish_scientist]

[0.1 + (0.1 * x) = 1.5] -> [x = 0.5] -> For anyone who's entire income is taxed by the first tax bracket, the change in effective tax is 50%.

I want a function that does that for all incomes.

[ucim]

I have no idea what you're saying. Your math is off by a factor of ten, if you're solving for x. And I have no idea what x represents.

Jose

[Zamfir]

So, you are looking for

f(X) =[ Enew (X) - Ecurrent(X)] / Ecurrent(X)

With E (X)= P(X)/x

Correct? You don't need effective tax rates for that, you can just work with the tax itself, since the 1/x terms cancel out.

Therefore,
f(X) = [ Pnew (X) - Pcurrent(X)] / Pcurrent(X)

Now the numerator term is simple in this case:
[ Pnew (X) - Pcurrent(X)] =
{0.05x for X < B2
{ 0.05*B2 for X>= B2

Then plug in the appropriate expression
for Pcurrent, bracket by bracket. Or use max() functions to produce a one-liner formula, if that suits your esthetics better.

[ivnja]

I don't have a function, but I do have a pretty(?) chart and the spreadsheet I used to create it to illustrate the current state of affairs for federal income tax rates in the US*. I used the tax bracket data I found here, specifically the rates for single filing:

In the attached spreadsheet, I used an income range of $100 to $500,000 and calculated the amount of income that falls in each tax bracket, the amount of tax that is subsequently owed in each bracket, and at the end the total tax owed and the effective total tax rate for each income. As a result, the setup is a little different than in the table above, but the sums work out the same way - mine is just a little more explicit about why, for example, the tax owed for anyone whose income is in the fourth bracket is "$18,713.75 plus 28% of the excess over $91,900." The way the formulas are entered, anyone who wants to should be able to play with bracket values and marginal rates to see what the effects on the totals would be. I worked in LibreOffice, so the .xls file may not come through quite right, particularly the graph, if you open that one.

Total effective tax rate increases rather more quickly at the lower end of total income levels (as a result of how Congress has chosen to set up the tax brackets - political decision), as well as at the lower end of each individual bracket (as an inescapable result of having a single marginal rate for each bracket).

For the second graph, I extended the income range another 100k and had the program calculate a best-fit line, which in this case was a power, with an equation that was 0.01122x^0.2525 with an R² of 0.97. That wasn't the cleanest number (and isn't quite the equation I ultimately used on the chart), but the resulting tax paid line (green) fits well enough with the existing line (orange) to show that if Congress wanted, they could choose a continuously-varying function like a power function as the tax rate rather than brackets without drastically altering anyone's tax burden.
[Edit to add a giant caveat: This is only true through about $1M of income with the current bracket setup - the slope of the bracketed tax rate "function" is constant as income goes to infinity and total effective tax burden in the bracketed system asymptotically approaches 39.6%, while the power function is concave up, so both the total effective tax rates and the resulting tax payments owed increasingly diverge as income increases. With the particular power function I used, total effective tax rate goes above 100% right around $50M of income.]

*with the exception of the exemptions - I know those have a significant effect at lower income levels, but I was mostly interested in looking at how the graph behaved around the edges of the brackets. I might add those in later.

[jewish_scientist]

How do you add attachments to a post?

[Thesh]

Below the submit button is a tab for attachments.

[ivnja]

How do you add attachments to a post?

With the Attachments tab directly below the "Post a reply" box* that you type your post into.

Hitting Place Inline, which shows up after you've uploaded the attachment, will let you put images in the body of your post (like the graphs in my previous post, or like this: )

Otherwise the attachments will show up at the end of the post, like the spreadsheets in my previous post.

*Edit: for whatever reason, the mobile version labels the actual text box with "Topic review," as shown in the image.